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<title>Bijection réciproque</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading">Bijection réciproque</h1>

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En <a href="Math%C3%A9matiques" title="Mathématiques">mathématiques</a>, la bijection réciproque (ou fonction réciproque ou réciproque) d'une <a href="Bijection" title="Bijection">bijection</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> est l'application qui associe à chaque élément de l'<a href="Ensemble_d'arriv%C3%A9e" title="Ensemble d'arrivée">ensemble d'arrivée</a> son unique <a href="Ant%C3%A9c%C3%A9dent_(math%C3%A9matiques)" title="Antécédent (mathématiques)">antécédent</a> par <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy">. Elle se note <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}}">

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<annotation encoding="application/x-tex">{\displaystyle f^{-1}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3e5cfa2f5c08d6fe7d046b73faa6e3f213acc802.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.653ex; height:3.009ex;" alt="{\displaystyle f^{-1}}" loading="lazy">.

<div class="mw-heading mw-heading2"><h2 id="Exemple">Exemple</h2></div>

On considère<a href="#cite_note-RacineCubique-1">[1]</a> l'<a href="Application_(math%C3%A9matiques)" title="Application (mathématiques)">application</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/786849c765da7a84dbc3cce43e96aad58a5868dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" loading="lazy"> vers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/786849c765da7a84dbc3cce43e96aad58a5868dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" loading="lazy"> définie par <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\left(x\right)=x^{3}}">

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<annotation encoding="application/x-tex">{\displaystyle f\left(x\right)=x^{3}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/76c7108e0b6cbd6170df9d862e9751a028cc2cdb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.287ex; height:3.176ex;" alt="{\displaystyle f\left(x\right)=x^{3}}" loading="lazy">.

Pour chaque réel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

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<annotation encoding="application/x-tex">{\displaystyle y}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy">, il y a un et un seul réel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> tel que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=x^{3}=f(x)}">

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<annotation encoding="application/x-tex">{\displaystyle y=x^{3}=f(x)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e04b21baa558633961006be00338a4dea9090baf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.154ex; height:3.176ex;" alt="{\displaystyle y=x^{3}=f(x)}" loading="lazy">, ainsi pour <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=8}">

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<annotation encoding="application/x-tex">{\displaystyle y=8}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0bb04da76e7f0bf95f90098eef10262cb68ced7f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.416ex; height:2.509ex;" alt="{\displaystyle y=8}" loading="lazy">, le seul <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> convenable est 2, en revanche, pour <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=-27}">

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<annotation encoding="application/x-tex">{\displaystyle y=-27}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c6252d3ba12b88bbabc44b60d6e3aa578fbd7791.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.387ex; height:2.509ex;" alt="{\displaystyle y=-27}" loading="lazy"> c'est –3. En termes mathématiques, on dit que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> est l'unique <a href="Ant%C3%A9c%C3%A9dent_(math%C3%A9matiques)" title="Antécédent (mathématiques)">antécédent</a> de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

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<annotation encoding="application/x-tex">{\displaystyle y}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"> et que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> est une <a href="Bijection" title="Bijection">bijection</a>.

On peut alors considérer l'application qui envoie <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

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<annotation encoding="application/x-tex">{\displaystyle y}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"> sur son antécédent, qu'on appelle dans cet exemple la <a href="Racine_cubique" title="Racine cubique">racine cubique</a> de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

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<annotation encoding="application/x-tex">{\displaystyle y}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"> : c'est elle qu'on nomme la « réciproque » de la bijection <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy">.

Si on tente d'effectuer la même construction pour la <a href="Racine_carr%C3%A9e" title="Racine carrée">racine carrée</a> et qu'on considère l'application <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

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<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"> de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/786849c765da7a84dbc3cce43e96aad58a5868dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" loading="lazy"> vers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/786849c765da7a84dbc3cce43e96aad58a5868dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" loading="lazy"> définie par <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x)=x^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle g(x)=x^{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/92746066d0381ea6189ffc725768840f81d83ba3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.737ex; height:3.176ex;" alt="{\displaystyle g(x)=x^{2}}" loading="lazy">, les choses ne se passent pas si simplement. En effet, pour certaines valeurs de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

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<annotation encoding="application/x-tex">{\displaystyle y}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy">, il y a deux valeurs de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> tels que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x)=y}">

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<annotation encoding="application/x-tex">{\displaystyle g(x)=y}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/665919413d9c38e2a3661752482e739472e1d8a8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.509ex; height:2.843ex;" alt="{\displaystyle g(x)=y}" loading="lazy"> ; ainsi, pour <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=4}">

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<annotation encoding="application/x-tex">{\displaystyle y=4}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b08f54bf2d9afb4eff307c97c349ab6a0c4c62eb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.416ex; height:2.509ex;" alt="{\displaystyle y=4}" loading="lazy">, on peut choisir <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=2}">

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<annotation encoding="application/x-tex">{\displaystyle x=2}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9f39b6e42e5ffb81ac7b051b9e48b9a91d0713c7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x=2}" loading="lazy"> mais aussi <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=-2}">

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<annotation encoding="application/x-tex">{\displaystyle x=-2}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3f3fcffdbfac20f62b54d9d9dca69c3f5ac1c871.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.399ex; height:2.343ex;" alt="{\displaystyle x=-2}" loading="lazy">, puisque 22 = 4 mais aussi (–2)2 = 4.

À l'inverse, pour d'autres choix de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle y}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy">, aucun <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> ne convient ; ainsi pour <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=-1}">

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<annotation encoding="application/x-tex">{\displaystyle y=-1}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d408b479252792fd0d5936e9eb0a269702521957.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.225ex; height:2.509ex;" alt="{\displaystyle y=-1}" loading="lazy">, l'équation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}=-1}">

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</mrow>

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<annotation encoding="application/x-tex">{\displaystyle x^{2}=-1}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/20d687ce8b19e0ef0d8f872a4e0bfdcf3a6665a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.453ex; height:2.843ex;" alt="{\displaystyle x^{2}=-1}" loading="lazy"> n'a aucune solution réelle. En termes mathématiques, on dit que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

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<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"> n'est ni <a href="Injection_(math%C3%A9matiques)" title="Injection (mathématiques)">injective</a> ni <a href="Surjection" title="Surjection">surjective</a>. Dans cet exemple, les définitions qui suivent ne permettent pas de parler de « bijection réciproque » (ni même d'« application réciproque ») de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

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<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy">.

<div class="mw-heading mw-heading2"><h2 id="Résultats_généraux">Résultats généraux</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Définition">Définition</h3></div>

Si <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> est une <a href="Bijection" title="Bijection">bijection</a> d'un ensemble <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {X} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {X} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a8d98cdc2216c7394d189ea3e09a479c826263b6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle \mathrm {X} }" loading="lazy"> vers un ensemble <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Y} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {Y} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/628b8dd0254a41a2803ecc14dc8848e786e05d1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle \mathrm {Y} }" loading="lazy">, cela veut dire (par définition des bijections) que tout élément <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"> de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Y} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {Y} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/628b8dd0254a41a2803ecc14dc8848e786e05d1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle \mathrm {Y} }" loading="lazy"> possède un antécédent et un seul par <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy">. On peut donc définir une application <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"> allant de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Y} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {Y} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/628b8dd0254a41a2803ecc14dc8848e786e05d1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle \mathrm {Y} }" loading="lazy"> vers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {X} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {X} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a8d98cdc2216c7394d189ea3e09a479c826263b6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle \mathrm {X} }" loading="lazy">, qui à <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"> associe son unique antécédent, c'est-à-dire que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f{\big (}g\left(y\right){\big )}=y}">

</mrow>

<mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f{\big (}g\left(y\right){\big )}=y}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ee4936cc968e3769a0ca3392f62ed874363f5f19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.517ex; height:3.176ex;" alt="{\displaystyle f{\big (}g\left(y\right){\big )}=y}" loading="lazy">.

L'application <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"> est une bijection, appelée bijection réciproque de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy">.

La bijection réciproque de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> est souvent notée<a href="#cite_note-Notation-1-2">[2]</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{-1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3e5cfa2f5c08d6fe7d046b73faa6e3f213acc802.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.653ex; height:3.009ex;" alt="{\displaystyle f^{-1}}" loading="lazy">, en prenant garde à la confusion possible avec la notation des <a href="Exposant_n%C3%A9gatif" class="mw-redirect" title="Exposant négatif">exposants négatifs</a>, pour laquelle on a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{-1}=1/x}">

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x^{-1}=1/x}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ce8f6e99ac485f23f1157232928b717e4a2b5360.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.416ex; height:3.176ex;" alt="{\displaystyle x^{-1}=1/x}" loading="lazy">.

<div class="mw-heading mw-heading3"><h3 id="Propriétés">Propriétés</h3></div>

<div class="mw-heading mw-heading4"><h4 id="Caractérisation">Caractérisation</h4></div>

Si <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> est une application d'un ensemble <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {X} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {X} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a8d98cdc2216c7394d189ea3e09a479c826263b6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle \mathrm {X} }" loading="lazy"> vers un ensemble <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Y} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {Y} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/628b8dd0254a41a2803ecc14dc8848e786e05d1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle \mathrm {Y} }" loading="lazy"> et s'il existe une application <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"> de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Y} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {Y} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/628b8dd0254a41a2803ecc14dc8848e786e05d1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle \mathrm {Y} }" loading="lazy"> vers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {X} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {X} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a8d98cdc2216c7394d189ea3e09a479c826263b6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle \mathrm {X} }" loading="lazy"> telle que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\circ f=\operatorname {id} _{\mathrm {X} }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g\circ f=\operatorname {id} _{\mathrm {X} }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a656999da672a7bacab50867bbfad0807826b7d4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.092ex; height:2.509ex;" alt="{\displaystyle g\circ f=\operatorname {id} _{\mathrm {X} }}" loading="lazy"> et <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\circ g=\operatorname {id} _{\mathrm {Y} }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\circ g=\operatorname {id} _{\mathrm {Y} }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/de1facbcd87f7d24a9fbb336b5daf0c0f941f79c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.092ex; height:2.509ex;" alt="{\displaystyle f\circ g=\operatorname {id} _{\mathrm {Y} }}" loading="lazy">, alors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> et <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"> sont des bijections, et <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"> est la bijection réciproque de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy">.

<div class="mw-heading mw-heading4"><h4 id="Réciproque_de_la_réciproque">Réciproque de la réciproque</h4></div>

La double propriété <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}\circ f=\operatorname {Id} _{\mathrm {X} }}">

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{-1}\circ f=\operatorname {Id} _{\mathrm {X} }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5b170174e989b44540ff6fab7ed36176437d9138.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.822ex; height:3.009ex;" alt="{\displaystyle f^{-1}\circ f=\operatorname {Id} _{\mathrm {X} }}" loading="lazy"> et <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\circ f^{-1}=\operatorname {Id} _{\mathrm {Y} }}">

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\circ f^{-1}=\operatorname {Id} _{\mathrm {Y} }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0b40833c66433aa73222cee7040fe2a1bcdb0bea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.822ex; height:3.009ex;" alt="{\displaystyle f\circ f^{-1}=\operatorname {Id} _{\mathrm {Y} }}" loading="lazy"> montre que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> est aussi la bijection réciproque de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{-1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3e5cfa2f5c08d6fe7d046b73faa6e3f213acc802.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.653ex; height:3.009ex;" alt="{\displaystyle f^{-1}}" loading="lazy">, c'est-à-dire que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(f^{-1}\right)^{-1}=f}">

<msup>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left(f^{-1}\right)^{-1}=f}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0e579eeb363e82c897128e2a978d1b4bcd49b63e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.493ex; height:3.843ex;" alt="{\displaystyle \left(f^{-1}\right)^{-1}=f}" loading="lazy">.

<div class="mw-heading mw-heading4"><h4 id="Réciproque_d'une_composée">Réciproque d'une composée</h4></div>

La réciproque de la composée de deux bijections est donnée par la formule : <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (g\circ f)^{-1}=f^{-1}\circ g^{-1}}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (g\circ f)^{-1}=f^{-1}\circ g^{-1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aa7feda06bf3a320235c07b30a97fb226f693183.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.129ex; height:3.176ex;" alt="{\displaystyle (g\circ f)^{-1}=f^{-1}\circ g^{-1}}" loading="lazy">.

On peut remarquer que l'ordre de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> et <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"> a été inversé ; pour « défaire » <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> suivi de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy">, il faut d'abord « défaire » <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"> puis « défaire » <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy">.

<div class="mw-heading mw-heading4"><h4 id="Involution">Involution</h4></div>

Certaines bijections de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {E} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {E} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/be1811407dea8b43727d28dbe8da7251985b03e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle \mathrm {E} }" loading="lazy"> vers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {E} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {E} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/be1811407dea8b43727d28dbe8da7251985b03e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle \mathrm {E} }" loading="lazy"> sont leur propre réciproque, c'est le cas par exemple de l'<a href="Fonction_inverse" title="Fonction inverse">application inverse</a> :<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{matrix}f:&\mathbb {R} ^{*}&\to &\mathbb {R} ^{*}\\&x&\mapsto &{\frac {1}{x}}\end{matrix}}}">

<mtr>

<mtd>

</mtd>

<mtd>

<msup>

</mrow>

</mrow>

</msup>

</mtd>

<mtd>

</mtd>

<mtd>

<msup>

</mrow>

</mrow>

</msup>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

<mfrac>

</mfrac>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}f:&\mathbb {R} ^{*}&\to &\mathbb {R} ^{*}\\&x&\mapsto &{\frac {1}{x}}\end{matrix}}}</annotation>

</semantics>

</math>

ou de toute <a href="Sym%C3%A9trie_(transformation_g%C3%A9om%C3%A9trique)#Symétrie_axiale_ou_orthogonale_par_rapport_à_une_droite" title="Symétrie (transformation géométrique)">symétrie orthogonale</a> dans le plan.

De telles applications sont dites <a href="Involution_(math%C3%A9matiques)" title="Involution (mathématiques)">involutives</a>.

<div class="mw-heading mw-heading2"><h2 id="Réciproque_d'une_fonction_numérique">Réciproque d'une fonction numérique</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Existence">Existence</h3></div>

Le <a href="Th%C3%A9or%C3%A8me_des_valeurs_interm%C3%A9diaires" title="Théorème des valeurs intermédiaires">théorème des valeurs intermédiaires</a> et son corollaire, le <a href="Th%C3%A9or%C3%A8me_de_la_bijection" title="Théorème de la bijection">théorème de la bijection</a>, assurent que toute application continue strictement monotone sur un intervalle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle I}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/535ea7fc4134a31cbe2251d9d3511374bc41be9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}" loading="lazy"> détermine une bijection de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle I}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/535ea7fc4134a31cbe2251d9d3511374bc41be9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}" loading="lazy"> sur <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(I)=J}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(I)=J}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7b8b3542a3360ee5ffecad43d0d30089554c4183.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.829ex; height:2.843ex;" alt="{\displaystyle f(I)=J}" loading="lazy"> et que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle J}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/359e4f407b49910e02c27c2f52e87a36cd74c053.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.471ex; height:2.176ex;" alt="{\displaystyle J}" loading="lazy"> est aussi un intervalle. Cela signifie qu'une telle fonction possède une application réciproque définie sur <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle J}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/359e4f407b49910e02c27c2f52e87a36cd74c053.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.471ex; height:2.176ex;" alt="{\displaystyle J}" loading="lazy"> à valeurs dans <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle I}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/535ea7fc4134a31cbe2251d9d3511374bc41be9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}" loading="lazy">.

Cette propriété permet la création de nouvelles fonctions définies comme application réciproque de fonctions usuelles.

<div class="mw-heading mw-heading3"><h3 id="Exemples">Exemples</h3></div>

<th align="center">Fonction <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/202945cce41ecebb6f643f31d119c514bec7a074.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}" loading="lazy">

</th>

<th align="center">Départ et arrivée

</th>

<th align="center">Fonction réciproque

</th>

<th align="center">Départ et arrivée

</th>

<th align="center">Notes

</th></tr>

<tr>

<td align="center">Puissance <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=x^{n}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)=x^{n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8d09a7e3b534fe963d72c17a19e5bdd8b09eebc1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.064ex; height:2.843ex;" alt="{\displaystyle f(x)=x^{n}}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [0,+\infty [\to [0,+\infty [}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5ec1d6c9fbd9e6824490729363fbedcd54d4d526.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.213ex; height:2.843ex;" alt="{\displaystyle [0,+\infty [\to [0,+\infty [}" loading="lazy">

</td>

<td align="center">Racine n-ième <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}(x)={\sqrt[{n}]{x}}=x^{\frac {1}{n}}}">

<msup>

</mrow>

</msup>

<mroot>

</mrow>

</mroot>

</mrow>

<msup>

<mfrac>

</mfrac>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{-1}(x)={\sqrt[{n}]{x}}=x^{\frac {1}{n}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/89f95fdf3ca98ead7affadba04fbb23f75086df1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.453ex; height:4.009ex;" alt="{\displaystyle f^{-1}(x)={\sqrt[{n}]{x}}=x^{\frac {1}{n}}}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [0,+\infty [\to [0,+\infty [}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5ec1d6c9fbd9e6824490729363fbedcd54d4d526.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.213ex; height:2.843ex;" alt="{\displaystyle [0,+\infty [\to [0,+\infty [}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> entier naturel non nul

</td></tr>

<tr>

<td align="center">Exponentielle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=\mathrm {e} ^{x}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)=\mathrm {e} ^{x}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0e64365630f6d294ef495f1b532a73e3c3bfb95e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.721ex; height:2.843ex;" alt="{\displaystyle f(x)=\mathrm {e} ^{x}}" loading="lazy">

</td>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} \to ]0,+\infty [}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b3bca7e85e76d838638dc78b4263e0df1b04dced.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.269ex; height:2.843ex;" alt="{\displaystyle \mathbb {R} \to ]0,+\infty [}" loading="lazy">

</td>

<td align="center">Logarithme naturel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}(x)=\ln(x)}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{-1}(x)=\ln(x)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4752c8591586354d72ed0943e1eecdd1f61ad053.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.969ex; height:3.176ex;" alt="{\displaystyle f^{-1}(x)=\ln(x)}" loading="lazy">

</td>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ]0,+\infty [\to \mathbb {R} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6eb0bd1603b2f60da5f91c7c29756c5b0a346d0f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.269ex; height:2.843ex;" alt="{\displaystyle ]0,+\infty [\to \mathbb {R} }" loading="lazy">

</td>

<td>

</td></tr>

<tr>

<td align="center">Exponentielle de base <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=a^{x}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)=a^{x}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7d8e0c87bf84722c5b9b35197ea8f97a683a4310.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.918ex; height:2.843ex;" alt="{\displaystyle f(x)=a^{x}}" loading="lazy">

</td>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} \to ]0,+\infty [}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b3bca7e85e76d838638dc78b4263e0df1b04dced.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.269ex; height:2.843ex;" alt="{\displaystyle \mathbb {R} \to ]0,+\infty [}" loading="lazy">

</td>

<td align="center">Logarithme de base a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}(x)=\log _{a}(x)}">

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{-1}(x)=\log _{a}(x)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e03ae8a15adf41126ebb73488a03d20e78f63881.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.103ex; height:3.176ex;" alt="{\displaystyle f^{-1}(x)=\log _{a}(x)}" loading="lazy">

</td>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ]0,+\infty [\to \mathbb {R} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6eb0bd1603b2f60da5f91c7c29756c5b0a346d0f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.269ex; height:2.843ex;" alt="{\displaystyle ]0,+\infty [\to \mathbb {R} }" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> réel strictement positif

</td></tr>

<tr>

<td align="center">Puissance <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=x^{\alpha }}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)=x^{\alpha }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ab7193cbfb1970fed42d7ec2b6b1a08d8d35f9a6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.13ex; height:2.843ex;" alt="{\displaystyle f(x)=x^{\alpha }}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ]0,+\infty [\to ]0,+\infty [}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a216b826e064bcf9d4e52e2bef16362d85586c90.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.568ex; height:2.843ex;" alt="{\displaystyle ]0,+\infty [\to ]0,+\infty [}" loading="lazy">

</td>

<td align="center">Puissance 1/α <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}(x)=x^{1/\alpha }}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{-1}(x)=x^{1/\alpha }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/49653401a3e13883eab3c0e3e0772c0f9b5bafc4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.149ex; height:3.343ex;" alt="{\displaystyle f^{-1}(x)=x^{1/\alpha }}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ]0,+\infty [\to ]0,+\infty [}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a216b826e064bcf9d4e52e2bef16362d85586c90.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.568ex; height:2.843ex;" alt="{\displaystyle ]0,+\infty [\to ]0,+\infty [}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"> réel non nul

</td></tr>

<tr>

<td align="center">Sinus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=\sin(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)=\sin(x)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5fb1266b7f7718442e31e45eef3d81bef6a8b9af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.511ex; height:2.843ex;" alt="{\displaystyle f(x)=\sin(x)}" loading="lazy">

</td>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [-\pi /2,\pi /2]\to [-1,1]}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/137cca97ae666f1cf121676f2b4b86aff79f0914.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.524ex; height:2.843ex;" alt="{\displaystyle [-\pi /2,\pi /2]\to [-1,1]}" loading="lazy">

</td>

<td align="center">Arc sinus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}(x)=\arcsin(x)}">

<msup>

</mrow>

</msup>

<mi>arcsin</mi>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{-1}(x)=\arcsin(x)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7b4ec1aa3341565afda66d818b30c2b5756eccc4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.992ex; height:3.176ex;" alt="{\displaystyle f^{-1}(x)=\arcsin(x)}" loading="lazy">

</td>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [-1,1]\to [-\pi /2,\pi /2]}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1468039f72acd377adac81cfaf0b36835b07cf52.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.524ex; height:2.843ex;" alt="{\displaystyle [-1,1]\to [-\pi /2,\pi /2]}" loading="lazy">

</td>

<td>

</td></tr>

<tr>

<td align="center">Cosinus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=\cos(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)=\cos(x)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d8b2ebd101b76e22d42a1b63e34ae75d17f2bfa6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.766ex; height:2.843ex;" alt="{\displaystyle f(x)=\cos(x)}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [0,\pi ]\to [-1,1]}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/de4e4fd8a119ea334249e02c8e909b338e7ea47b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.897ex; height:2.843ex;" alt="{\displaystyle [0,\pi ]\to [-1,1]}" loading="lazy">

</td>

<td align="center">Arc cosinus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}(x)=\arccos(x)}">

<msup>

</mrow>

</msup>

<mi>arccos</mi>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{-1}(x)=\arccos(x)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/021aad3d0be562a5e26e802bce803c097b1d210c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.247ex; height:3.176ex;" alt="{\displaystyle f^{-1}(x)=\arccos(x)}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [-1,1]\to [0,\pi ]}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7346a7a0a131c16bb9e4037422b7001fa0212641.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.897ex; height:2.843ex;" alt="{\displaystyle [-1,1]\to [0,\pi ]}" loading="lazy">

</td>

<td>

</td></tr>

<tr>

<td align="center">Tangente <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=\tan(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)=\tan(x)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/69f203a2c591766f0a9f6492811e482366e2af34.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.015ex; height:2.843ex;" alt="{\displaystyle f(x)=\tan(x)}" loading="lazy">

</td>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ]-\pi /2,\pi /2[\to \mathbb {R} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/67e77c01ddb85b8a7506faad925399421aee691c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.097ex; height:2.843ex;" alt="{\displaystyle ]-\pi /2,\pi /2[\to \mathbb {R} }" loading="lazy">

</td>

<td align="center">Arc tangente <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}(x)=\arctan(x)}">

<msup>

</mrow>

</msup>

<mi>arctan</mi>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{-1}(x)=\arctan(x)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fbef89edff111f67137f02826e7ce852cf67198b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.496ex; height:3.176ex;" alt="{\displaystyle f^{-1}(x)=\arctan(x)}" loading="lazy">

</td>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} \to ]-\pi /2,\pi /2[}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d04e1b43b53f4e862d177ed006cd4e0097439b94.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.129ex; height:2.843ex;" alt="{\displaystyle \mathbb {R} \to ]-\pi /2,\pi /2[}" loading="lazy">

</td>

<td>

</td></tr>

</tbody></table>

À l'aide de ces fonctions, la recherche de l'application réciproque consiste à résoudre l'équation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)=y}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0a5080a8b0a963407ea74ffa50702563771518d1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.672ex; height:2.843ex;" alt="{\displaystyle f(x)=y}" loading="lazy">, d'inconnue <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy">  :

La fonction <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon x\mapsto x^{2}+3}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\colon x\mapsto x^{2}+3}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/271cf7d50e27726aab2d4153d9f3f0e9c7e3d0c8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.643ex; height:3.009ex;" alt="{\displaystyle f\colon x\mapsto x^{2}+3}" loading="lazy"> est une bijection de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ]-\infty ,0]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ]-\infty ,0]}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ee0b41e938421c21a8f7ad8963acdea1067e153d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.622ex; height:2.843ex;" alt="{\displaystyle ]-\infty ,0]}" loading="lazy"> sur <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [3,+\infty [}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [3,+\infty [}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/28adf9f9f5a95666a772c7f57e77628c3ce4ba19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.622ex; height:2.843ex;" alt="{\displaystyle [3,+\infty [}" loading="lazy"> et possède une application réciproque que l'on cherche à déterminer en résolvant, pour <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"> dans <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [3,+\infty [}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [3,+\infty [}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/28adf9f9f5a95666a772c7f57e77628c3ce4ba19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.622ex; height:2.843ex;" alt="{\displaystyle [3,+\infty [}" loading="lazy">, l'équation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}+3=y}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x^{2}+3=y}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8d6c89db5262d168e3c179f884cb65b0a65421ce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.641ex; height:3.009ex;" alt="{\displaystyle x^{2}+3=y}" loading="lazy">, ou encore <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}=y-3}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x^{2}=y-3}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/309d94b8df952b74f99548069fd2a6296280d827.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.641ex; height:3.009ex;" alt="{\displaystyle x^{2}=y-3}" loading="lazy">. Puisque <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\geq 3}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y\geq 3}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/db241d80394fb243a18b064fe03f52a71cf471bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.416ex; height:2.509ex;" alt="{\displaystyle y\geq 3}" loading="lazy">, cette équation possède deux solutions dont une seule appartenant à l'intervalle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ]-\infty ,0]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ]-\infty ,0]}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ee0b41e938421c21a8f7ad8963acdea1067e153d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.622ex; height:2.843ex;" alt="{\displaystyle ]-\infty ,0]}" loading="lazy"> : <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=-{\sqrt {y-3}}}">

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x=-{\sqrt {y-3}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e7524281af008ad36892f57ea12580f498f1583f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:13.718ex; height:3.509ex;" alt="{\displaystyle x=-{\sqrt {y-3}}}" loading="lazy">. Donc la réciproque de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> est <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{-1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3e5cfa2f5c08d6fe7d046b73faa6e3f213acc802.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.653ex; height:3.009ex;" alt="{\displaystyle f^{-1}}" loading="lazy"> définie par <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}(y)=-{\sqrt {y-3}}}">

<msup>

</mrow>

</msup>

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{-1}(y)=-{\sqrt {y-3}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d31e0662bdd153db749ad8cef0a5401f480f5480.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:19.007ex; height:3.509ex;" alt="{\displaystyle f^{-1}(y)=-{\sqrt {y-3}}}" loading="lazy">.

Cette recherche peut se révéler infructueuse et nécessiter la création d'une fonction nouvelle. Ainsi, la fonction <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon x\mapsto f(x)=x\mathrm {e} ^{x}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\colon x\mapsto f(x)=x\mathrm {e} ^{x}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f3155cca7f03008eaa8c20f51a4266b9b6bfa930.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.307ex; height:2.843ex;" alt="{\displaystyle f\colon x\mapsto f(x)=x\mathrm {e} ^{x}}" loading="lazy"> est une bijection de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,+\infty [}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [0,+\infty [}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/207d226525287a9b2ebb3ba52c61454a0df207b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.622ex; height:2.843ex;" alt="{\displaystyle [0,+\infty [}" loading="lazy"> vers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,+\infty [}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [0,+\infty [}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/207d226525287a9b2ebb3ba52c61454a0df207b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.622ex; height:2.843ex;" alt="{\displaystyle [0,+\infty [}" loading="lazy"> ; l'équation correspondante <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=x\mathrm {e} ^{x}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y=x\mathrm {e} ^{x}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fd7c4e6de47266383afce3999d1d6cc93af0a7c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.789ex; height:2.676ex;" alt="{\displaystyle y=x\mathrm {e} ^{x}}" loading="lazy"> n'a pas de solution exprimable à l'aide des fonctions usuelles, ce qui oblige, pour exprimer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=f^{-1}(y)}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x=f^{-1}(y)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b8c259d877499b841cadd9d19146f2413cec1882.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.046ex; height:3.176ex;" alt="{\displaystyle x=f^{-1}(y)}" loading="lazy">, à définir une nouvelle fonction, ici la <a href="Fonction_W_de_Lambert" title="Fonction W de Lambert">fonction W de Lambert</a>.

<div class="mw-heading mw-heading3"><h3 id="Graphe">Graphe</h3></div>

Lorsque deux fonctions sont réciproques l'une de l'autre, alors leurs représentations graphiques dans un plan muni d'un repère orthonormal sont symétriques l'une de l'autre par rapport à la droite <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (D)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (D)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5cbcb66c83c0cf2a2d0e26d456072eef521e6823.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.734ex; height:2.843ex;" alt="{\displaystyle (D)}" loading="lazy"> d'équation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y=x}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d0abe2e7da593fb7b41d44e87a97fefdd8998b77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.584ex; height:2.009ex;" alt="{\displaystyle y=x}" loading="lazy"> (appelée aussi <a href="Premi%C3%A8re_bissectrice" title="Première bissectrice">première bissectrice</a>).

En effet, si <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {M} (x,y)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {M} (x,y)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/29b9ee315908e99375eb5c4aebc762b4556eec4c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.459ex; height:2.843ex;" alt="{\displaystyle \mathrm {M} (x,y)}" loading="lazy"> est un point du graphe de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy">, alors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=f(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y=f(x)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2311a6a75c54b0ea085a381ba472c31d59321514.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.672ex; height:2.843ex;" alt="{\displaystyle y=f(x)}" loading="lazy"> donc <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=f^{-1}(y)}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x=f^{-1}(y)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b8c259d877499b841cadd9d19146f2413cec1882.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.046ex; height:3.176ex;" alt="{\displaystyle x=f^{-1}(y)}" loading="lazy"> donc <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {M'} (y,x)}">

<msup>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {M'} (y,x)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2d0dde83e46903823db5fdd3971fa489882fca42.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.144ex; height:3.009ex;" alt="{\displaystyle \mathrm {M'} (y,x)}" loading="lazy"> est un point du graphe de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{-1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3e5cfa2f5c08d6fe7d046b73faa6e3f213acc802.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.653ex; height:3.009ex;" alt="{\displaystyle f^{-1}}" loading="lazy">. Or le point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {M'} (y,x)}">

<msup>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {M'} (y,x)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2d0dde83e46903823db5fdd3971fa489882fca42.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.144ex; height:3.009ex;" alt="{\displaystyle \mathrm {M'} (y,x)}" loading="lazy"> est le symétrique du point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {M} (x,y)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {M} (x,y)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/29b9ee315908e99375eb5c4aebc762b4556eec4c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.459ex; height:2.843ex;" alt="{\displaystyle \mathrm {M} (x,y)}" loading="lazy"> par rapport à la droite <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (D)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (D)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5cbcb66c83c0cf2a2d0e26d456072eef521e6823.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.734ex; height:2.843ex;" alt="{\displaystyle (D)}" loading="lazy">, pour les deux raisons suivantes :

Le milieu du segment <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [\mathrm {MM'} ]}">

<msup>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [\mathrm {MM'} ]}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/147f0d1efc9fc64d6d7513aa82c852cf91ed1d00.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.24ex; height:3.009ex;" alt="{\displaystyle [\mathrm {MM'} ]}" loading="lazy"> est sur la droite <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (D)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (D)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5cbcb66c83c0cf2a2d0e26d456072eef521e6823.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.734ex; height:2.843ex;" alt="{\displaystyle (D)}" loading="lazy">, et d'autre part, le vecteur <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overrightarrow {\mathrm {MM} '}}}">

<mover>

<msup>

</mrow>

</msup>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {\mathrm {MM} '}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/17e60d61cdd683be262f6af4d5d93340c101109c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-top: -0.491ex; width:5.077ex; height:4.176ex;" alt="{\displaystyle {\overrightarrow {\mathrm {MM} '}}}" loading="lazy"> est orthogonal au vecteur de coordonnées <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1,1)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (1,1)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2c2a42feb07f4139bf871ae6856b11d4567bea23.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.168ex; height:2.843ex;" alt="{\displaystyle (1,1)}" loading="lazy">, qui est un <a href="Vecteur_directeur" title="Vecteur directeur">vecteur directeur</a> de la droite <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (D)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (D)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5cbcb66c83c0cf2a2d0e26d456072eef521e6823.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.734ex; height:2.843ex;" alt="{\displaystyle (D)}" loading="lazy"> (leur <a href="Produit_scalaire_canonique" title="Produit scalaire canonique">produit scalaire canonique</a> est nul).

On sait donc que le symétrique de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {M} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {M} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d1ec92b986053ec4967f418634cf062b9d980f9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.131ex; height:2.176ex;" alt="{\displaystyle \mathrm {M} }" loading="lazy"> par rapport à <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (D)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (D)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5cbcb66c83c0cf2a2d0e26d456072eef521e6823.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.734ex; height:2.843ex;" alt="{\displaystyle (D)}" loading="lazy"> est un point du graphe de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{-1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3e5cfa2f5c08d6fe7d046b73faa6e3f213acc802.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.653ex; height:3.009ex;" alt="{\displaystyle f^{-1}}" loading="lazy">. Un raisonnement analogue prouve que si <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {M} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {M} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d1ec92b986053ec4967f418634cf062b9d980f9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.131ex; height:2.176ex;" alt="{\displaystyle \mathrm {M} }" loading="lazy"> est un point du graphe de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{-1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3e5cfa2f5c08d6fe7d046b73faa6e3f213acc802.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.653ex; height:3.009ex;" alt="{\displaystyle f^{-1}}" loading="lazy">, alors son symétrique par rapport à <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (D)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (D)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5cbcb66c83c0cf2a2d0e26d456072eef521e6823.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.734ex; height:2.843ex;" alt="{\displaystyle (D)}" loading="lazy"> est un point du graphe de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy">.

<div class="mw-heading mw-heading3"><h3 id="Continuité">Continuité</h3></div>

En général, la réciproque d'une <a href="Continuit%C3%A9_(math%C3%A9matiques)" title="Continuité (mathématiques)">fonction continue</a> n'est pas continue mais la réciproque d'une fonction continue sur un intervalle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle I}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/535ea7fc4134a31cbe2251d9d3511374bc41be9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}" loading="lazy"> à valeurs dans un intervalle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle J}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/359e4f407b49910e02c27c2f52e87a36cd74c053.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.471ex; height:2.176ex;" alt="{\displaystyle J}" loading="lazy"> est une fonction continue sur <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle J}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/359e4f407b49910e02c27c2f52e87a36cd74c053.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.471ex; height:2.176ex;" alt="{\displaystyle J}" loading="lazy">, selon le <a href="Th%C3%A9or%C3%A8me_de_la_bijection" title="Théorème de la bijection">théorème de la bijection</a>.

<div class="mw-heading mw-heading3"><h3 id="Dérivabilité">Dérivabilité</h3></div>

Si <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> est une fonction continue sur un intervalle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle I}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/535ea7fc4134a31cbe2251d9d3511374bc41be9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}" loading="lazy"> à valeurs dans un intervalle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle J}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/359e4f407b49910e02c27c2f52e87a36cd74c053.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.471ex; height:2.176ex;" alt="{\displaystyle J}" loading="lazy"> et si <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{-1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3e5cfa2f5c08d6fe7d046b73faa6e3f213acc802.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.653ex; height:3.009ex;" alt="{\displaystyle f^{-1}}" loading="lazy"> est sa réciproque, la fonction <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{-1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3e5cfa2f5c08d6fe7d046b73faa6e3f213acc802.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.653ex; height:3.009ex;" alt="{\displaystyle f^{-1}}" loading="lazy"> est dérivable en tout point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f11423fbb2e967f986e36804a8ae4271734917c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" loading="lazy"> tant que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> admet en <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}(b)}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{-1}(b)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/32607d1dc67910b73ab3afc938932c73af809948.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.46ex; height:3.176ex;" alt="{\displaystyle f^{-1}(b)}" loading="lazy"> une dérivée non nulle.

La dérivée en <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f11423fbb2e967f986e36804a8ae4271734917c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" loading="lazy"> de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{-1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3e5cfa2f5c08d6fe7d046b73faa6e3f213acc802.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.653ex; height:3.009ex;" alt="{\displaystyle f^{-1}}" loading="lazy"> est alors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{f'\left(f^{-1}(b)\right)}}}">

<mfrac>

<mrow>

<msup>

</msup>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{f'\left(f^{-1}(b)\right)}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ef4d7fb870e5d26f61269a036b0ea8672f830f78.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:11.498ex; height:6.009ex;" alt="{\displaystyle {\frac {1}{f'\left(f^{-1}(b)\right)}}}" loading="lazy">.

Un moyen simple de comprendre ce résultat, mais non de le démontrer, est d'utiliser les notations différentielles et de remarquer que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {{\rm {d}}x}{{\rm {d}}y}}={\frac {1}{{\rm {d}}y/{\rm {d}}x}}}">

<mfrac>

<mrow>

</mrow>

</mrow>

<mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {{\rm {d}}x}{{\rm {d}}y}}={\frac {1}{{\rm {d}}y/{\rm {d}}x}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c886aed8306ce29afb24f3b4b186cc1037bf8cdc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:13.626ex; height:6.176ex;" alt="{\displaystyle {\frac {{\rm {d}}x}{{\rm {d}}y}}={\frac {1}{{\rm {d}}y/{\rm {d}}x}}}" loading="lazy">.

On trouve une démonstration dans l'article <a class="external text" href="https://fr.wikiversity.org/wiki/fr:Fonctions_d%27une_variable_r%C3%A9elle/D%C3%A9rivabilit%C3%A9#Dérivée_et_opérations">Dérivée et opérations</a> sur Wikiversité.

<div class="mw-heading mw-heading3"><h3 id="Recherche_graphique_ou_numérique_d'une_réciproque">Recherche graphique ou numérique d'une réciproque</h3></div>

Il n'est pas toujours possible de déterminer la réciproque de manière analytique : on sait calculer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/202945cce41ecebb6f643f31d119c514bec7a074.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}" loading="lazy">, mais on ne sait pas calculer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}(y)}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{-1}(y)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8b357745fa4a2178733a502b4432072be8222fd4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.618ex; height:3.176ex;" alt="{\displaystyle f^{-1}(y)}" loading="lazy">. On peut alors utiliser une méthode graphique ou une approximation <a href="M%C3%A9thode_num%C3%A9rique" class="mw-redirect" title="Méthode numérique">numérique</a>.

La méthode graphique consiste à tracer la courbe représentative <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=f(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y=f(x)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2311a6a75c54b0ea085a381ba472c31d59321514.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.672ex; height:2.843ex;" alt="{\displaystyle y=f(x)}" loading="lazy">. Pour rechercher <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}(y)}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{-1}(y)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8b357745fa4a2178733a502b4432072be8222fd4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.618ex; height:3.176ex;" alt="{\displaystyle f^{-1}(y)}" loading="lazy">, on cherche le point de la courbe dont l'ordonnée est <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"> et on lit son abscisse. Pour ce faire, on trace la droite d'ordonnée <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"> concernée, on recherche l'intersection de cette droite avec la courbe, et l'on trace la droite parallèle à l'axe des ordonnées passant par cette intersection. Le point d'intersection de cette droite avec l'axe des abscisses donne la valeur <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> recherchée. C'est le principe d'un grand nombre d'<a href="Nomogramme#Abaques" title="Nomogramme">abaques</a>.

Numériquement, rechercher <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}(y)}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{-1}(y)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8b357745fa4a2178733a502b4432072be8222fd4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.618ex; height:3.176ex;" alt="{\displaystyle f^{-1}(y)}" loading="lazy"> revient à rechercher les racines de la fonction <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x)=f(x)-y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g(x)=f(x)-y}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ec1f36ddd9d26426979ceac37ced86863fdabe7b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.767ex; height:2.843ex;" alt="{\displaystyle g(x)=f(x)-y}" loading="lazy">.

Si l'on sait que le domaine de recherche — intervalle des <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> possibles — est « restreint » et que la fonction est dérivable sur cet intervalle, on peut linéariser la fonction, c'est-à-dire la remplacer par une <a href="Fonction_affine" title="Fonction affine">fonction affine</a> obtenue par un <a href="D%C3%A9veloppement_limit%C3%A9" title="Développement limité">développement limité</a> au voisinage d'un point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/86f21d0e31751534cd6584264ecf864a6aa792cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}" loading="lazy"> de cet intervalle :

<msub>

</mrow>

</msub>

<msup>

</msup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g(x)\simeq g(x_{0})+g'(x_{0})(x-x_{0})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a8be3a09203cc4c84e031b411f173a2bd0cafcd7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.863ex; height:3.009ex;" alt="{\displaystyle g(x)\simeq g(x_{0})+g'(x_{0})(x-x_{0})}" loading="lazy"></dd></dl>

On a ainsi une approximation de la solution, si <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g'(x_{0})\neq 0}">

<msup>

</msup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g'(x_{0})\neq 0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/adc6314572776ed8d2688aa748985cc00d251be4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.257ex; height:3.009ex;" alt="{\displaystyle g'(x_{0})\neq 0}" loading="lazy"> :<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}(y)\simeq x_{0}-{\frac {g(x_{0})}{g'(x_{0})}}}">

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

<msup>

</msup>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{-1}(y)\simeq x_{0}-{\frac {g(x_{0})}{g'(x_{0})}}}</annotation>

</semantics>

</math>

C'est la démarche de l'algorithme de Newton, mais avec une seule <a href="It%C3%A9ration" title="Itération">itération</a>.

On peut également utiliser une fonction d'approximation plus complexe mais néanmoins inversible.

<div class="mw-heading mw-heading2"><h2 id="Exemples_de_réciproques_de_transformations_du_plan">Exemples de réciproques de transformations du plan</h2></div>

Les transformations du plan sont les applications bijectives du plan ; il est donc intéressant d'en connaître les réciproques, du moins pour les transformations de référence.

<th align="center">Transformation

</th>

<th align="center">Transformation réciproque

</th></tr>

<tr>

<td align="center"><a href="Translation" title="Translation">Translation</a> de vecteur <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {u}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {u}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/89c41e9cf70c5e5b56e2128a136985a75f90ba43.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:2.343ex;" alt="{\displaystyle {\vec {u}}}" loading="lazy">

</td>

<td align="center">Translation de vecteur <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\vec {u}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -{\vec {u}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/efe25b4bd6bdc781b9316fc224cbc90009363f99.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.138ex; height:2.509ex;" alt="{\displaystyle -{\vec {u}}}" loading="lazy">

</td></tr>

<tr>

<td align="center"><a href="Sym%C3%A9trie_(transformation_g%C3%A9om%C3%A9trique)" title="Symétrie (transformation géométrique)">Symétrie</a> de centre <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {O} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {O} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f3d6d4173d32feed308e80dbaf00e1274f40702d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \mathrm {O} }" loading="lazy"> ou d'axe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (D)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (D)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5cbcb66c83c0cf2a2d0e26d456072eef521e6823.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.734ex; height:2.843ex;" alt="{\displaystyle (D)}" loading="lazy">

</td>

<td align="center">Symétrie de centre <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {O} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {O} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f3d6d4173d32feed308e80dbaf00e1274f40702d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \mathrm {O} }" loading="lazy"> ou d'axe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (D)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (D)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5cbcb66c83c0cf2a2d0e26d456072eef521e6823.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.734ex; height:2.843ex;" alt="{\displaystyle (D)}" loading="lazy">

</td></tr>

<tr>

<td align="center"><a href="Homoth%C3%A9tie" title="Homothétie">Homothétie</a> de centre <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {C} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {C} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a22ea6dc0369a8826d0f3ad63159f1604ab645c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathrm {C} }" loading="lazy"> et de rapport k

</td>

<td align="center">Homothétie de centre <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {C} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {C} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a22ea6dc0369a8826d0f3ad63159f1604ab645c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathrm {C} }" loading="lazy"> et de rapport 1/k

</td></tr>

<tr>

<td align="center">Rotation de centre <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {C} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {C} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a22ea6dc0369a8826d0f3ad63159f1604ab645c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathrm {C} }" loading="lazy"> et d'angle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \theta }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }" loading="lazy">

</td>

<td align="center">Rotation de centre <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {C} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {C} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a22ea6dc0369a8826d0f3ad63159f1604ab645c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathrm {C} }" loading="lazy"> et d'angle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\theta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -\theta }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1d715f4b31b39f52768eba6405b658be5dc573f5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.899ex; height:2.343ex;" alt="{\displaystyle -\theta }" loading="lazy">

</td></tr>

<tr>

<td align="center"><a href="Similitude_(g%C3%A9om%C3%A9trie)" title="Similitude (géométrie)">Similitude</a> directe de centre <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {C} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {C} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a22ea6dc0369a8826d0f3ad63159f1604ab645c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathrm {C} }" loading="lazy">, de rapport <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> et d'angle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \theta }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }" loading="lazy">

</td>

<td align="center">Similitude directe de centre <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {C} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {C} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a22ea6dc0369a8826d0f3ad63159f1604ab645c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathrm {C} }" loading="lazy">, de rapport <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/k}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1/k}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a7e9fedad8c70c6331b2640b56c23cef8c884e1f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.536ex; height:2.843ex;" alt="{\displaystyle 1/k}" loading="lazy"> et d'angle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\theta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -\theta }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1d715f4b31b39f52768eba6405b658be5dc573f5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.899ex; height:2.343ex;" alt="{\displaystyle -\theta }" loading="lazy">

</td></tr>

<tr>

<td align="center">Similitude indirecte de centre <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {C} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {C} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a22ea6dc0369a8826d0f3ad63159f1604ab645c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathrm {C} }" loading="lazy">, de rapport <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> et d'axe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (D)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (D)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5cbcb66c83c0cf2a2d0e26d456072eef521e6823.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.734ex; height:2.843ex;" alt="{\displaystyle (D)}" loading="lazy">

</td>

<td align="center">Similitude indirecte de centre <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {C} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {C} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a22ea6dc0369a8826d0f3ad63159f1604ab645c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathrm {C} }" loading="lazy">, de rapport <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/k}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1/k}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a7e9fedad8c70c6331b2640b56c23cef8c884e1f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.536ex; height:2.843ex;" alt="{\displaystyle 1/k}" loading="lazy"> et d'axe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (D)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (D)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5cbcb66c83c0cf2a2d0e26d456072eef521e6823.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.734ex; height:2.843ex;" alt="{\displaystyle (D)}" loading="lazy">

</td></tr>

<tr>

<td align="center">Symétrie glissée d'axe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (D)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (D)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5cbcb66c83c0cf2a2d0e26d456072eef521e6823.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.734ex; height:2.843ex;" alt="{\displaystyle (D)}" loading="lazy"> et de vecteur <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {u}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {u}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/89c41e9cf70c5e5b56e2128a136985a75f90ba43.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:2.343ex;" alt="{\displaystyle {\vec {u}}}" loading="lazy">

</td>

<td align="center">Symétrie glissée d'axe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (D)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (D)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5cbcb66c83c0cf2a2d0e26d456072eef521e6823.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.734ex; height:2.843ex;" alt="{\displaystyle (D)}" loading="lazy"> et de vecteur <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\vec {u}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -{\vec {u}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/efe25b4bd6bdc781b9316fc224cbc90009363f99.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.138ex; height:2.509ex;" alt="{\displaystyle -{\vec {u}}}" loading="lazy">

</td></tr>

<tr>

<td align="center"><a href="Affinit%C3%A9_(math%C3%A9matiques)" title="Affinité (mathématiques)">Affinité</a> d'axe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (D)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (D)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5cbcb66c83c0cf2a2d0e26d456072eef521e6823.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.734ex; height:2.843ex;" alt="{\displaystyle (D)}" loading="lazy"> de direction <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (D')}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (D')}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/01b7487fbaca75c0a5483b60a761c12ff3d9020e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:3.009ex;" alt="{\displaystyle (D')}" loading="lazy"> et de rapport <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">

</td>

<td align="center">Affinité d'axe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (D)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (D)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5cbcb66c83c0cf2a2d0e26d456072eef521e6823.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.734ex; height:2.843ex;" alt="{\displaystyle (D)}" loading="lazy"> de direction <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (D')}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (D')}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/01b7487fbaca75c0a5483b60a761c12ff3d9020e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:3.009ex;" alt="{\displaystyle (D')}" loading="lazy"> et de rapport <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/k}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1/k}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a7e9fedad8c70c6331b2640b56c23cef8c884e1f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.536ex; height:2.843ex;" alt="{\displaystyle 1/k}" loading="lazy">

</td></tr></tbody></table>

<div class="mw-heading mw-heading2"><h2 id="Réciproques_en_algèbre">Réciproques en algèbre</h2></div>

En algèbre, un <a href="Morphisme" title="Morphisme">morphisme</a> bijectif de groupes, d'anneaux, de corps, d'espaces vectoriels admet une application réciproque qui est aussi un morphisme de même type. L'application et sa réciproque sont appelés des <a href="Isomorphisme" title="Isomorphisme">isomorphismes</a>.

Dans le cas d'une application <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> <a href="Application_lin%C3%A9aire" title="Application linéaire">linéaire</a> d'un <a href="Espace_vectoriel" title="Espace vectoriel">espace vectoriel</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy"> vers un espace vectoriel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy">, tous deux de dimension finie et munis de bases, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> est bijective si et seulement si <a href="Matrice_d'une_application_lin%C3%A9aire" title="Matrice d'une application linéaire">sa matrice</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> dans les bases fixées est une <a href="Matrice_(math%C3%A9matiques)" title="Matrice (mathématiques)">matrice carrée</a> inversible. La matrice dans ces bases de la réciproque de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> est alors la <a href="Matrice_inversible" title="Matrice inversible">matrice inverse</a> de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy">, notée <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M^{-1}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M^{-1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5d09e5d114e22862ebd24581f86d697b457db05a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.832ex; height:2.676ex;" alt="{\displaystyle M^{-1}}" loading="lazy">.

<div class="mw-heading mw-heading2"><h2 id="Quelques_concepts_apparentés">Quelques concepts apparentés</h2></div>

Soit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon X\mapsto Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\colon X\mapsto Y}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ac6a918e1322cc2731b499c950f2c6cba4636038.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.68ex; height:2.509ex;" alt="{\displaystyle f\colon X\mapsto Y}" loading="lazy"> une application.

<ul><li>Même lorsque <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> n'est pas bijective, il est possible de définir une <a href="Relation_binaire" title="Relation binaire">relation binaire</a> réciproque, de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> dans <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">, qui à tout élément de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> associe ses antécédents par <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> (donc rien si cet élément n'a pas d'antécédents). On parle alors de <a href="Fonction_multivalu%C3%A9e" title="Fonction multivaluée">réciproque multiforme</a>. L'application <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> est bijective si et seulement si cette relation réciproque est une application, et dans ce cas, cette application est bien l'application réciproque de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy">. On définit plus généralement la <a href="Fonction_multivalu%C3%A9e#Multifonction_réciproque" title="Fonction multivaluée">réciproque d'une multifonction</a> quelconque ou, ce qui revient au même, la <a href="Relation_binaire#Réciproque" title="Relation binaire">réciproque d'une relation binaire</a>.</li>

<li>Pour qu'il existe des inverses à gauche de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy">, c'est-à-dire des applications <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"> telles que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\circ f=\operatorname {Id} _{X}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g\circ f=\operatorname {Id} _{X}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b92353be1b61436c3bd6346d93b3192d4d6b92c9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.452ex; height:2.509ex;" alt="{\displaystyle g\circ f=\operatorname {Id} _{X}}" loading="lazy">, il faut et il suffit que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> soit injective. Pour qu'il existe des inverses à droite de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy">, c'est-à-dire des applications <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"> telles que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\circ g=\operatorname {Id} _{Y}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\circ g=\operatorname {Id} _{Y}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/51400c485a770d7a3b391efa7579e55286b677ad.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.306ex; height:2.509ex;" alt="{\displaystyle f\circ g=\operatorname {Id} _{Y}}" loading="lazy">, il faut et (en admettant l'<a href="Axiome_du_choix" title="Axiome du choix">axiome du choix</a>) il suffit que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> soit surjective.</li></ul>

La fonction réciproque d'une fonction <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> ne doit pas être confondue avec la <a href="Fonction_inverse" title="Fonction inverse">fonction inverse</a> de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy">. Cette confusion est fréquente du fait de la notation<a href="#cite_note-Notation-1-2">[2]</a> commune <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{-1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3e5cfa2f5c08d6fe7d046b73faa6e3f213acc802.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.653ex; height:3.009ex;" alt="{\displaystyle f^{-1}}" loading="lazy">, et parce que le terme anglais reciprocal se traduit souvent par inverse en français, tandis que l'adjectif anglais inverse se traduit parfois par réciproque en français.

<div class="mw-heading mw-heading2"><h2 id="Théorème_d'inversion_locale">Théorème d'inversion locale</h2></div>

Le <a href="Th%C3%A9or%C3%A8me_d'inversion_locale" title="Théorème d'inversion locale">théorème d'inversion locale</a> précise les <a href="Condition_d'existence" title="Condition d'existence">conditions d'existence</a> locale d'une application réciproque pour une fonction <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy">. C'est une généralisation d'un théorème simple sur les fonctions de la variable réelle.

Théorème — Si <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> est définie sur un intervalle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle I}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/535ea7fc4134a31cbe2251d9d3511374bc41be9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}" loading="lazy"> et si <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> est un élément de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle I}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/535ea7fc4134a31cbe2251d9d3511374bc41be9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}" loading="lazy">, si <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> possède en <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> une dérivée continue non nulle, alors il existe un intervalle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{a}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle I_{a}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/83b5cb24870dadbb0efdf2e6d864ef853a0f1f89.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.125ex; height:2.509ex;" alt="{\displaystyle I_{a}}" loading="lazy"> autour de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy">, un intervalle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{f(a)}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle J_{f(a)}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/29f90628c34b5d2d47ec5bbfae1c4a6d0e300644.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:4.576ex; height:3.009ex;" alt="{\displaystyle J_{f(a)}}" loading="lazy"> autour de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(a)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(a)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/368cb4b81ba5754d7a354a4ce49c2f1084bdaace.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.318ex; height:2.843ex;" alt="{\displaystyle f(a)}" loading="lazy"> et une fonction <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{-1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3e5cfa2f5c08d6fe7d046b73faa6e3f213acc802.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.653ex; height:3.009ex;" alt="{\displaystyle f^{-1}}" loading="lazy"> définie sur <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{f(a)}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle J_{f(a)}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/29f90628c34b5d2d47ec5bbfae1c4a6d0e300644.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:4.576ex; height:3.009ex;" alt="{\displaystyle J_{f(a)}}" loading="lazy"> qui soit l'application réciproque de la restriction de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> à <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{a}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle I_{a}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/83b5cb24870dadbb0efdf2e6d864ef853a0f1f89.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.125ex; height:2.509ex;" alt="{\displaystyle I_{a}}" loading="lazy">.

Cette application réciproque est aussi dérivable en <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(a)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(a)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/368cb4b81ba5754d7a354a4ce49c2f1084bdaace.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.318ex; height:2.843ex;" alt="{\displaystyle f(a)}" loading="lazy">.

</div>

Le théorème d'inversion locale généralise cette propriété à des fonctions définies sur des espaces vectoriels réels de dimension finie. La condition « <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f'(a)}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f'(a)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d98c61769b4906b6c0bd72f9de874410c7bf0d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.044ex; height:3.009ex;" alt="{\displaystyle f'(a)}" loading="lazy"> non nulle » est alors remplacée par « le <a href="Jacobien" class="mw-redirect" title="Jacobien">jacobien</a> de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> en <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> est non nul ». De plus, si <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> est de <a href="Classe_de_r%C3%A9gularit%C3%A9" title="Classe de régularité">classe</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {C}}^{k}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}^{k}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/119cdf8d90948dbf32e0db74eb837ba3ae4fa170.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.332ex; height:2.676ex;" alt="{\displaystyle {\mathcal {C}}^{k}}" loading="lazy">, l'application réciproque l'est aussi.

<div class="mw-heading mw-heading2"><h2 id="Notes_et_références">Notes et références</h2></div>

<li id="cite_note-RacineCubique-1"><a href="#cite_ref-RacineCubique_1-0">↑</a> L'exemple de la racine cubique est celui choisi par <a href="Jacques_Dixmier" title="Jacques Dixmier">Jacques Dixmier</a> dans son Cours de mathématiques du <abbr class="abbr" title="Premier">1er</abbr> cycle, Gauthier-Villars, 1967, p. 9.

</li>

<li id="cite_note-Notation-1-2">Ce choix de notation s'explique par le fait que la <a href="Loi_de_composition" title="Loi de composition">loi de composition</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \circ }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \circ }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/99add39d2b681e2de7ff62422c32704a05c7ec31.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.125ex; margin-bottom: -0.297ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle \circ }" loading="lazy">, restreinte aux <a href="Permutation" title="Permutation">permutations</a> d'un ensemble, est une loi de <a href="Groupe_(math%C3%A9matiques)" title="Groupe (mathématiques)">groupe</a>, et que <a href="Groupe_sym%C3%A9trique" title="Groupe symétrique">ce groupe</a> est noté multiplicativement. C'est cependant une ambiguïté de notation assez gênante pour que les <a href="Syst%C3%A8me_de_calcul_formel" title="Système de calcul formel">logiciels de calcul formel</a> séparent ces deux notions ; ainsi, <a href="Maple" title="Maple">Maple</a> note l'inverse <code>f^(-1)</code> et la bijection réciproque <code>f@@(–1)</code>.

</li>

</ol></div>

</div>

<div class="mw-heading mw-heading2"><h2 id="Articles_connexes">Articles connexes</h2></div>

<ul><li><a href="Fonction_circulaire_r%C3%A9ciproque" title="Fonction circulaire réciproque">Fonction circulaire réciproque</a></li>

<li><a href="Fonction_logistique_(Verhulst)" title="Fonction logistique (Verhulst)">Fonction logistique (Verhulst)</a></li></ul>

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