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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Image d'une application</span></h1>
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<p>On appelle <b>image d'une <a href="Application_(math%C3%A9matiques)" title="Application (mathématiques)">application</a></b> <span class="texhtml mvar" style="font-style:italic;">f</span> (d'un <a href="Ensemble" title="Ensemble">ensemble</a> <span class="texhtml mvar" style="font-style:italic;">A</span> vers un ensemble <span class="texhtml mvar" style="font-style:italic;">B</span>) l'<a href="Image_directe" title="Image directe">image directe</a> par <span class="texhtml mvar" style="font-style:italic;">f</span> de l'<a href="Ensemble_de_d%C3%A9part" class="mw-redirect" title="Ensemble de départ">ensemble de départ</a> <span class="texhtml mvar" style="font-style:italic;">A</span><sup id="cite_ref-Liret_1-0" class="reference"><a href="#cite_note-Liret-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>. C'est donc le <a href="Sous-ensemble" class="mw-redirect" title="Sous-ensemble">sous-ensemble</a> de <span class="texhtml mvar" style="font-style:italic;">B</span> contenant les <a href="Image_(math%C3%A9matiques)" title="Image (mathématiques)">images</a> de tous les <a href="Appartenance_(math%C3%A9matiques)" title="Appartenance (mathématiques)">éléments</a> de <span class="texhtml mvar" style="font-style:italic;">A</span>, et uniquement ces images. On le note <span class="texhtml">Im(<i>f</i>)</span>.
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Im} (f)=\{y\in B\mid \exists x\in A\quad f(x)=y\}=\{f(x)\mid x\in A\}=f(A)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Im</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<mi>f</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo fence="false" stretchy="false">{</mo>
<mi>y</mi>
<mo>∈<!-- ∈ --></mo>
<mi>B</mi>
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<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
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<mo>=</mo>
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<mo>=</mo>
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<mi>f</mi>
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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Im} (f)=\{y\in B\mid \exists x\in A\quad f(x)=y\}=\{f(x)\mid x\in A\}=f(A)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/878ed32aab17e11fe263877ba38c975e0bb0c247.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:62.805ex; height:2.843ex;" alt="{\displaystyle \operatorname {Im} (f)=\{y\in B\mid \exists x\in A\quad f(x)=y\}=\{f(x)\mid x\in A\}=f(A)}" loading="lazy"></span>.</dd></dl>
<p><br>
Exemple&nbsp;: <span class="citation">«&nbsp;L'image de la <a href="Fonction_sinus" class="mw-redirect" title="Fonction sinus">fonction sinus</a> est le <a href="Intervalle_compact" class="mw-redirect" title="Intervalle compact">segment</a> <span class="texhtml">[–1, 1]</span><sup id="cite_ref-Liret_1-1" class="reference"><a href="#cite_note-Liret-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>.&nbsp;»</span><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>Note 1<span class="cite-bracket">]</span></a></sup>
</p><p>Une application est <a href="Surjective" class="mw-redirect" title="Surjective">surjective</a> si et seulement si son image coïncide avec son <a href="Ensemble_d'arriv%C3%A9e" title="Ensemble d'arrivée">ensemble d'arrivée</a>.
</p><p>Une application est dite <a href="Injection_(math%C3%A9matiques)" title="Injection (mathématiques)">injective</a> si tout élément de son <a href="Ensemble_d'arriv%C3%A9e" title="Ensemble d'arrivée">ensemble d'arrivée</a> a <a href="Unicit%C3%A9_(math%C3%A9matiques)" title="Unicité (mathématiques)">au plus</a> un <a href="Ant%C3%A9c%C3%A9dent_(math%C3%A9matiques)" title="Antécédent (mathématiques)">antécédent</a> par <i>f.</i>
</p><p>Une application est dite <a href="Bijection" title="Bijection">bijective</a> si elle est à la fois surjective et injective, ce qui signifie que chaque élément de l'ensemble d'arrivée a un antécédent et que celui-ci est unique.
</p><p>On peut aussi parler d'<a href="Image_r%C3%A9ciproque" title="Image réciproque">image réciproque</a> d'une fonction qui est définie par:
</p><p><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Im} (f^{-1})=\{x\in A\mid f(x)\in B\}=f^{-1}(B)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Im</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
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<mi>f</mi>
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<mo>−<!-- − --></mo>
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<mi>x</mi>
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<mi>f</mi>
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<mi>x</mi>
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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Im} (f^{-1})=\{x\in A\mid f(x)\in B\}=f^{-1}(B)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cb89da0247181bce8dfa423ae0656f932fa52af2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.859ex; height:3.176ex;" alt="{\displaystyle \operatorname {Im} (f^{-1})=\{x\in A\mid f(x)\in B\}=f^{-1}(B)}" loading="lazy"></span>
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<div class="mw-heading mw-heading2"><h2 id="Notes_et_références"><span id="Notes_et_r.C3.A9f.C3.A9rences"></span>Notes et références</h2></div>
<div class="mw-heading mw-heading3"><h3 id="Notes">Notes</h3></div>
<div class="references-small decimal" style=""><div class="mw-references-wrap"><ol class="references" data-mw-group="Note">
<li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a> </span><span class="reference-text">Cette affirmation n'est vraie que si l'ensemble de départ est l'ensemble des nombres réels <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">
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<mrow class="MJX-TeXAtom-ORD">
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<mi mathvariant="double-struck">R</mi>
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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation>
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</math></span><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"></span> et est incorrecte si on généralise à l'ensemble des nombres complexes <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }">
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<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">C</mi>
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<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f9add4085095b9b6d28d045fd9c92c2c09f549a7.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 {C} }" loading="lazy"></span>.</span>
</li>
</ol></div>
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<div class="mw-heading mw-heading3"><h3 id="Références"><span id="R.C3.A9f.C3.A9rences"></span>Références</h3></div>
<div class="references-small decimal" style=""><div class="mw-references-wrap"><ol class="references">
<li id="cite_note-Liret-1"><span class="reference-text"><span class="ouvrage" id="Liret2006"><span class="ouvrage" id="François_Liret2006">François Liret, <cite class="italique">Maths en pratique&nbsp;: À l'usage des étudiants</cite>, <a href="Dunod" class="mw-redirect" title="Dunod">Dunod</a>, <time>2006</time>, 600&nbsp;<abbr class="abbr" title="pages">p.</abbr> <small style="line-height:1em;">(<a href="International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&nbsp;<span class="nowrap">978-2100496297</span>, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=-0X3hYe38jsC&amp;pg=PA13">lire en ligne</a>)</small>, <abbr class="abbr" title="page">p.</abbr>&nbsp;13<span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Maths+en+pratique&amp;rft.pub=Dunod&amp;rft.stitle=%C3%80+l%27usage+des+%C3%A9tudiants&amp;rft.aulast=Liret&amp;rft.aufirst=Fran%C3%A7ois&amp;rft.date=2006&amp;rft.pages=13&amp;rft.tpages=600&amp;rft.isbn=978-2100496297&amp;rft_id=%2F%2Fbooks.google.com%2Fbooks%3Fid%3D-0X3hYe38jsC%26pg%3DPA13&amp;rfr_id=info%3Asid%2Ffr.wikipedia.org%3AImage+d%27une+application"></span></span></span></span>
</li>
</ol></div>
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<div class="mw-heading mw-heading2"><h2 id="Articles_connexes">Articles connexes</h2></div>
<ul><li><a href="Application_lin%C3%A9aire#Noyau_et_image" title="Application linéaire">Image d'une application linéaire</a></li>
<li><a href="Lemme_des_noyaux" title="Lemme des noyaux">Lemme des noyaux</a></li>
<li><a href="Cat%C3%A9gorie_ab%C3%A9lienne" title="Catégorie abélienne">Catégorie abélienne</a></li>
<li><a href="Limite_projective" title="Limite projective">Limite projective</a></li>
<li><a href="Noyau_(alg%C3%A8bre)" title="Noyau (algèbre)">Noyau (algèbre)</a></li>
<li><a href="Fonction_multivalu%C3%A9e#Domaine,_image,_sélection" title="Fonction multivaluée">Image d'une fonction multivaluée</a> (autrement dit&nbsp;: d'une <a href="Relation_binaire" title="Relation binaire">relation binaire</a>)</li></ul>
<ul id="bandeau-portail" class="bandeau-portail"><li><span class="bandeau-portail-element"><span class="bandeau-portail-icone"><span class="noviewer" typeof="mw:File"></span></span> <span class="bandeau-portail-texte">Portail des mathématiques</span> </span></li> </ul></div><!--htdig_noindex--><div><div class="zim-footer">
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