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<title>Inverse function</title>

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Inverse function

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</style><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="Multiplicative_inverse" title="Multiplicative inverse">multiplicative inverse</a> or <a href="Additive_inverse" title="Additive inverse">additive inverse</a>.</div>

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</style><table class="sidebar nomobile nowraplinks"><tbody><tr><th class="sidebar-title" style="letter-spacing:0.0125em; background-color:#FFCC99"><a href="Function_(mathematics)" title="Function (mathematics)">Function</a></th></tr><tr><td class="sidebar-image">x ↦ f (x)</td></tr><tr><th class="sidebar-heading" style="font-size: 117%; letter-spacing: 0.0125em; font-weight: 500; border-top: 1px solid black; padding: 5px 0 3px">

<a href="History_of_the_function_concept" title="History of the function concept">History of the function concept</a></th></tr><tr><th class="sidebar-heading" style="font-size: 117%; letter-spacing: 0.0125em; font-weight: 500; border-top: 1px solid black; padding: 5px 0 3px">

Types by <a href="Domain_of_a_function" title="Domain of a function">domain</a> and <a href="Codomain" title="Codomain">codomain</a></th></tr><tr><td class="sidebar-content">

𝔹

→ <var>X</var></a></li>

𝔹<var>n</var>

→ <var>X</var></a></li>

→ ℤ</a></li>

ℤ

→ <var>X</var></a></li>

→ ℝ</a></li>

ℝ

→ <var>X</var></a></li>

ℝ<var>n</var>

→ <var>X</var></a></li>

→ ℂ</a></li>

ℂ

→ <var>X</var></a></li>

ℂ<var>n</var>

→ <var>X</var></a></li></ul>

</div></td>

</tr><tr><th class="sidebar-heading" style="font-size: 117%; letter-spacing: 0.0125em; font-weight: 500; border-top: 1px solid black; padding: 5px 0 3px">

<a href="List_of_types_of_functions" title="List of types of functions">Classes/properties</a> </th></tr><tr><td class="sidebar-content">

<ul><li><a href="Constant_function" title="Constant function">Constant</a></li>

<li><a href="Identity_function" title="Identity function">Identity</a></li>

<li><a href="Linear_map" title="Linear map">Linear</a></li>

<li><a href="Polynomial" title="Polynomial">Polynomial</a></li>

<li><a href="Rational_function" title="Rational function">Rational</a></li>

<li><a href="Algebraic_function" title="Algebraic function">Algebraic</a></li>

<li><a href="Analytic_function" title="Analytic function">Analytic</a></li>

<li><a href="Smooth_function" class="mw-redirect" title="Smooth function">Smooth</a></li>

<li><a href="Continuous_function" title="Continuous function">Continuous</a></li>

<li><a href="Measurable_function" title="Measurable function">Measurable</a></li>

<li><a href="Injective_function" title="Injective function">Injective</a></li>

<li><a href="Surjective_function" title="Surjective function">Surjective</a></li>

<li><a href="Bijection" title="Bijection">Bijective</a></li></ul>

</div></td>

</tr><tr><th class="sidebar-heading" style="font-size: 117%; letter-spacing: 0.0125em; font-weight: 500; border-top: 1px solid black; padding: 5px 0 3px">

Constructions</th></tr><tr><td class="sidebar-content">

<ul><li><a href="Restriction_(mathematics)" title="Restriction (mathematics)">Restriction</a></li>

<li><a href="Function_composition" title="Function composition">Composition</a></li>

</ul>

</div></td>

</tr><tr><th class="sidebar-heading" style="font-size: 117%; letter-spacing: 0.0125em; font-weight: 500; border-top: 1px solid black; padding: 5px 0 3px">

Generalizations </th></tr><tr><td class="sidebar-content">

<ul><li><a href="Relation_(mathematics)" title="Relation (mathematics)">Relation</a> (<a href="Binary_relation" title="Binary relation">Binary relation</a>)</li>

<li><a href="Set-valued_function" title="Set-valued function">Set-valued</a></li>

<li><a href="Multivalued_function" title="Multivalued function">Multivalued</a></li>

<li><a href="Partial_function" title="Partial function">Partial</a></li>

<li><a href="Implicit_function" title="Implicit function">Implicit</a></li>

<li><a href="Function_space" title="Function space">Space</a></li>

<li><a href="Higher-order_function" title="Higher-order function">Higher-order</a></li>

<li><a href="Morphism" title="Morphism">Morphism</a></li>

<li><a href="Functor" title="Functor">Functor</a></li></ul>

</div></td>

</tr><tr><th class="sidebar-heading" style="font-size: 117%; letter-spacing: 0.0125em; font-weight: 500; border-top: 1px solid black; padding: 5px 0 3px">

<a href="List_of_mathematical_functions" title="List of mathematical functions">List of specific functions</a></th></tr><tr><td class="sidebar-navbar"><style data-mw-deduplicate="TemplateStyles:r1239400231">

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In <a href="Mathematics" title="Mathematics">mathematics</a>, the inverse function of a <a href="Function_(mathematics)" title="Function (mathematics)">function</a> f (also called the inverse of f) is a function that undoes the operation of f. The inverse of f exists <a href="If_and_only_if" title="If and only if">if and only if</a> f is <a href="Bijection" title="Bijection">bijective</a>, and if it exists, is denoted by <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="./63d2b575826d75cfdbc1bea2f34ccfa71f1c59b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.3ex; height:3.009ex;" alt="{\displaystyle f^{-1}.}" loading="lazy">

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

</mstyle>

</mrow>

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

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</math><img src="./07b9ff205beb51e7899846aeae788ae5e5546a3e.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\to Y}" loading="lazy">, its inverse <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}\colon Y\to X}">

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

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</math><img src="./31a1bc0edf199414feff53da55c19b265bc5015a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.055ex; height:3.009ex;" alt="{\displaystyle f^{-1}\colon Y\to X}" loading="lazy"> admits an explicit description: it sends each element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\in Y}">

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

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</math><img src="./cee1c0ec36a82f33f5e3d7434d5667881b4ec323.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.769ex; height:2.509ex;" alt="{\displaystyle y\in Y}" loading="lazy"> to the unique element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X}">

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

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</math><img src="./3e580967f68f36743e894aa7944f032dda6ea01d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}" loading="lazy"> such that f(x) = y.

As an example, consider the <a href="Real_number" title="Real number">real-valued</a> function of a real variable given by f(x) = 5x − 7. One can think of f as the function which multiplies its input by 5 then subtracts 7 from the result. To undo this, one adds 7 to the input, then divides the result by 5. Therefore, the inverse of f is the function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}\colon \mathbb {R} \to \mathbb {R} }">

<msup>

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

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</math><img src="./45fe95e7e82c9eea2cd86f6b9789fd811a56bac4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.657ex; height:3.009ex;" alt="{\displaystyle f^{-1}\colon \mathbb {R} \to \mathbb {R} }" loading="lazy"> defined by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}(y)={\frac {y+7}{5}}.}">

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<annotation encoding="application/x-tex">{\displaystyle f^{-1}(y)={\frac {y+7}{5}}.}</annotation>

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</math><img src="./8fee2ee9786aa2e2f5f84330f6a71e297ab6a087.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:16.358ex; height:5.343ex;" alt="{\displaystyle f^{-1}(y)={\frac {y+7}{5}}.}" loading="lazy">

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

Let f be a function whose <a href="Domain_of_a_function" title="Domain of a function">domain</a> is the <a href="Set_(mathematics)" title="Set (mathematics)">set</a> X, and whose <a href="Codomain" title="Codomain">codomain</a> is the set Y. Then f is invertible if there exists a function g from Y to X such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(f(x))=x}">

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

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</math><img src="./d12c5b0b3b9b020bfc9f5e330c074b809cc54ebc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.771ex; height:2.843ex;" alt="{\displaystyle g(f(x))=x}" loading="lazy"> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X}">

</mstyle>

</mrow>

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

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</math><img src="./3e580967f68f36743e894aa7944f032dda6ea01d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(g(y))=y}">

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

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</math><img src="./e774231d71a255aa573dfbb55f22e4d9137bbeb3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.423ex; height:2.843ex;" alt="{\displaystyle f(g(y))=y}" loading="lazy"> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\in Y}">

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

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</math><img src="./cee1c0ec36a82f33f5e3d7434d5667881b4ec323.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.769ex; height:2.509ex;" alt="{\displaystyle y\in Y}" loading="lazy">.<a href="#cite_note-:2-1">[1]</a>

If f is invertible, then there is exactly one function g satisfying this property. The function g is called the inverse of f, and is usually denoted as f −1, a notation introduced by <a href="John_Frederick_William_Herschel" class="mw-redirect" title="John Frederick William Herschel">John Frederick William Herschel</a> in 1813.<a href="#cite_note-Herschel_1813-2">[2]</a><a href="#cite_note-Herschel_1820-3">[3]</a><a href="#cite_note-Peirce_1852-4">[4]</a><a href="#cite_note-Peano_1903-5">[5]</a><a href="#cite_note-Cajori_1929-6">[6]</a><a href="#cite_note-NB2-7">[nb 1]</a>

The function f is invertible if and only if it is bijective. This is because the condition <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(f(x))=x}">

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

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</math><img src="./d12c5b0b3b9b020bfc9f5e330c074b809cc54ebc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.771ex; height:2.843ex;" alt="{\displaystyle g(f(x))=x}" loading="lazy"> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X}">

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

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</math><img src="./3e580967f68f36743e894aa7944f032dda6ea01d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}" loading="lazy"> implies that f is <a href="Injective_function" title="Injective function">injective</a>, and the condition <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(g(y))=y}">

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

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</math><img src="./e774231d71a255aa573dfbb55f22e4d9137bbeb3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.423ex; height:2.843ex;" alt="{\displaystyle f(g(y))=y}" loading="lazy"> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\in Y}">

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

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</math><img src="./cee1c0ec36a82f33f5e3d7434d5667881b4ec323.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.769ex; height:2.509ex;" alt="{\displaystyle y\in Y}" loading="lazy"> implies that f is <a href="Surjective_function" title="Surjective function">surjective</a>.

The inverse function f −1 to f can be explicitly described as the function

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<mtext>the unique element </mtext>

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<annotation encoding="application/x-tex">{\displaystyle f^{-1}(y)=({\text{the unique element }}x\in X{\text{ such that }}f(x)=y)}</annotation>

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</math><img src="./4d37723845dfe5e67688edf6766c65c25cf8d00d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:56.52ex; height:3.176ex;" alt="{\displaystyle f^{-1}(y)=({\text{the unique element }}x\in X{\text{ such that }}f(x)=y)}" loading="lazy">.</dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Inverses_and_composition">Inverses and composition</h3></div>

<div role="note" class="hatnote navigation-not-searchable">See also: <a href="Inverse_element" title="Inverse element">Inverse element</a></div>

Recall that if f is an invertible function with domain X and codomain Y, then

<msup>

</mrow>

</msup>

<mrow>

<mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./2605eaf6c0af2d9d58e445c320556ce22f27e80c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.695ex; height:3.176ex;" alt="{\displaystyle f^{-1}\left(f(x)\right)=x}" loading="lazy">, for every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./3e580967f68f36743e894aa7944f032dda6ea01d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\left(f^{-1}(y)\right)=y}">

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./02df4d47e92336ff81902d7bc9ddc9306850c9ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.668ex; height:3.343ex;" alt="{\displaystyle f\left(f^{-1}(y)\right)=y}" loading="lazy"> for every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\in Y}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./cee1c0ec36a82f33f5e3d7434d5667881b4ec323.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.769ex; height:2.509ex;" alt="{\displaystyle y\in Y}" loading="lazy">.</dd></dl>

Using the <a href="Composition_of_functions" class="mw-redirect" title="Composition of functions">composition of functions</a>, this statement can be rewritten to the following equations between functions:

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./dc32e927e523d18d2c49b6ad29a6be645bad0436.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.797ex; height:3.009ex;" alt="{\displaystyle f^{-1}\circ f=\operatorname {id} _{X}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\circ f^{-1}=\operatorname {id} _{Y},}">

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./5628f161d115223516ed21f9e0470e070a3fe4af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.297ex; height:3.009ex;" alt="{\displaystyle f\circ f^{-1}=\operatorname {id} _{Y},}" loading="lazy"></dd></dl>

where idX is the <a href="Identity_function" title="Identity function">identity function</a> on the set X; that is, the function that leaves its argument unchanged. In <a href="Category_theory" title="Category theory">category theory</a>, this statement is used as the definition of an inverse <a href="Morphism" title="Morphism">morphism</a>.

Considering function composition helps to understand the notation f −1. Repeatedly composing a function f: X→X with itself is called <a href="Iterated_function" title="Iterated function">iteration</a>. If f is applied n times, starting with the value x, then this is written as f n(x); so f 2(x) = f (f (x)), etc. Since f −1(f (x)) = x, composing f −1 and f n yields f n−1, "undoing" the effect of one application of f.

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

While the notation f −1(x) might be misunderstood,<a href="#cite_note-:2-1">[1]</a> (f(x))−1 certainly denotes the <a href="Multiplicative_inverse" title="Multiplicative inverse">multiplicative inverse</a> of f(x) and has nothing to do with the inverse function of f.<a href="#cite_note-Cajori_1929-6">[6]</a> The notation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{\langle -1\rangle }}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./9b64f0c9fac3a319ae96603c0e4ff9ea91fba163.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.933ex; height:3.176ex;" alt="{\displaystyle f^{\langle -1\rangle }}" loading="lazy"> might be used for the inverse function to avoid ambiguity with the <a href="Multiplicative_inverse" title="Multiplicative inverse">multiplicative inverse</a>.<a href="#cite_note-8">[7]</a>

In keeping with the general notation, some English authors use expressions like sin−1(x) to denote the inverse of the sine function applied to x (actually a <a href="#Partial_inverses">partial inverse</a>; see below).<a href="#cite_note-9">[8]</a><a href="#cite_note-Cajori_1929-6">[6]</a> Other authors feel that this may be confused with the notation for the multiplicative inverse of sin (x), which can be denoted as (sin (x))−1.<a href="#cite_note-Cajori_1929-6">[6]</a> To avoid any confusion, an <a href="Inverse_trigonometric_function" class="mw-redirect" title="Inverse trigonometric function">inverse trigonometric function</a> is often indicated by the prefix "<a href="Arc_(function_prefix)" class="mw-redirect" title="Arc (function prefix)">arc</a>" (for Latin arcus).<a href="#cite_note-Korn_2000-10">[9]</a><a href="#cite_note-Atlas_2009-11">[10]</a> For instance, the inverse of the sine function is typically called the <a href="Arcsine" class="mw-redirect" title="Arcsine">arcsine</a> function, written as <a href="Arcsin" class="mw-redirect" title="Arcsin">arcsin</a>(x).<a href="#cite_note-Korn_2000-10">[9]</a><a href="#cite_note-Atlas_2009-11">[10]</a> Similarly, the inverse of a <a href="Hyperbolic_function" class="mw-redirect" title="Hyperbolic function">hyperbolic function</a> is indicated by the prefix "<a href="Ar_(function_prefix)" class="mw-redirect" title="Ar (function prefix)">ar</a>" (for Latin ārea).<a href="#cite_note-Atlas_2009-11">[10]</a> For instance, the inverse of the <a href="Hyperbolic_sine" class="mw-redirect" title="Hyperbolic sine">hyperbolic sine</a> function is typically written as <a href="Arsinh" class="mw-redirect" title="Arsinh">arsinh</a>(x).<a href="#cite_note-Atlas_2009-11">[10]</a> The expressions like sin−1(x) can still be useful to distinguish the <a href="Multivalued_function" title="Multivalued function">multivalued</a> inverse from the partial inverse: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin ^{-1}(x)=\{(-1)^{n}\arcsin(x)+\pi n:n\in \mathbb {Z} \}}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<mi>arcsin</mi>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sin ^{-1}(x)=\{(-1)^{n}\arcsin(x)+\pi n:n\in \mathbb {Z} \}}</annotation>

</semantics>

</math><img src="./cba167d85a7d685371195b4137f66e69ed68480a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:42.527ex; height:3.176ex;" alt="{\displaystyle \sin ^{-1}(x)=\{(-1)^{n}\arcsin(x)+\pi n:n\in \mathbb {Z} \}}" loading="lazy">. Other inverse special functions are sometimes prefixed with the prefix "inv", if the ambiguity of the f −1 notation should be avoided.<a href="#cite_note-Hall_1909-12">[11]</a><a href="#cite_note-Atlas_2009-11">[10]</a>

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

<div class="mw-heading mw-heading3"><h3 id="Squaring_and_square_root_functions">Squaring and square root functions</h3></div>

The function f: R → [0,∞) given by f(x) = x2 is not injective because <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-x)^{2}=x^{2}}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./9e6f96a4f1afdf835fc9f919e238b606c142fa21.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.484ex; height:3.176ex;" alt="{\displaystyle (-x)^{2}=x^{2}}" loading="lazy"> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./a9c6d458566aec47a7259762034790c8981aefab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.848ex; height:2.176ex;" alt="{\displaystyle x\in \mathbb {R} }" loading="lazy">. Therefore, f is not invertible.

If the domain of the function is restricted to the nonnegative reals, that is, we take the function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon [0,\infty )\to [0,\infty );\ x\mapsto x^{2}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\colon [0,\infty )\to [0,\infty );\ x\mapsto x^{2}}</annotation>

</semantics>

</math><img src="./f8fab9fbfd45eb1a75de491de40f3c6f8b42a8e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.012ex; height:3.176ex;" alt="{\displaystyle f\colon [0,\infty )\to [0,\infty );\ x\mapsto x^{2}}" loading="lazy"> with the same rule as before, then the function is bijective and so, invertible.<a href="#cite_note-13">[12]</a> The inverse function here is called the (positive) square root function and is denoted by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\mapsto {\sqrt {x}}}">

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./2159a46cc4bbf43fa8ac0fc605d91960d9b50aa0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.209ex; height:3.009ex;" alt="{\displaystyle x\mapsto {\sqrt {x}}}" loading="lazy">.

<div class="mw-heading mw-heading3"><h3 id="Standard_inverse_functions">Standard inverse functions</h3></div>

The following table shows several standard functions and their inverses:

<caption>Inverse arithmetic functions

</caption>

<th scope="col" align="center">Function f(x)

</th>

<th scope="col" align="center">Inverse f −1(y)

</th>

<th scope="col" align="center">Notes

</th></tr>

<tr>

<td align="center">x <a href="Addition" title="Addition">+</a> a

</td>

<td align="center">y <a href="Subtraction" title="Subtraction">−</a> a

</td>

<td>

</td></tr>

<tr>

<td align="center">a − x

</td>

<td align="center">a − y

</td>

<td>

</td></tr>

<tr>

</td>

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/* end https://en.wikipedia.org/ */

</style>⁠y/m⁠

</td>

<td>m ≠ 0

</td></tr>

<tr>

<td align="center">⁠1/x⁠ (i.e. x−1)

</td>

<td align="center">⁠1/y⁠ (i.e. y−1)

</td>

<td>x, y ≠ 0

</td></tr>

<tr>

<td align="center">xp

</td>

<mroot>

</mrow>

</mroot>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./cb87b7b6c3375274b556888ab6f0c90f95fd91b9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.091ex; height:3.009ex;" alt="{\displaystyle {\sqrt[{p}]{y}}}" loading="lazy"> (i.e. y1/p)

</td>

<td>integer p > 0; x, y ≥ 0 if p is even

</td></tr>

<tr>

<td align="center">ax

</td>

<td align="center"><a href="Logarithm" title="Logarithm">log</a>a y

</td>

<td>y > 0 and a > 0 and a ≠ 1

</td></tr>

<tr>

<td align="center">x<a href="E_(mathematical_constant)" title="E (mathematical constant)">e</a>x

</td>

<td align="center"><a href="Lambert_W_function" title="Lambert W function">W</a> (y)

</td>

<td>x ≥ −1 and y ≥ −1/e

</td></tr>

<tr>

<td align="center"><a href="Trigonometric_function" class="mw-redirect" title="Trigonometric function">trigonometric functions</a>

</td>

<td align="center"><a href="Inverse_trigonometric_function" class="mw-redirect" title="Inverse trigonometric function">inverse trigonometric functions</a>

</td>

<td>various restrictions (see table below)

</td></tr>

<tr>

<td align="center"><a href="Hyperbolic_function" class="mw-redirect" title="Hyperbolic function">hyperbolic functions</a>

</td>

<td align="center"><a href="Inverse_hyperbolic_function" class="mw-redirect" title="Inverse hyperbolic function">inverse hyperbolic functions</a>

</td>

<td>various restrictions

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

<div class="mw-heading mw-heading3"><h3 id="Formula_for_the_inverse">Formula for the inverse</h3></div>

Many functions given by algebraic formulas possess a formula for their inverse. This is because the inverse <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="./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"> of an invertible function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon \mathbb {R} \to \mathbb {R} }">

</mrow>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./b1cacd5f7bbe1027cc75fbe2fbd9cb5e79485302.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.283ex; height:2.509ex;" alt="{\displaystyle f\colon \mathbb {R} \to \mathbb {R} }" loading="lazy"> has an explicit description as

<msup>

</mrow>

</msup>

<mtext>the unique element </mtext>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{-1}(y)=({\text{the unique element }}x\in \mathbb {R} {\text{ such that }}f(x)=y)}</annotation>

</semantics>

</math><img src="./0602aa88084351cb2f725cafe24948e50a798e87.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:56.219ex; height:3.176ex;" alt="{\displaystyle f^{-1}(y)=({\text{the unique element }}x\in \mathbb {R} {\text{ such that }}f(x)=y)}" loading="lazy">.</dd></dl>

This allows one to easily determine inverses of many functions that are given by algebraic formulas. For example, if f is the function

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)=(2x+8)^{3}}</annotation>

</semantics>

</math><img src="./b206747b18386ca06771b1c558d5834c7fc2ac44.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.875ex; height:3.176ex;" alt="{\displaystyle f(x)=(2x+8)^{3}}" loading="lazy"></dd></dl>

then to determine <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="./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"> for a real number y, one must find the unique real number x such that (2x + 8)3 = y. This equation can be solved:

<mtr>

<mtd>

</mtd>

<mtd>

<msup>

</mrow>

</msup>

</mtd>

</mtr>

<mtr>

<mtd>

<mroot>

</mrow>

</mroot>

</mrow>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

<mroot>

</mrow>

</mroot>

</mrow>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

<mfrac>

<mrow>

<mroot>

</mrow>

</mroot>

</mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}y&=(2x+8)^{3}\\{\sqrt[{3}]{y}}&=2x+8\\{\sqrt[{3}]{y}}-8&=2x\\{\dfrac {{\sqrt[{3}]{y}}-8}{2}}&=x.\end{aligned}}}</annotation>

</semantics>

</math><img src="./3107b63d271dd3e5b2f928a036c04525c896fbda.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.505ex; width:21.139ex; height:16.176ex;" alt="{\displaystyle {\begin{aligned}y&=(2x+8)^{3}\\{\sqrt[{3}]{y}}&=2x+8\\{\sqrt[{3}]{y}}-8&=2x\\{\dfrac {{\sqrt[{3}]{y}}-8}{2}}&=x.\end{aligned}}}" loading="lazy"></dd></dl>

Thus the inverse function f −1 is given by the formula

<msup>

</mrow>

</msup>

<mfrac>

<mrow>

<mroot>

</mrow>

</mroot>

</mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{-1}(y)={\frac {{\sqrt[{3}]{y}}-8}{2}}.}</annotation>

</semantics>

</math><img src="./82ff457f789a7a6b985416c9d82f04d89a8cf686.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:18.294ex; height:5.843ex;" alt="{\displaystyle f^{-1}(y)={\frac {{\sqrt[{3}]{y}}-8}{2}}.}" loading="lazy"></dd></dl>

Sometimes, the inverse of a function cannot be expressed by a <a href="Closed-form_formula" class="mw-redirect" title="Closed-form formula">closed-form formula</a>. For example, if f is the function

</mstyle>

</mrow>

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

</semantics>

</math><img src="./ac1a4561ee60693ae976396bf541ccf0573044d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.905ex; height:2.843ex;" alt="{\displaystyle f(x)=x-\sin x,}" loading="lazy"></dd></dl>

then f is a bijection, and therefore possesses an inverse function f −1. The <a href="Kepler's_equation#Inverse_Kepler_equation" title="Kepler's equation">formula for this inverse</a> has an expression as an infinite sum:

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</munderover>

<mfrac>

<msup>

</mrow>

</mrow>

</msup>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</munder>

<mrow>

<mrow>

<mfrac>

<msup>

</mrow>

</mrow>

</msup>

<mrow>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<msup>

<mrow>

<mfrac>

<mroot>

<mrow>

</mrow>

</mrow>

</mroot>

</mfrac>

</mrow>

</mrow>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{-1}(y)=\sum _{n=1}^{\infty }{\frac {y^{n/3}}{n!}}\lim _{\theta \to 0}\left({\frac {\mathrm {d} ^{\,n-1}}{\mathrm {d} \theta ^{\,n-1}}}\left({\frac {\theta }{\sqrt[{3}]{\theta -\sin(\theta )}}}\right)^{n}\right).}</annotation>

</semantics>

</math><img src="./1a3c7e1ed9b1858e0cfa36100898930fdc62c2dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:51.714ex; height:7.509ex;" alt="{\displaystyle f^{-1}(y)=\sum _{n=1}^{\infty }{\frac {y^{n/3}}{n!}}\lim _{\theta \to 0}\left({\frac {\mathrm {d} ^{\,n-1}}{\mathrm {d} \theta ^{\,n-1}}}\left({\frac {\theta }{\sqrt[{3}]{\theta -\sin(\theta )}}}\right)^{n}\right).}" loading="lazy"></dd></dl>

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

Since a function is a special type of <a href="Binary_relation" title="Binary relation">binary relation</a>, many of the properties of an inverse function correspond to properties of <a href="Converse_relation" title="Converse relation">converse relations</a>.

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

If an inverse function exists for a given function f, then it is unique.<a href="#cite_note-Wolf72-14">[13]</a> This follows since the inverse function must be the converse relation, which is completely determined by f.

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

There is a symmetry between a function and its inverse. Specifically, if f is an invertible function with domain X and codomain Y, then its inverse f −1 has domain Y and image X, and the inverse of f −1 is the original function f. In symbols, for functions f:X → Y and f−1:Y → X,<a href="#cite_note-Wolf72-14">[13]</a>

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

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</math><img src="./dc32e927e523d18d2c49b6ad29a6be645bad0436.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.797ex; height:3.009ex;" alt="{\displaystyle f^{-1}\circ f=\operatorname {id} _{X}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\circ f^{-1}=\operatorname {id} _{Y}.}">

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

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</math><img src="./4a955bf47a6543870712b482a1af85b5deb98142.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.297ex; height:3.009ex;" alt="{\displaystyle f\circ f^{-1}=\operatorname {id} _{Y}.}" loading="lazy"></dd></dl>

This statement is a consequence of the implication that for f to be invertible it must be bijective. The <a href="Involution_(mathematics)" title="Involution (mathematics)">involutory</a> nature of the inverse can be concisely expressed by<a href="#cite_note-15">[14]</a>

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

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</math><img src="./176400473f9e8b87b91180839e5a7f4fbff08fb6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.14ex; height:3.843ex;" alt="{\displaystyle \left(f^{-1}\right)^{-1}=f.}" loading="lazy"></dd></dl>

The inverse of a composition of functions is given by<a href="#cite_note-16">[15]</a>

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

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</math><img src="./1e07d77c80674d7dffba861da07b7813adbf10bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.776ex; height:3.176ex;" alt="{\displaystyle (g\circ f)^{-1}=f^{-1}\circ g^{-1}.}" loading="lazy"></dd></dl>

Notice that the order of g and f have been reversed; to undo f followed by g, we must first undo g, and then undo f.

For example, let f(x) = 3x and let g(x) = x + 5. Then the composition g ∘ f is the function that first multiplies by three and then adds five,

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<annotation encoding="application/x-tex">{\displaystyle (g\circ f)(x)=3x+5.}</annotation>

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</math><img src="./8fe48829eae122101a9178edd25d805a2ebe35db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.778ex; height:2.843ex;" alt="{\displaystyle (g\circ f)(x)=3x+5.}" loading="lazy"></dd></dl>

To reverse this process, we must first subtract five, and then divide by three,

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<annotation encoding="application/x-tex">{\displaystyle (g\circ f)^{-1}(x)={\tfrac {1}{3}}(x-5).}</annotation>

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</math><img src="./f8fd51df789f5a167d0e7f5aeefb92f264ac3d46.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:24.416ex; height:3.676ex;" alt="{\displaystyle (g\circ f)^{-1}(x)={\tfrac {1}{3}}(x-5).}" loading="lazy"></dd></dl>

This is the composition

(f −1 ∘ g −1)(x).

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

If X is a set, then the <a href="Identity_function" title="Identity function">identity function</a> on X is its own inverse:

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<annotation encoding="application/x-tex">{\displaystyle {\operatorname {id} _{X}}^{-1}=\operatorname {id} _{X}.}</annotation>

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</math><img src="./860fb56df8d0c1e23aecf97d2e40fa276b64d1b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.221ex; height:3.009ex;" alt="{\displaystyle {\operatorname {id} _{X}}^{-1}=\operatorname {id} _{X}.}" loading="lazy"></dd></dl>

More generally, a function f : X → X is equal to its own inverse, if and only if the composition f ∘ f is equal to idX. Such a function is called an <a href="Involution_(mathematics)" title="Involution (mathematics)">involution</a>.

<div class="mw-heading mw-heading3"><h3 id="Graph_of_the_inverse">Graph of the inverse</h3></div>

If f is invertible, then the graph of the function

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

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</math><img src="./915a5e50073dee92572eb12065ae0324abbeb200.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 y=f^{-1}(x)}" loading="lazy"></dd></dl>

is the same as the graph of the equation

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

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</math><img src="./e70c5ecbfa32ede7a43eb3eca01bbe0fc0fef84f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.318ex; height:2.843ex;" alt="{\displaystyle x=f(y).}" loading="lazy"></dd></dl>

This is identical to the equation y = f(x) that defines the graph of f, except that the roles of x and y have been reversed. Thus the graph of f −1 can be obtained from the graph of f by switching the positions of the x and y axes. This is equivalent to <a href="Reflection_(mathematics)" title="Reflection (mathematics)">reflecting</a> the graph across the line

y = x.<a href="#cite_note-17">[16]</a><a href="#cite_note-:2-1">[1]</a>

<div class="mw-heading mw-heading3"><h3 id="Inverses_and_derivatives">Inverses and derivatives</h3></div>

By the <a href="Inverse_function_theorem" title="Inverse function theorem">inverse function theorem</a>, a <a href="Continuous_function" title="Continuous function">continuous function</a> of a single variable <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon A\to \mathbb {R} }">

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

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</math><img src="./e825429becf7730acee7e5a9da089c6b98ca759d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.348ex; height:2.509ex;" alt="{\displaystyle f\colon A\to \mathbb {R} }" loading="lazy"> (where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\subseteq \mathbb {R} }">

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

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</math><img src="./e286ff36e0ad1e0e1e70be50b1cbcfe232b94867.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.52ex; height:2.343ex;" alt="{\displaystyle A\subseteq \mathbb {R} }" loading="lazy">) is invertible on its range (image) if and only if it is either strictly <a href="Monotonic_function" title="Monotonic function">increasing or decreasing</a> (with no local <a href="Maxima_and_minima" class="mw-redirect" title="Maxima and minima">maxima or minima</a>). For example, the function

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

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</math><img src="./5700617329ea47740d6a6d15965ddc8088ececc4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.07ex; height:3.176ex;" alt="{\displaystyle f(x)=x^{3}+x}" loading="lazy"></dd></dl>

is invertible, since the <a href="Derivative" title="Derivative">derivative</a>

f′(x) = 3x2 + 1 is always positive.

If the function f is <a href="Differentiable_function" title="Differentiable function">differentiable</a> on an interval I and f′(x) ≠ 0 for each x ∈ I, then the inverse f −1 is differentiable on f(I).<a href="#cite_note-18">[17]</a> If y = f(x), the derivative of the inverse is given by the inverse function theorem,

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

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</math><img src="./abdf8c72eb45cf5f37d747d02389f2978a7afc77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:19.545ex; height:6.009ex;" alt="{\displaystyle \left(f^{-1}\right)^{\prime }(y)={\frac {1}{f'\left(x\right)}}.}" loading="lazy"></dd></dl>

Using <a href="Leibniz's_notation" title="Leibniz's notation">Leibniz's notation</a> the formula above can be written as

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<annotation encoding="application/x-tex">{\displaystyle {\frac {dx}{dy}}={\frac {1}{dy/dx}}.}</annotation>

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</math><img src="./5cb027b717f777d6028d963dc86342ba3a49f8f7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:14.043ex; height:6.176ex;" alt="{\displaystyle {\frac {dx}{dy}}={\frac {1}{dy/dx}}.}" loading="lazy"></dd></dl>

This result follows from the <a href="Chain_rule" title="Chain rule">chain rule</a> (see the article on <a href="Inverse_functions_and_differentiation" class="mw-redirect" title="Inverse functions and differentiation">inverse functions and differentiation</a>).

The inverse function theorem can be generalized to functions of several variables. Specifically, a continuously differentiable <a href="Real_multivariable_function" class="mw-redirect" title="Real multivariable function">multivariable function</a> f : Rn → Rn is invertible in a neighborhood of a point p as long as the <a href="Jacobian_matrix_and_determinant" title="Jacobian matrix and determinant">Jacobian matrix</a> of f at p is <a href="Invertible_matrix" title="Invertible matrix">invertible</a>. In this case, the Jacobian of f −1 at f(p) is the <a href="Matrix_inverse" class="mw-redirect" title="Matrix inverse">matrix inverse</a> of the Jacobian of f at p.

<div class="mw-heading mw-heading2"><h2 id="Real-world_examples">Real-world examples</h2></div>

<ul><li>Let f be the function that converts a temperature in degrees <a href="Celsius" title="Celsius">Celsius</a> to a temperature in degrees <a href="Fahrenheit" title="Fahrenheit">Fahrenheit</a>, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F=f(C)={\tfrac {9}{5}}C+32;}">

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<annotation encoding="application/x-tex">{\displaystyle F=f(C)={\tfrac {9}{5}}C+32;}</annotation>

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</math> then its inverse function converts degrees Fahrenheit to degrees Celsius, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C=f^{-1}(F)={\tfrac {5}{9}}(F-32),}">

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<annotation encoding="application/x-tex">{\displaystyle C=f^{-1}(F)={\tfrac {5}{9}}(F-32),}</annotation>

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</math><a href="#cite_note-:1-19">[18]</a> since <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}f^{-1}(f(C))={}&f^{-1}\left({\tfrac {9}{5}}C+32\right)={\tfrac {5}{9}}\left(({\tfrac {9}{5}}C+32)-32\right)=C,\\&{\text{for every value of }}C,{\text{ and }}\\[6pt]f\left(f^{-1}(F)\right)={}&f\left({\tfrac {5}{9}}(F-32)\right)={\tfrac {9}{5}}\left({\tfrac {5}{9}}(F-32)\right)+32=F,\\&{\text{for every value of }}F.\end{aligned}}}">

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<mtext>for every value of </mtext>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}f^{-1}(f(C))={}&f^{-1}\left({\tfrac {9}{5}}C+32\right)={\tfrac {5}{9}}\left(({\tfrac {9}{5}}C+32)-32\right)=C,\\&{\text{for every value of }}C,{\text{ and }}\\[6pt]f\left(f^{-1}(F)\right)={}&f\left({\tfrac {5}{9}}(F-32)\right)={\tfrac {9}{5}}\left({\tfrac {5}{9}}(F-32)\right)+32=F,\\&{\text{for every value of }}F.\end{aligned}}}</annotation>

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<li>Suppose f assigns each child in a family its birth year. An inverse function would output which child was born in a given year. However, if the family has children born in the same year (for instance, twins or triplets, etc.) then the output cannot be known when the input is the common birth year. As well, if a year is given in which no child was born then a child cannot be named. But if each child was born in a separate year, and if we restrict attention to the three years in which a child was born, then we do have an inverse function. For example, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}f({\text{Allan}})&=2005,\quad &f({\text{Brad}})&=2007,\quad &f({\text{Cary}})&=2001\\f^{-1}(2005)&={\text{Allan}},\quad &f^{-1}(2007)&={\text{Brad}},\quad &f^{-1}(2001)&={\text{Cary}}\end{aligned}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}f({\text{Allan}})&=2005,\quad &f({\text{Brad}})&=2007,\quad &f({\text{Cary}})&=2001\\f^{-1}(2005)&={\text{Allan}},\quad &f^{-1}(2007)&={\text{Brad}},\quad &f^{-1}(2001)&={\text{Cary}}\end{aligned}}}</annotation>

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<li>Let R be the function that leads to an x percentage rise of some quantity, and F be the function producing an x percentage fall. Applied to $100 with x = 10%, we find that applying the first function followed by the second does not restore the original value of $100, demonstrating the fact that, despite appearances, these two functions are not inverses of each other.</li>

<li>The formula to calculate the pH of a solution is pH = −log10[H+]. In many cases we need to find the concentration of acid from a pH measurement. The inverse function [H+] = 10−pH is used.</li></ul>

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

<div class="mw-heading mw-heading3"><h3 id="Partial_inverses">Partial inverses</h3></div>

Even if a function f is not one-to-one, it may be possible to define a partial inverse of f by <a href="Function_(mathematics)#Restrictions_and_extensions" title="Function (mathematics)">restricting</a> the domain. For example, the function

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

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is not one-to-one, since x2 = (−x)2. However, the function becomes one-to-one if we restrict to the domain x ≥ 0, in which case

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(If we instead restrict to the domain x ≤ 0, then the inverse is the negative of the square root of y.)

<div class="mw-heading mw-heading3"><h3 id="Full_inverses">Full inverses</h3></div>

Alternatively, there is no need to restrict the domain if we are content with the inverse being a <a href="Multivalued_function" title="Multivalued function">multivalued function</a>:

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Sometimes, this multivalued inverse is called the full inverse of f, and the portions (such as √x and −√x) are called branches. The most important branch of a multivalued function (e.g. the positive square root) is called the <a href="Principal_branch" title="Principal branch">principal branch</a>, and its value at y is called the principal value of f −1(y).

For a continuous function on the real line, one branch is required between each pair of <a href="Minima_and_maxima" class="mw-redirect" title="Minima and maxima">local extrema</a>. For example, the inverse of a <a href="Cubic_function" title="Cubic function">cubic function</a> with a local maximum and a local minimum has three branches (see the adjacent picture).

<div class="mw-heading mw-heading3"><h3 id="Trigonometric_inverses">Trigonometric inverses</h3></div>

The above considerations are particularly important for defining the inverses of <a href="Trigonometric_functions" title="Trigonometric functions">trigonometric functions</a>. For example, the <a href="Sine_function" class="mw-redirect" title="Sine function">sine function</a> is not one-to-one, since

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<annotation encoding="application/x-tex">{\displaystyle \sin(x+2\pi )=\sin(x)}</annotation>

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</math><img src="./a5e286836d2acae12c3b9fe6dac144cc438d3a7f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.423ex; height:2.843ex;" alt="{\displaystyle \sin(x+2\pi )=\sin(x)}" loading="lazy"></dd></dl>

for every real x (and more generally sin(x + 2πn) = sin(x) for every <a href="Integer" title="Integer">integer</a> n). However, the sine is one-to-one on the interval

[−⁠π/2⁠, ⁠π/2⁠], and the corresponding partial inverse is called the <a href="Arcsine" class="mw-redirect" title="Arcsine">arcsine</a>. This is considered the principal branch of the inverse sine, so the principal value of the inverse sine is always between −⁠π/2⁠ and ⁠π/2⁠. The following table describes the principal branch of each inverse trigonometric function:<a href="#cite_note-20">[19]</a>

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</th>

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</th></tr>

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<td>arcsin</td>

<td>−⁠π/2⁠ ≤ sin−1(x) ≤ ⁠π/2⁠

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<tr>

<td>arccos</td>

<td>0 ≤ cos−1(x) ≤ π

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<tr>

<td>arctan</td>

<td>−⁠π/2⁠ < tan−1(x) < ⁠π/2⁠

</td></tr>

<tr>

<td>arccot</td>

<td>0 < cot−1(x) < π

</td></tr>

<tr>

<td>arcsec</td>

<td>0 ≤ sec−1(x) ≤ π

</td></tr>

<tr>

<td>arccsc</td>

<td>−⁠π/2⁠ ≤ csc−1(x) ≤ ⁠π/2⁠

</td></tr>

</tbody></table>

<div class="mw-heading mw-heading3"><h3 id="Left_and_right_inverses">Left and right inverses</h3></div>

<a href="Function_composition" title="Function composition">Function composition</a> on the left and on the right need not coincide. In general, the conditions

<ol><li>"There exists g such that g(f(x))=x" and</li>

<li>"There exists g such that f(g(x))=x"</li></ol>

imply different properties of f. For example, let f: R → [0, ∞) denote the squaring map, such that f(x) = x2 for all x in R, and let g: [0, ∞) → R denote the square root map, such that g(x) = √x for all x ≥ 0. Then f(g(x)) = x for all x in [0, ∞); that is, g is a right inverse to f. However, g is not a left inverse to f, since, e.g., g(f(−1)) = 1 ≠ −1.

<div class="mw-heading mw-heading4"><h4 id="Left_inverses">Left inverses</h4></div>

If f: X → Y, a left inverse for f (or <a href="Retract_(category_theory)" class="mw-redirect" title="Retract (category theory)">retraction</a> of f ) is a function g: Y → X such that composing f with g from the left gives the identity function<a href="#cite_note-21">[20]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\circ f=\operatorname {id} _{X}{\text{.}}}">

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math> That is, the function g satisfies the rule

The function g must equal the inverse of f on the image of f, but may take any values for elements of Y not in the image.

A function f with nonempty domain is injective if and only if it has a left inverse.<a href="#cite_note-22">[21]</a> An elementary proof runs as follows:

<ul><li>If g is the left inverse of f, and f(x) = f(y), then g(f(x)) = g(f(y)) = x = y.</li>

<li>If nonempty f: X → Y is injective, construct a left inverse g: Y → X as follows: for all y ∈ Y, if y is in the image of f, then there exists x ∈ X such that f(x) = y. Let g(y) = x; this definition is unique because f is injective. Otherwise, let g(y) be an arbitrary element of X.For all x ∈ X, f(x) is in the image of f. By construction, g(f(x)) = x, the condition for a left inverse.</li></ul>

In classical mathematics, every injective function f with a nonempty domain necessarily has a left inverse; however, this may fail in <a href="Constructive_mathematics" class="mw-redirect" title="Constructive mathematics">constructive mathematics</a>. For instance, a left inverse of the <a href="Inclusion_map" title="Inclusion map">inclusion</a> {0,1} → R of the two-element set in the reals violates <a href="Indecomposability_(constructive_mathematics)" class="mw-redirect" title="Indecomposability (constructive mathematics)">indecomposability</a> by giving a <a href="Retract_(category_theory)" class="mw-redirect" title="Retract (category theory)">retraction</a> of the real line to the set {0,1}.<a href="#cite_note-23">[22]</a>

<div class="mw-heading mw-heading4"><h4 id="Right_inverses">Right inverses</h4></div>

A right inverse for f (or <a href="Section_(category_theory)" title="Section (category theory)">section</a> of f ) is a function h: Y → X such that

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./fec35e5cc6885c4ec7493267cc24ee4606a77cee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.983ex; height:2.509ex;" alt="{\displaystyle f\circ h=\operatorname {id} _{Y}.}" loading="lazy"></dd></dl>

That is, the function h satisfies the rule

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f0f158a9f64615ffaecb0a03c2b1f87c90701dbc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.732ex; height:2.843ex;" alt="{\displaystyle \displaystyle h(y)=x}" loading="lazy">, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \displaystyle f(x)=y.}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./e8e7ed08295dd7a7f584cb88f5e4ccb549d32155.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.318ex; height:2.843ex;" alt="{\displaystyle \displaystyle f(x)=y.}" loading="lazy"></dd></dl>

Thus, h(y) may be any of the elements of X that map to y under f.

A function f has a right inverse if and only if it is <a href="Surjective_function" title="Surjective function">surjective</a> (though constructing such an inverse in general requires the <a href="Axiom_of_choice" title="Axiom of choice">axiom of choice</a>).

<dl><dd>If h is the right inverse of f, then f is surjective. For all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\in Y}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./cee1c0ec36a82f33f5e3d7434d5667881b4ec323.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.769ex; height:2.509ex;" alt="{\displaystyle y\in Y}" loading="lazy">, there is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=h(y)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./04605d887de339e440c9b6a6026f3bc8c53d70ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.732ex; height:2.843ex;" alt="{\displaystyle x=h(y)}" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=f(h(y))=y}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./c74a6a13e2c6834606d0487cf3a5d61ad5cafa42.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.162ex; height:2.843ex;" alt="{\displaystyle f(x)=f(h(y))=y}" loading="lazy">.</dd>

<dd>If f is surjective, f has a right inverse h, which can be constructed as follows: for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\in Y}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./cee1c0ec36a82f33f5e3d7434d5667881b4ec323.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.769ex; height:2.509ex;" alt="{\displaystyle y\in Y}" loading="lazy">, there is at least one <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./3e580967f68f36743e894aa7944f032dda6ea01d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}" loading="lazy"> such that <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="./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"> (because f is surjective), so we choose one to be the value of h(y).<a href="#cite_note-24">[23]</a></dd></dl>

<div class="mw-heading mw-heading4"><h4 id="Two-sided_inverses">Two-sided inverses</h4></div>

An inverse that is both a left and right inverse (a two-sided inverse), if it exists, must be unique. In fact, if a function has a left inverse and a right inverse, they are both the same two-sided inverse, so it can be called the inverse.

</mstyle>

</mrow>

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

</semantics>

</math><img src="./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"> is a left inverse and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./b26be3e694314bc90c3215047e4a2010c6ee184a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}" loading="lazy"> a right inverse of <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="./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">, for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\in Y}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./cee1c0ec36a82f33f5e3d7434d5667881b4ec323.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.769ex; height:2.509ex;" alt="{\displaystyle y\in Y}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(y)=g(f(h(y))=h(y)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./79ce4417038ceee9354b1a7ea99a9498578fa086.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.994ex; height:2.843ex;" alt="{\displaystyle g(y)=g(f(h(y))=h(y)}" loading="lazy">.</dd></dl>

A function has a two-sided inverse if and only if it is bijective.

<dl><dd>A bijective function f is injective, so it has a left inverse (if f is the empty function, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon \varnothing \to \varnothing }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./4414d3f0d1bb64f5e32ba9552a0df8f3f9dc5024.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.543ex; height:2.509ex;" alt="{\displaystyle f\colon \varnothing \to \varnothing }" loading="lazy"> is its own left inverse). f is surjective, so it has a right inverse. By the above, the left and right inverse are the same.</dd>

<dd>If f has a two-sided inverse g, then g is a left inverse and right inverse of f, so f is injective and surjective.</dd></dl>

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

If f: X → Y is any function (not necessarily invertible), the preimage (or inverse image) of an element y ∈ Y is defined to be the set of all elements of X that map to y:

<msup>

</mrow>

</msup>

<mrow>

<mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./dc382af754a32000c914c3ee03d2162d966ca68a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.834ex; height:3.176ex;" alt="{\displaystyle f^{-1}(y)=\left\{x\in X:f(x)=y\right\}.}" loading="lazy"></dd></dl>

The preimage of y can be thought of as the <a href="Image_(mathematics)" title="Image (mathematics)">image</a> of y under the (multivalued) full inverse of the function f.

The notion can be generalized to subsets of the range. Specifically, if S is any <a href="Subset" title="Subset">subset</a> of Y, the preimage of S, denoted by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}(S)}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./c994fcc06dde13bed6ddd5c653fdd17f51bbaecb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.962ex; height:3.176ex;" alt="{\displaystyle f^{-1}(S)}" loading="lazy">, is the set of all elements of X that map to S:

<msup>

</mrow>

</msup>

<mrow>

<mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./a5f30edd275f0470fe7f86e9a41f8c5afd4b538b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.264ex; height:3.176ex;" alt="{\displaystyle f^{-1}(S)=\left\{x\in X:f(x)\in S\right\}.}" loading="lazy"></dd></dl>

For example, take the function f: R → R; x ↦ x2. This function is not invertible as it is not bijective, but preimages may be defined for subsets of the codomain, e.g.

<msup>

</mrow>

</msup>

<mrow>

<mrow>

</mrow>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{-1}(\left\{1,4,9,16\right\})=\left\{-4,-3,-2,-1,1,2,3,4\right\}}</annotation>

</semantics>

</math><img src="./ca5dcc32044757ef73de33c14e8862692dcc78e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:45.895ex; height:3.176ex;" alt="{\displaystyle f^{-1}(\left\{1,4,9,16\right\})=\left\{-4,-3,-2,-1,1,2,3,4\right\}}" loading="lazy">.</dd></dl>

The original notion and its generalization are related by the identity <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}(y)=f^{-1}(\{y\}),}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./aab0071347d4450a3b9bf408ab312fd0ec61d626.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.306ex; height:3.176ex;" alt="{\displaystyle f^{-1}(y)=f^{-1}(\{y\}),}" loading="lazy"> The preimage of a single element y ∈ Y – a <a href="Singleton_set" class="mw-redirect" title="Singleton set">singleton set</a> {y} – is sometimes called the <a href="Fiber_(mathematics)" title="Fiber (mathematics)">fiber</a> of y. When Y is the set of real numbers, it is common to refer to f −1({y}) as a <a href="Level_set" title="Level set">level set</a>.

<ul><li><a href="Lagrange_inversion_theorem" title="Lagrange inversion theorem">Lagrange inversion theorem</a>, gives the Taylor series expansion of the inverse function of an analytic function</li>

<li><a href="Integral_of_inverse_functions" title="Integral of inverse functions">Integral of inverse functions</a></li>

<li><a href="Inverse_Fourier_transform" class="mw-redirect" title="Inverse Fourier transform">Inverse Fourier transform</a></li>

<li><a href="Reversible_computing" title="Reversible computing">Reversible computing</a></li></ul>

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

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<li id="cite_note-NB2-7"><a href="#cite_ref-NB2_7-0">^</a> Not to be confused with numerical exponentiation such as taking the multiplicative inverse of a nonzero real number.

</li>

</ol></div></div>

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

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</style><cite id="CITEREFWeisstein" class="citation web cs1">Weisstein, Eric W. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/InverseFunction.html">"Inverse Function"</a>. mathworld.wolfram.com. Retrieved 2020-09-08.</cite>

</li>

<li id="cite_note-Herschel_1813-2"><a href="#cite_ref-Herschel_1813_2-0">^</a> <cite id="CITEREFHerschel1813" class="citation journal cs1"><a href="John_Frederick_William_Herschel" class="mw-redirect" title="John Frederick William Herschel">Herschel, John Frederick William</a> (1813) [1812-11-12]. <a rel="nofollow" class="external text" href="https://doi.org/10.1098%2Frstl.1813.0005">"On a Remarkable Application of Cotes's Theorem"</a>. <a href="Philosophical_Transactions_of_the_Royal_Society_of_London" class="mw-redirect" title="Philosophical Transactions of the Royal Society of London">Philosophical Transactions of the Royal Society of London</a>. 103 (Part 1). London: <a href="Royal_Society_of_London" class="mw-redirect" title="Royal Society of London">Royal Society of London</a>, printed by W. Bulmer and Co., Cleveland-Row, St. James's, sold by G. and W. Nicol, Pall-Mall: 8–26 [10]. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1098%2Frstl.1813.0005">10.1098/rstl.1813.0005</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/107384">107384</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:118124706">118124706</a>.</cite>

</li>

<li id="cite_note-Herschel_1820-3"><a href="#cite_ref-Herschel_1820_3-0">^</a> <cite id="CITEREFHerschel1820" class="citation book cs1"><a href="John_Frederick_William_Herschel" class="mw-redirect" title="John Frederick William Herschel">Herschel, John Frederick William</a> (1820). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=PWcSAAAAIAAJ&pg=PA5">"Part III. Section I. Examples of the Direct Method of Differences"</a>. A Collection of Examples of the Applications of the Calculus of Finite Differences. Cambridge, UK: Printed by J. Smith, sold by J. Deighton & sons. pp. 1–13 [5–6]. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20200804031020/https://books.google.de/books?hl=de&id=PWcSAAAAIAAJ&jtp=5">Archived</a> from the original on 2020-08-04. Retrieved 2020-08-04.</cite> <a rel="nofollow" class="external autonumber" href="https://archive.org/details/acollectionexam00lacrgoog">[1]</a> (NB. Inhere, Herschel refers to his <a href="#CITEREFHerschel1813">1813 work</a> and mentions <a href="Hans_Heinrich_B%C3%BCrmann" title="Hans Heinrich Bürmann">Hans Heinrich Bürmann</a>'s older work.)

</li>

<li id="cite_note-Peirce_1852-4"><a href="#cite_ref-Peirce_1852_4-0">^</a> <cite id="CITEREFPeirce1852" class="citation book cs1"><a href="Benjamin_Peirce" title="Benjamin Peirce">Peirce, Benjamin</a> (1852). Curves, Functions and Forces. Vol. I (new ed.). Boston, USA. p. 203.</cite><code class="cs1-code">{{cite book}}</code>: CS1 maint: location missing publisher (link)

</li>

<li id="cite_note-Peano_1903-5"><a href="#cite_ref-Peano_1903_5-0">^</a> <cite id="CITEREFPeano1903" class="citation book cs1 cs1-prop-foreign-lang-source"><a href="Giuseppe_Peano" title="Giuseppe Peano">Peano, Giuseppe</a> (1903). Formulaire mathématique (in French). Vol. IV. p. 229.</cite>

</li>

<li id="cite_note-Cajori_1929-6">^ <a href="#cite_ref-Cajori_1929_6-0">a</a> <a href="#cite_ref-Cajori_1929_6-1">b</a> <a href="#cite_ref-Cajori_1929_6-2">c</a> <a href="#cite_ref-Cajori_1929_6-3">d</a> <cite id="CITEREFCajori1952" class="citation book cs1"><a href="Florian_Cajori" title="Florian Cajori">Cajori, Florian</a> (1952) [March 1929]. "§472. The power of a logarithm / §473. Iterated logarithms / §533. John Herschel's notation for inverse functions / §535. Persistence of rival notations for inverse functions / §537. Powers of trigonometric functions". <a rel="nofollow" class="external text" href="https://books.google.com/books?id=bT5suOONXlgC">A History of Mathematical Notations</a>. Vol. 2 (3rd corrected printing of 1929 issue, 2nd ed.). Chicago, USA: <a href="Open_court_publishing_company" class="mw-redirect" title="Open court publishing company">Open court publishing company</a>. pp. 108, 176–179, 336, 346. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-60206-714-1</bdi>. Retrieved 2016-01-18. <q>[...] §473. Iterated logarithms [...] We note here the symbolism used by <a href="Alfred_Pringsheim" title="Alfred Pringsheim">Pringsheim</a> and <a href="Jules_Molk" title="Jules Molk">Molk</a> in their joint Encyclopédie article: "2logb a = logb (logb a), ..., k+1logb a = logb (klogb a)." [...] §533. <a href="John_Frederick_William_Herschel" class="mw-redirect" title="John Frederick William Herschel">John Herschel</a>'s notation for inverse functions, sin−1 x, tan−1 x, etc., was published by him in the <a href="Philosophical_Transactions_of_London" class="mw-redirect" title="Philosophical Transactions of London">Philosophical Transactions of London</a>, for the year 1813. He says (<a href="#CITEREFHerschel1813">p. 10</a>): "This notation cos.−1 e must not be understood to signify 1/cos. e, but what is usually written thus, arc (cos.=e)." He admits that some authors use cos.m A for (cos. A)m, but he justifies his own notation by pointing out that since d2 x, Δ3 x, Σ2 x mean dd x, ΔΔΔ x, ΣΣ x, we ought to write sin.2 x for sin. sin. x, log.3 x for log. log. log. x. Just as we write d−n V=∫n V, we may write similarly sin.−1 x=arc (sin.=x), log.−1 x.=cx. Some years later Herschel explained that in 1813 he used fn(x), f−n(x), sin.−1 x, etc., "as he then supposed for the first time. The work of a German Analyst, <a href="Hans_Heinrich_B%C3%BCrmann" title="Hans Heinrich Bürmann">Burmann</a>, has, however, within these few months come to his knowledge, in which the same is explained at a considerably earlier date. He[Burmann], however, does not seem to have noticed the convenience of applying this idea to the inverse functions tan−1, etc., nor does he appear at all aware of the inverse calculus of functions to which it gives rise." Herschel adds, "The symmetry of this notation and above all the new and most extensive views it opens of the nature of analytical operations seem to authorize its universal adoption."<a href="#CITEREFHerschel1820">[a]</a> [...] §535. Persistence of rival notations for inverse function.— [...] The use of Herschel's notation underwent a slight change in <a href="Benjamin_Peirce" title="Benjamin Peirce">Benjamin Peirce</a>'s books, to remove the chief objection to them; Peirce wrote: "cos[−1] x," "log[−1] x."<a href="#CITEREFPeirce1852">[b]</a> [...] §537. Powers of trigonometric functions.—Three principal notations have been used to denote, say, the square of sin x, namely, (sin x)2, sin x2, sin2 x. The prevailing notation at present is sin2 x, though the first is least likely to be misinterpreted. In the case of sin2 x two interpretations suggest themselves; first, sin x · sin x; second,<a href="#CITEREFPeano1903">[c]</a> sin (sin x). As functions of the last type do not ordinarily present themselves, the danger of misinterpretation is very much less than in case of log2 x, where log x · log x and log (log x) are of frequent occurrence in analysis. [...] The notation sinn x for (sin x)n has been widely used and is now the prevailing one. [...]</q></cite> <code class="cs1-code">{{cite book}}</code>: ISBN / Date incompatibility (help) (xviii+367+1 pages including 1 addenda page) (NB. ISBN and link for reprint of 2nd edition by Cosimo, Inc., New York, USA, 2013.)

</li>

<li id="cite_note-8"><a href="#cite_ref-8">^</a> Helmut Sieber und Leopold Huber: Mathematische Begriffe und Formeln für Sekundarstufe I und II der Gymnasien. Ernst Klett Verlag.

</li>

<li id="cite_note-9"><a href="#cite_ref-9">^</a> <a href="#CITEREFThomas1972">Thomas 1972</a>, pp. 304–309

</li>

<li id="cite_note-Korn_2000-10">^ <a href="#cite_ref-Korn_2000_10-0">a</a> <a href="#cite_ref-Korn_2000_10-1">b</a> <cite id="CITEREFKornKorn2000" class="citation book cs1">Korn, Grandino Arthur; <a href="Theresa_M._Korn" title="Theresa M. Korn">Korn, Theresa M.</a> (2000) [1961]. "21.2.-4. Inverse Trigonometric Functions". <a rel="nofollow" class="external text" href="https://archive.org/details/mathematicalhand00korn_849">Mathematical handbook for scientists and engineers: Definitions, theorems, and formulars for reference and review</a> (3 ed.). Mineola, New York, USA: <a href="Dover_Publications%2C_Inc." class="mw-redirect" title="Dover Publications, Inc.">Dover Publications, Inc.</a> p. <a rel="nofollow" class="external text" href="https://archive.org/details/mathematicalhand00korn_849/page/n828">811</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-486-41147-7</bdi>.</cite>

</li>

<li id="cite_note-Atlas_2009-11">^ <a href="#cite_ref-Atlas_2009_11-0">a</a> <a href="#cite_ref-Atlas_2009_11-1">b</a> <a href="#cite_ref-Atlas_2009_11-2">c</a> <a href="#cite_ref-Atlas_2009_11-3">d</a> <a href="#cite_ref-Atlas_2009_11-4">e</a> <cite id="CITEREFOldhamMylandSpanier2009" class="citation book cs1">Oldham, Keith B.; Myland, Jan C.; Spanier, Jerome (2009) [1987]. An Atlas of Functions: with Equator, the Atlas Function Calculator (2 ed.). <a href="Springer_Science%2BBusiness_Media%2C_LLC" class="mw-redirect" title="Springer Science+Business Media, LLC">Springer Science+Business Media, LLC</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-0-387-48807-3">10.1007/978-0-387-48807-3</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-48806-6</bdi>. <a href="LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <a rel="nofollow" class="external text" href="https://lccn.loc.gov/2008937525">2008937525</a>.</cite>

</li>

<li id="cite_note-Hall_1909-12"><a href="#cite_ref-Hall_1909_12-0">^</a> <cite id="CITEREFHallFrink1909" class="citation book cs1 cs1-prop-location-test">Hall, Arthur Graham; Frink, Fred Goodrich (1909). <a rel="nofollow" class="external text" href="https://archive.org/stream/planetrigonometr00hallrich#page/n30/mode/1up">"Article 14: Inverse trigonometric functions"</a>. Written at Ann Arbor, Michigan, USA. Plane Trigonometry. New York: <a href="Henry_Holt_%26_Company" class="mw-redirect" title="Henry Holt & Company">Henry Holt & Company</a>. pp. 15–16. Retrieved 2017-08-12. <q>α = arcsin m This notation is universally used in Europe and is fast gaining ground in this country. A less desirable symbol, α = sin-1 m, is still found in English and American texts. The notation α = inv sin m is perhaps better still on account of its general applicability. [...] A similar symbolic relation holds for the other <a href="Trigonometric_function" class="mw-redirect" title="Trigonometric function">trigonometric functions</a>. It is frequently read 'arc-sine m' or 'anti-sine m', since two mutually inverse functions are said each to be the anti-function of the other.</q></cite>

</li>

<li id="cite_note-13"><a href="#cite_ref-13">^</a> <a href="#CITEREFLay2006">Lay 2006</a>, p. 69, Example 7.24

</li>

<li id="cite_note-Wolf72-14">^ <a href="#cite_ref-Wolf72_14-0">a</a> <a href="#cite_ref-Wolf72_14-1">b</a> <a href="#CITEREFWolf1998">Wolf 1998</a>, p. 208, Theorem 7.2

</li>

<li id="cite_note-15"><a href="#cite_ref-15">^</a> <a href="#CITEREFSmithEggenSt._Andre2006">Smith, Eggen & St. Andre 2006</a>, pg. 141 Theorem 3.3(a)

</li>

<li id="cite_note-16"><a href="#cite_ref-16">^</a> <a href="#CITEREFLay2006">Lay 2006</a>, p. 71, Theorem 7.26

</li>

<li id="cite_note-17"><a href="#cite_ref-17">^</a> <a href="#CITEREFBriggsCochran2011">Briggs & Cochran 2011</a>, pp. 28–29

</li>

<li id="cite_note-18"><a href="#cite_ref-18">^</a> <a href="#CITEREFLay2006">Lay 2006</a>, p. 246, Theorem 26.10

</li>

<li id="cite_note-:1-19"><a href="#cite_ref-:1_19-0">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.mathsisfun.com/sets/function-inverse.html">"Inverse Functions"</a>. www.mathsisfun.com. Retrieved 2020-09-08.</cite>

</li>

<li id="cite_note-20"><a href="#cite_ref-20">^</a> <a href="#CITEREFBriggsCochran2011">Briggs & Cochran 2011</a>, pp. 39–42

</li>

<li id="cite_note-21"><a href="#cite_ref-21">^</a> <cite id="CITEREFDummitFoote" class="citation book cs1">Dummit; Foote. Abstract Algebra.</cite>

</li>

<li id="cite_note-22"><a href="#cite_ref-22">^</a> <cite id="CITEREFMac_Lane" class="citation book cs1">Mac Lane, Saunders. Categories for the Working Mathematician.</cite>

</li>

<li id="cite_note-23"><a href="#cite_ref-23">^</a> <cite id="CITEREFFraenkel1954" class="citation journal cs1">Fraenkel (1954). <a rel="nofollow" class="external text" href="https://doi.org/10.1038%2F173967a0">"Abstract Set Theory"</a>. Nature. 173 (4412): 967. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1954Natur.173..967C">1954Natur.173..967C</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2F173967a0">10.1038/173967a0</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:7735523">7735523</a>.</cite>

</li>

<li id="cite_note-24"><a href="#cite_ref-24">^</a> <cite id="CITEREFLoehr2019" class="citation book cs1">Loehr, Nicholas A. (2019-11-20). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=mGUIEQAAQBAJ&pg=PA272">An Introduction to Mathematical Proofs</a>. CRC Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-000-70962-9</bdi>.</cite>

</li>

</ol></div></div>

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

<ul><li><cite id="CITEREFBriggsCochran2011" class="citation book cs1">Briggs, William; Cochran, Lyle (2011). <a rel="nofollow" class="external text" href="https://archive.org/details/calculusearlytra0000brig">Calculus / Early Transcendentals Single Variable</a>. <a href="Addison-Wesley" title="Addison-Wesley">Addison-Wesley</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-321-66414-3</bdi>.</cite></li>

<li><cite id="CITEREFDevlin2004" class="citation book cs1"><a href="Keith_J._Devlin" class="mw-redirect" title="Keith J. Devlin">Devlin, Keith J.</a> (2004). Sets, Functions, and Logic / An Introduction to Abstract Mathematics (3 ed.). <a href="Chapman_%26_Hall" title="Chapman & Hall">Chapman & Hall</a> / <a href="CRC_Mathematics" class="mw-redirect" title="CRC Mathematics">CRC Mathematics</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-58488-449-1</bdi>.</cite></li>

<li><cite id="CITEREFFletcherPatty1988" class="citation book cs1">Fletcher, Peter; Patty, C. Wayne (1988). Foundations of Higher Mathematics. PWS-Kent. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-87150-164-3</bdi>.</cite></li>

<li><cite id="CITEREFLay2006" class="citation book cs1">Lay, Steven R. (2006). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=k4k_AQAAIAAJ">Analysis / With an Introduction to Proof</a> (4 ed.). <a href="Pearson_(publisher)" class="mw-redirect" title="Pearson (publisher)">Pearson</a> / <a href="Prentice_Hall" title="Prentice Hall">Prentice Hall</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-13-148101-5</bdi>.</cite></li>

<li><cite id="CITEREFSmithEggenSt._Andre2006" class="citation book cs1">Smith, Douglas; Eggen, Maurice; St. Andre, Richard (2006). A Transition to Advanced Mathematics (6 ed.). <a href="Thompson_Brooks/Cole" class="mw-redirect" title="Thompson Brooks/Cole">Thompson Brooks/Cole</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-534-39900-9</bdi>.</cite></li>

<li><cite id="CITEREFThomas1972" class="citation book cs1">Thomas Jr., George Brinton (1972). Calculus and Analytic Geometry Part 1: Functions of One Variable and Analytic Geometry (Alternate ed.). <a href="Addison-Wesley" title="Addison-Wesley">Addison-Wesley</a>.</cite></li>

<li><cite id="CITEREFWolf1998" class="citation book cs1">Wolf, Robert S. (1998). Proof, Logic, and Conjecture / The Mathematician's Toolbox. <a href="W._H._Freeman_and_Co." class="mw-redirect" title="W. H. Freeman and Co.">W. H. Freeman and Co.</a> <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-7167-3050-7</bdi>.</cite></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

<ul><li><cite id="CITEREFAmazigoRubenfeld1980" class="citation book cs1">Amazigo, John C.; Rubenfeld, Lester A. (1980). "Implicit Functions; Jacobians; Inverse Functions". <a rel="nofollow" class="external text" href="https://archive.org/details/advancedcalculus0000amaz">Advanced Calculus and its Applications to the Engineering and Physical Sciences</a>. New York: Wiley. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/advancedcalculus0000amaz/page/n114">103</a>–120. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-471-04934-4</bdi>.</cite></li>

<li><cite id="CITEREFBinmore1983" class="citation book cs1"><a href="Ken_Binmore" class="mw-redirect" title="Ken Binmore">Binmore, Ken G.</a> (1983). "Inverse Functions". Calculus. New York: <a href="Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. pp. 161–197. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-521-28952-1</bdi>.</cite></li>

<li><cite id="CITEREFSpivak1994" class="citation book cs1">Spivak, Michael (1994). Calculus (3 ed.). Publish or Perish. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-914098-89-6</bdi>.</cite></li>

<li><cite id="CITEREFStewart2002" class="citation book cs1"><a href="James_Stewart_(mathematician)" title="James Stewart (mathematician)">Stewart, James</a> (2002). <a rel="nofollow" class="external text" href="https://archive.org/details/calculus0000stew">Calculus</a> (5 ed.). <a href="Brooks_Cole" class="mw-redirect" title="Brooks Cole">Brooks Cole</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-534-39339-7</bdi>.</cite></li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

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<ul><li><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Inverse_function">"Inverse function"</a>, <a href="Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a>, <a href="European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite></li></ul>

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