View source for Inverse function

{{short description|Mathematical concept}}
{{Distinguish|multiplicative inverse|additive inverse}}
{{use dmy dates|date=August 2020|cs1-dates=y}}
[[Image:Inverse Function.png|thumb|right|A function {{mvar|f}} and its inverse {{math|''f''<sup> −1</sup>}}. Because {{mvar|f}} maps {{mvar|a}} to 3, the inverse {{math|''f''<sup> −1</sup>}} maps 3 back to {{mvar|a}}.]]
{{Functions}}
In [[mathematics]], the '''inverse function''' of a [[Function (mathematics)|function]] {{Mvar|f}} (also called the '''inverse''' of {{Mvar|f}}) is a [[function (mathematics)|function]] that undoes the operation of {{Mvar|f}}. The inverse of {{Mvar|f}} exists [[if and only if]] {{Mvar|f}} is [[Bijection|bijective]], and if it exists, is denoted by <math>f^{-1} .</math>

For a function <math>f\colon X\to Y</math>, its inverse <math>f^{-1}\colon Y\to X</math> admits an explicit description:  it sends each element <math>y\in Y</math> to the unique element <math>x\in X</math> such that {{Math|1=''f''(''x'') = ''y''}}.

As an example, consider the [[Real number|real-valued]] function of a real variable given by {{math|1=''f''(''x'') = 5''x'' − 7}}. One can think of {{Mvar|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 {{Mvar|f}} is the function <math>f^{-1}\colon \R\to\R</math> defined by <math>f^{-1}(y) = \frac{y+7}{5} .</math>

==Definitions==
[[Image:Inverse Functions Domain and Range.png|thumb|right|240px|If {{mvar|f}} maps {{mvar|X}} to {{mvar|Y}}, then {{math|''f''<sup> −1</sup>}} maps {{mvar|Y}} back to {{mvar|X}}.]]

Let {{mvar|f}} be a function whose [[domain of a function|domain]] is the [[Set (mathematics)|set]] {{mvar|X}}, and whose [[codomain]] is the set {{mvar|Y}}. Then {{mvar|f}} is ''invertible'' if there exists a function {{mvar|g}} from {{mvar|Y}} to {{mvar|X}} such that <math>g(f(x))=x</math> for all <math>x\in X</math> and <math>f(g(y))=y</math> for all <math>y\in Y</math>.<ref name=":2">{{Cite web|last=Weisstein|first=Eric W.|title=Inverse Function|url=https://mathworld.wolfram.com/InverseFunction.html|access-date=2020-09-08|website=mathworld.wolfram.com|language=en}}</ref>

If {{mvar|f}} is invertible, then there is exactly one function {{mvar|g}} satisfying this property. The function {{mvar|g}} is called the inverse of {{mvar|f}}, and is usually denoted as {{math|''f''<sup> −1</sup>}}, a notation introduced by [[John Frederick William Herschel]] in 1813.<ref name="Herschel_1813"/><ref name="Herschel_1820"/><ref name="Peirce_1852"/><ref name="Peano_1903"/><ref name="Cajori_1929"/><ref group="nb" name="NB2"/>

The function {{mvar|f}} is invertible if and only if it is bijective. This is because the condition <math>g(f(x))=x</math> for all <math>x\in X</math> implies that {{mvar|f}} is [[Injective function|injective]], and the condition <math>f(g(y))=y</math> for all <math>y\in Y</math> implies that {{mvar|f}} is [[Surjective function|surjective]].

The inverse function {{math|''f''<sup> −1</sup>}} to {{mvar|f}} can be explicitly described as the function

:<math>f^{-1}(y)=(\text{the unique element }x\in X\text{ such that }f(x)=y)</math>.

==={{anchor|Compositional inverse}}Inverses and composition===
{{See also|Inverse element}}

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

: <math> f^{-1}\left(f(x)\right) = x</math>, for every <math>x \in X</math> and <math> f\left(f^{-1}(y)\right) = y</math> for every <math>y \in Y </math>.

Using the [[composition of functions]], this statement can be rewritten to the following equations between functions:

: <math> f^{-1} \circ f = \operatorname{id}_X</math> and <math>f \circ f^{-1} = \operatorname{id}_Y, </math>

where {{math|id<sub>''X''</sub>}} is the [[identity function]] on the set {{mvar|X}}; that is, the function that leaves its argument unchanged. In [[category theory]], this statement is used as the definition of an inverse [[morphism]].

Considering function composition helps to understand the notation {{math|''f''<sup> −1</sup>}}. Repeatedly composing a function {{math|''f'': ''X''→''X''}} with itself is called [[iterated function|iteration]]. If {{mvar|f}} is applied {{mvar|n}} times, starting with the value {{mvar|x}}, then this is written as {{math|''f''<sup> ''n''</sup>(''x'')}}; so {{math|''f''<sup> 2</sup>(''x'') {{=}} ''f'' (''f'' (''x''))}}, etc. Since {{math|''f''<sup> −1</sup>(''f'' (''x'')) {{=}} ''x''}}, composing {{math|''f''<sup> −1</sup>}} and {{math|''f''<sup> ''n''</sup>}} yields {{math|''f''<sup> ''n''−1</sup>}}, "undoing" the effect of one application of {{mvar|f}}.

===Notation===
While the notation {{math|''f''<sup> −1</sup>(''x'')}} might be misunderstood,<ref name=":2" /> {{math|(''f''(''x''))<sup>−1</sup>}} certainly denotes the [[multiplicative inverse]] of {{math|''f''(''x'')}} and has nothing to do with the inverse function of {{mvar|f}}.<ref name="Cajori_1929"/> The notation <math>f^{\langle -1\rangle}</math> might be used for the inverse function to avoid ambiguity with the [[multiplicative inverse]].<ref>Helmut Sieber und Leopold Huber: ''Mathematische Begriffe und Formeln für Sekundarstufe I und II der Gymnasien.'' Ernst Klett Verlag.</ref>

In keeping with the general notation, some English authors use expressions like {{math|sin<sup>−1</sup>(''x'')}} to denote the inverse of the sine function applied to {{mvar|x}} (actually a [[#Partial inverses|partial inverse]]; see below).<ref>{{harvnb|Thomas|1972|loc=pp. 304–309}}</ref><ref name="Cajori_1929"/> Other authors feel that this may be confused with the notation for the multiplicative inverse of {{math|sin (''x'')}}, which can be denoted as {{math|(sin (''x''))<sup>−1</sup>}}.<ref name="Cajori_1929"/> To avoid any confusion, an [[inverse trigonometric function]] is often indicated by the prefix "[[arc (function prefix)|arc]]" (for Latin {{lang|la|arcus}}).<ref name="Korn_2000"/><ref name="Atlas_2009"/> For instance, the inverse of the sine function is typically called the [[arcsine]] function, written as {{math|[[arcsin]](''x'')}}.<ref name="Korn_2000"/><ref name="Atlas_2009"/> Similarly, the inverse of a [[hyperbolic function]] is indicated by the prefix "[[ar (function prefix)|ar]]" (for Latin {{lang|la|ārea}}).<ref name="Atlas_2009"/> For instance, the inverse of the [[hyperbolic sine]] function is typically written as {{math|[[arsinh]](''x'')}}.<ref name="Atlas_2009"/> The expressions like {{math|sin<sup>−1</sup>(''x'')}} can still be useful to distinguish the [[Multivalued function|multivalued]] inverse from the partial inverse: <math>\sin^{-1}(x) = \{(-1)^n \arcsin(x) + \pi n : n \in \mathbb Z\}</math>. Other inverse special functions are sometimes prefixed with the prefix "inv", if the ambiguity of the {{math|''f''<sup> −1</sup>}} notation should be avoided.<ref name="Hall_1909"/><ref name="Atlas_2009"/>

== Examples ==

===Squaring and square root functions===
The function {{math|''f'': '''R''' → [0,∞)}} given by {{math|1=''f''(''x'') = ''x''<sup>2</sup>}} is not injective because <math>(-x)^2=x^2</math> for all <math>x\in\R</math>. Therefore, {{Mvar|f}} is not invertible.

If the domain of the function is restricted to the nonnegative reals, that is, we take the function <math>f\colon [0,\infty)\to [0,\infty);\ x\mapsto x^2</math> with the same ''rule'' as before, then the function is bijective and so, invertible.<ref>{{harvnb|Lay|2006|loc=p. 69, Example 7.24}}</ref> The inverse function here is called the ''(positive) square root function'' and is denoted by <math>x\mapsto\sqrt x</math>.
<!-- Repetitive. To be held for a short time until refactor is finished. 
===Inverses in higher mathematics===
The definition given above is commonly adopted in [[set theory]] and [[calculus]]. In higher mathematics, the notation
:<math>f\colon X \to Y </math>
means "{{mvar|f}} is a function mapping elements of a set {{mvar|X}} to elements of a set {{mvar|Y }}". The source, {{mvar|X}}, is called the domain of {{mvar|f}}, and the target, {{mvar|Y}}, is called the [[codomain]]. The codomain contains the range of {{mvar|f}} as a [[subset]], and is part of the definition of {{mvar|f}}.<ref>{{harvnb|Smith|Eggen|St. Andre|2006|loc=p. 179}}</ref>

When using codomains, the inverse of a function {{math| ''f'': ''X'' → ''Y''}} is required to have domain {{mvar|Y}} and codomain {{mvar|X}}. For the inverse to be defined on all of {{mvar|Y}}, every element of {{mvar|Y}} must lie in the range of the function {{mvar|f}}. A function with this property is called ''onto'' or ''[[Surjective function|surjective]]''. Thus, a function with a codomain is invertible if and only if it is both ''[[Injective function|injective]]'' (one-to-one) and surjective (onto). Such a function is called a one-to-one correspondence or a [[bijection]], and has the property that every element {{math| ''y'' ∈ ''Y''}} corresponds to exactly one element {{math| ''x'' ∈ ''X''}}.
-->

=== Standard inverse functions ===
The following table shows several standard functions and their inverses:
{| class="wikitable" align="center"
|+Inverse arithmetic functions
|-
!scope="col" align="center" | Function {{math|''f''(''x'')}}
!scope="col" align="center" | Inverse {{math|''f''<sup> −1</sup>(''y'')}}
!scope="col" align="center" | Notes
|-
| align="center" | {{math|''x'' [[addition|+]] ''a''}}
| align="center" | {{math|''y'' [[subtraction|−]] ''a''}}
|
|-
| align="center" | {{math|''a'' − ''x''}}
| align="center" | {{math|''a'' − ''y''}}
|
|-
| align="center" | {{math|[[multiplication|''mx'']]}}
| align="center" | {{sfrac|{{mvar|y}}|{{mvar|m}}}}
| {{math|''m'' ≠ 0}}
|-
| align="center" | {{sfrac|1|{{mvar|x}}}} (i.e. {{math|''x''<sup>−1</sup>}})
| align="center" | {{sfrac|1|{{mvar|y}}}} (i.e. {{math|''y''<sup>−1</sup>}})
| {{math|''x'', ''y'' ≠ 0}}
|-
| align="center" | {{math|''x''<sup>''p''</sup>}}
| align="center" | <math>\sqrt[p]y</math> (i.e. {{math|''y''<sup>1/''p''</sup>}})
| integer {{math|''p'' > 0}}; {{math|''x'', ''y'' ≥ 0}} if {{math|p}} is even
|-
| align="center" | {{math|''a''<sup>''x''</sup>}}
| align="center" | {{math|[[logarithm|log]]<sub>''a''</sub> ''y''}}
| {{math|''y'' > 0}} and {{math|''a'' > 0}} and {{math|''a'' ≠ 1}}
|-
| align="center" | {{math|''x''[[e (mathematical constant)|''e'']]<sup>''x''</sup>}}
| align="center" | {{math|[[Lambert W function|W]] (''y'')}}
| {{math|''x'' ≥ −1}} and {{math|''y'' ≥ −1/''e''}}
|-
| align="center" | [[trigonometric function]]s
| align="center" | [[inverse trigonometric function]]s
| various restrictions (see table below)
|-
| align="center" | [[hyperbolic function]]s
| align="center" | [[inverse hyperbolic function]]s
| various restrictions
|}

=== Formula for the inverse ===
Many functions given by algebraic formulas possess a formula for their inverse. This is because the inverse <math>f^{-1} </math> of an invertible function <math>f\colon\R\to\R</math> has an explicit description as

: <math>f^{-1}(y)=(\text{the unique element }x\in \R\text{ such that }f(x)=y)</math>.

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

: <math>f(x) = (2x + 8)^3 </math>

then to determine <math>f^{-1}(y) </math> for a real number {{Mvar|y}}, one must find the unique real number {{mvar|x}} such that {{math|1= (2''x'' + 8)<sup>3</sup> = ''y''}}. This equation can be solved:

: <math>\begin{align}
      y         & = (2x+8)^3 \\
  \sqrt[3]{y}   & = 2x + 8   \\
\sqrt[3]{y} - 8 & = 2x       \\
\dfrac{\sqrt[3]{y} - 8}{2} & = x .
\end{align}</math>

Thus the inverse function {{math|''f''<sup> −1</sup>}} is given by the formula

: <math>f^{-1}(y) = \frac{\sqrt[3]{y} - 8} 2.</math>

Sometimes, the inverse of a function cannot be expressed by a [[closed-form formula]]. For example, if {{mvar|f}} is the function

: <math>f(x) = x - \sin x ,</math>

then {{mvar|f}} is a bijection, and therefore possesses an inverse function {{math|''f''<sup> −1</sup>}}. The [[Kepler's equation#Inverse Kepler equation|formula for this inverse]] has an expression as an infinite sum:

: <math> 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).
</math>

==Properties==
Since a function is a special type of [[binary relation]], many of the properties of an inverse function correspond to properties of [[converse relation]]s.

===Uniqueness===
If an inverse function exists for a given function {{mvar|f}}, then it is unique.<ref name="Wolf72">{{harvnb|Wolf|1998|loc=p. 208, Theorem 7.2}}</ref> This follows since the inverse function must be the converse relation, which is completely determined by {{mvar|f}}.

===Symmetry===
There is a symmetry between a function and its inverse. Specifically, if {{mvar|f}} is an invertible function with domain {{mvar|X}} and codomain {{mvar|Y}}, then its inverse {{math|''f''<sup> −1</sup>}} has domain {{mvar|Y}} and image {{mvar|X}}, and the inverse of {{math|''f''<sup> −1</sup>}} is the original function {{mvar|f}}. In symbols, for functions {{math|''f'':''X'' → ''Y''}}  and {{math|''f''<sup>−1</sup>:''Y'' → ''X''}},<ref name=Wolf72 />

:<math>f^{-1}\circ f = \operatorname{id}_X </math> and <math> f \circ f^{-1} = \operatorname{id}_Y.</math>

This statement is a consequence of the implication that for {{mvar|f}} to be invertible it must be bijective. The [[involution (mathematics)|involutory]] nature of the inverse can be concisely expressed by<ref>{{harvnb|Smith|Eggen|St. Andre|2006|loc=pg. 141 Theorem 3.3(a)}}</ref> 
:<math>\left(f^{-1}\right)^{-1} = f.</math>

[[Image:Composition of Inverses.png|thumb|right|240px|The inverse of {{math| ''g'' ∘ ''f'' }} is {{math| ''f''<sup> −1</sup> ∘ ''g''<sup> −1</sup>}}.]]
The inverse of a composition of functions is given by<ref>{{harvnb|Lay|2006|loc=p. 71, Theorem 7.26}}</ref> 
:<math>(g \circ f)^{-1} = f^{-1} \circ g^{-1}.</math>
Notice that the order of {{mvar|g}} and {{mvar|f}} have been reversed; to undo {{mvar|f}} followed by {{mvar|g}}, we must first undo {{mvar|g}}, and then undo {{mvar|f}}.

For example, let {{math|1= ''f''(''x'') = 3''x''}} and let {{math|1= ''g''(''x'') = ''x'' + 5}}. Then the composition {{math| ''g'' ∘ ''f''}} is the function that first multiplies by three and then adds five,

: <math>(g \circ f)(x) = 3x + 5.</math>

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

: <math>(g \circ f)^{-1}(x) = \tfrac13(x - 5).</math>

This is the composition
{{math| (''f''<sup> −1</sup> ∘ ''g''<sup> −1</sup>)(''x'')}}.

===Self-inverses===
If {{mvar|X}} is a set, then the [[identity function]] on {{mvar|X}} is its own inverse:

: <math>{\operatorname{id}_X}^{-1} = \operatorname{id}_X.</math>

More generally, a function {{math| ''f'' : ''X'' → ''X''}} is equal to its own inverse, if and only if the composition {{math| ''f'' ∘ ''f''}} is equal to {{math|id<sub>''X''</sub>}}. Such a function is called an [[Involution (mathematics)|involution]].

===Graph of the inverse===
[[Image:Inverse Function Graph.png|thumb|right|The graphs of {{math|1= ''y'' = ''f''(''x'') }} and {{math|1= ''y'' = ''f''<sup> −1</sup>(''x'')}}. The dotted line is {{math|1= ''y'' = ''x''}}.]]
If  {{mvar|f}} is invertible, then the graph of the function

: <math>y = f^{-1}(x)</math>

is the same as the graph of the equation

: <math>x = f(y) .</math>

This is identical to the equation {{math|1= ''y'' = ''f''(''x'')}} that defines the graph of {{mvar|f}}, except that the roles of {{mvar|x}} and {{mvar|y}} have been reversed. Thus the graph of {{math|''f''<sup> −1</sup>}} can be obtained from the graph of {{mvar|f}} by switching the positions of the {{mvar|x}} and {{mvar|y}} axes. This is equivalent to [[Reflection (mathematics)|reflecting]] the graph across the line
{{math|1= ''y'' = ''x''}}.<ref>{{harvnb|Briggs|Cochran|2011|loc=pp. 28–29}}</ref><ref name=":2" />

===Inverses and derivatives===
By the [[inverse function theorem]], a [[continuous function]] of a single variable <math>f\colon A\to\mathbb{R}</math> (where <math>A\subseteq\mathbb{R}</math>) is invertible on its range (image) if and only if it is either strictly [[monotonic function|increasing or decreasing]] (with no local [[maxima and minima|maxima or minima]]). For example, the function

: <math>f(x) = x^3 + x</math>

is invertible, since the [[derivative]]
{{math|1= ''f&prime;''(''x'') = 3''x''<sup>2</sup> + 1 }} is always positive.

If the function {{mvar|f}} is [[Differentiable function|differentiable]] on an interval {{mvar|I}} and {{math| ''f&prime;''(''x'') ≠ 0}} for each {{math|''x'' ∈ ''I''}}, then the inverse {{math|''f''<sup> −1</sup>}} is differentiable on {{math|''f''(''I'')}}.<ref>{{harvnb|Lay|2006|loc=p. 246, Theorem 26.10}}</ref>  If {{math|1= ''y'' = ''f''(''x'')}}, the derivative of the inverse is given by the inverse function theorem,

: <math>\left(f^{-1}\right)^\prime (y)  = \frac{1}{f'\left(x \right)}. </math>

Using [[Leibniz's notation]] the formula above can be written as

: <math>\frac{dx}{dy} = \frac{1}{dy / dx}. </math>

This result follows from the [[chain rule]] (see the article on [[inverse functions and differentiation]]).

The inverse function theorem can be generalized to functions of several variables. Specifically, a continuously differentiable [[real multivariable function|multivariable function]] {{math| ''f '': '''R'''<sup>''n''</sup> → '''R'''<sup>''n''</sup>}} is invertible in a neighborhood of a point {{mvar|p}} as long as the [[Jacobian matrix and determinant|Jacobian matrix]] of {{mvar|f}} at {{mvar|p}} is [[invertible matrix|invertible]]. In this case, the Jacobian of {{math|''f''<sup> −1</sup>}} at {{math|''f''(''p'')}} is the [[matrix inverse]] of the Jacobian of {{mvar|f}} at {{mvar|p}}.

== Real-world examples ==
* Let {{mvar|f}} be the function that converts a temperature in degrees [[Celsius]] to a temperature in degrees [[Fahrenheit]], <math display="block"> F = f(C) = \tfrac95 C + 32 ;</math> then its inverse function converts degrees Fahrenheit to degrees Celsius, <math display="block"> C = f^{-1}(F) = \tfrac59 (F - 32) ,</math><ref name=":1">{{Cite web|title=Inverse Functions|url=https://www.mathsisfun.com/sets/function-inverse.html|access-date=2020-09-08|website=www.mathsisfun.com}}</ref> since <math display="block">
\begin{align}
f^{-1} (f(C)) = {} & f^{-1}\left(  \tfrac95 C + 32 \right) = \tfrac59 \left( (\tfrac95 C + 32 ) - 32 \right) =  C, \\
& \text{for every value of } C, \text{ and } \\[6pt]
f\left(f^{-1}(F)\right) = {} & f\left(\tfrac59 (F - 32)\right) = \tfrac95 \left(\tfrac59 (F - 32)\right) + 32 = F, \\
& \text{for every value of } F.
\end{align}
</math>
* Suppose {{mvar|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">\begin{align}
 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{align}
</math>
* Let {{mvar|R}} be the function that leads to an {{mvar|x}} percentage rise of some quantity, and {{mvar|F}} be the function producing an {{mvar|x}} percentage fall. Applied to $100 with {{mvar|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.
* The formula to calculate the pH of a solution is {{math|1=pH = −log<sub>10</sub>[H<sup>+</sup>]}}. In many cases we need to find the concentration of acid from a pH measurement. The inverse function {{math|1=[H<sup>+</sup>] = 10<sup>−pH</sup>}} is used.

==Generalizations==
===Partial inverses===
[[Image:Inverse square graph.svg|thumb|right|The square root of {{mvar|x}} is a partial inverse to {{math|1= ''f''(''x'') = ''x''<sup>2</sup>}}.]]
Even if a function {{mvar|f}} is not one-to-one, it may be possible to define a '''partial inverse''' of {{mvar|f}} by [[Function (mathematics)#Restrictions and extensions|restricting]] the domain. For example, the function

: <math>f(x) = x^2</math>

is not one-to-one, since {{math|1= ''x''<sup>2</sup> = (−''x'')<sup>2</sup>}}. However, the function becomes one-to-one if we restrict to the domain {{math| ''x'' ≥ 0}}, in which case

: <math>f^{-1}(y) = \sqrt{y} . </math>

(If we instead restrict to the domain {{math| ''x'' ≤ 0}}, then the inverse is the negative of the square root of {{mvar|y}}.) 

===Full inverses===
[[File:Inversa d'una cúbica gràfica.png|right|thumb|The inverse of this [[cubic function]] has three branches.]]

Alternatively, there is no need to restrict the domain if we are content with the inverse being a [[multivalued function]]:

: <math>f^{-1}(y) = \pm\sqrt{y} . </math>

Sometimes, this multivalued inverse is called the '''full inverse''' of {{mvar|f}}, and the portions (such as {{sqrt|{{mvar|x}}}} and −{{sqrt|{{mvar|x}}}}) are called ''branches''. The most important branch of a multivalued function (e.g. the positive square root) is called the ''[[principal branch]]'', and its value at {{mvar|y}} is called the ''principal value'' of {{math|''f''<sup> −1</sup>(''y'')}}.

For a continuous function on the real line, one branch is required between each pair of [[minima and maxima|local extrema]]. For example, the inverse of a [[cubic function]] with a local maximum and a local minimum has three branches (see the adjacent picture).

===Trigonometric inverses===
[[Image:Gràfica del arcsinus.png|right|thumb|The [[arcsine]] is a partial inverse of the [[sine]] function.]]

The above considerations are particularly important for defining the inverses of [[trigonometric functions]]. For example, the [[sine function]] is not one-to-one, since

: <math>\sin(x + 2\pi) = \sin(x)</math>

for every real {{mvar|x}} (and more generally {{math|1= sin(''x'' + 2{{pi}}''n'') = sin(''x'')}} for every [[integer]] {{mvar|n}}). However, the sine is one-to-one on the interval
{{closed-closed|−{{sfrac|{{pi}}|2}}, {{sfrac|{{pi}}|2}}}}, and the corresponding partial inverse is called the [[arcsine]]. This is considered the principal branch of the inverse sine, so the principal value of the inverse sine is always between −{{sfrac|{{pi}}|2}} and {{sfrac|{{pi}}|2}}. The following table describes the principal branch of each inverse trigonometric function:<ref>{{harvnb|Briggs|Cochran|2011|loc=pp. 39–42}}</ref>
{| class="wikitable" style="text-align:center"
|-
!function
!Range of usual [[principal value]]
|-
| arcsin || {{math|−{{sfrac|{{pi}}|2}} ≤ sin<sup>−1</sup>(''x'') ≤ {{sfrac|{{pi}}|2}}}}
|-
| arccos || {{math|0 ≤ cos<sup>−1</sup>(''x'') ≤ {{pi}}}}
|-
| arctan || {{math|−{{sfrac|π|2}} < tan<sup>−1</sup>(''x'') < {{sfrac|{{pi}}|2}}}}
|-
| arccot || {{math|0 < cot<sup>−1</sup>(''x'') < {{pi}}}}
|-
| arcsec || {{math|0 ≤ sec<sup>−1</sup>(''x'') ≤ {{pi}}}}
|-
| arccsc || {{math|−{{sfrac|{{pi}}|2}} ≤ csc<sup>−1</sup>(''x'') ≤ {{sfrac|{{pi}}|2}}}}
|-
|}

===Left and right inverses===
[[Function composition]] on the left and on the right need not coincide.  In general, the conditions 
# "There exists {{mvar|g}} such that {{math|''g''(''f''(''x'')){{=}}''x''}}" and 
# "There exists {{mvar|g}} such that {{math|''f''(''g''(''x'')){{=}}''x''}}"
imply different properties of {{mvar|f}}.  For example, let {{math|''f'': '''R''' → {{closed-open|0, ∞}}}} denote the squaring map, such that {{math|1=''f''(''x'') = ''x''<sup>2</sup>}} for all {{mvar|x}} in {{math|'''R'''}}, and let  {{math|{{mvar|g}}: {{closed-open|0, ∞}} → '''R'''}} denote the square root map, such that {{math|''g''(''x'') {{=}} }}{{radic|{{mvar|x}}}} for all {{math|''x'' ≥ 0}}. Then {{math|1=''f''(''g''(''x'')) = ''x''}} for all {{mvar|x}} in {{closed-open|0, ∞}}; that is, {{mvar|g}} is a right inverse to {{mvar|f}}. However, {{mvar|g}} is not a left inverse to {{mvar|f}}, since, e.g., {{math|1=''g''(''f''(−1)) = 1 ≠ −1}}.

====Left inverses====
If {{math|''f'': ''X'' → ''Y''}}, a '''left inverse''' for {{mvar|f}} (or ''[[retract (category theory)|retraction]]'' of {{mvar|f}} ) is a function {{math| ''g'': ''Y'' → ''X''}} such that composing {{mvar|f}} with {{mvar|g}} from the left gives the identity function<ref>{{cite book|last1=Dummit|last2=Foote|title=Abstract Algebra}}</ref> <math display="block">g \circ f = \operatorname{id}_X\text{.}</math>  That is, the function {{mvar|g}} satisfies the rule
: If {{math|''f''(''x''){{=}}''y''}}, then {{math|''g''(''y''){{=}}''x''}}.

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

A function {{mvar|f}} with nonempty domain is injective if and only if it has a left inverse.<ref>{{cite book|last=Mac&nbsp;Lane|first=Saunders|title=Categories for the Working Mathematician}}</ref>  An elementary proof runs as follows:
* If {{mvar|g}} is the left inverse of {{mvar|f}}, and {{math|1=''f''(''x'') = ''f''(''y'')}}, then {{math|1=''g''(''f''(''x'')) = ''g''(''f''(''y'')) = ''x'' = ''y''}}.
* <p>If nonempty {{math|''f'': ''X'' → ''Y''}} is injective, construct a left inverse {{math|''g'': ''Y'' → ''X''}} as follows: for all {{math|''y'' ∈ ''Y''}}, if {{mvar|y}} is in the image of {{mvar|f}}, then there exists {{math|''x'' ∈ ''X''}} such that {{math|1=''f''(''x'') = ''y''}}.  Let {{math|1=''g''(''y'') = ''x''}}; this definition is unique because {{mvar|f}} is injective.  Otherwise, let {{math|''g''(''y'')}} be an arbitrary element of {{mvar|X}}.</p><p>For all {{math|''x'' ∈ ''X''}}, {{math|''f''(''x'')}} is in the image of {{mvar|f}}.  By construction, {{math|1=''g''(''f''(''x'')) = ''x''}}, the condition for a left inverse.</p>

In classical mathematics, every injective function {{mvar|f}} with a nonempty domain necessarily has a left inverse; however, this may fail in [[constructive mathematics]]. For instance, a left inverse of the [[Inclusion map|inclusion]] {{math|{0,1} → '''R'''}} of the two-element set in the reals violates [[indecomposability (constructive mathematics)|indecomposability]] by giving a [[Retract (category theory)|retraction]] of the real line to the set {{math|{0,1{{)}}}}.<ref>{{cite journal|last=Fraenkel|title=Abstract Set Theory|journal=Nature |year=1954 |volume=173 |issue=4412 |page=967 |doi=10.1038/173967a0 |bibcode=1954Natur.173..967C |s2cid=7735523 |doi-access=free }}</ref>

====Right inverses====
[[File:Right inverse with surjective function.svg|thumb|Example of '''right inverse''' with non-injective, surjective function]]
A '''right inverse''' for {{mvar|f}} (or ''[[section (category theory)|section]]'' of {{mvar|f}} ) is a function {{math| ''h'': ''Y'' → ''X''}} such that

: <math>f \circ h = \operatorname{id}_Y . </math>

That is, the function {{mvar|h}} satisfies the rule

: If <math>\displaystyle h(y) = x</math>, then <math>\displaystyle f(x) = y .</math>

Thus, {{math|''h''(''y'')}} may be any of the elements of {{mvar|X}} that map to {{mvar|y}} under {{mvar|f}}.

A function {{mvar|f}} has a right inverse if and only if it is [[surjective function|surjective]] (though constructing such an inverse in general requires the [[axiom of choice]]).

: If {{mvar|h}} is the right inverse of {{mvar|f}}, then {{mvar|f}} is surjective. For all <math>y \in Y</math>, there is <math>x = h(y)</math> such that <math>f(x) = f(h(y)) = y</math>.
: If {{mvar|f}} is surjective, {{mvar|f}} has a right inverse {{mvar|h}}, which can be constructed as follows: for all <math>y \in Y</math>, there is at least one <math>x \in X</math> such that <math>f(x) = y</math> (because {{mvar|f}} is surjective), so we choose one to be the value of {{math|''h''(''y'')}}.{{Citation needed|date=February 2021}}

====Two-sided inverses====
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'''.

: If <math>g</math> is a left inverse and <math>h</math> a right inverse of <math>f</math>, for all <math>y \in Y</math>, <math>g(y) = g(f(h(y)) = h(y)</math>.

A function has a two-sided inverse if and only if it is bijective.
: A bijective function {{mvar|f}} is injective, so it has a left inverse (if {{mvar|f}} is the empty function, <math>f \colon \varnothing \to \varnothing</math> is its own left inverse). {{mvar|f}} is surjective, so it has a right inverse. By the above, the left and right inverse are the same.
: If {{mvar|f}} has a two-sided inverse {{mvar|g}}, then {{mvar|g}} is a left inverse and right inverse of {{mvar|f}}, so {{mvar|f}} is injective and surjective.

===Preimages===
If {{math|''f'': ''X'' → ''Y''}} is any function (not necessarily invertible), the '''preimage''' (or '''inverse image''') of an element {{math| ''y'' &isin; ''Y''}} is defined to be the set of all elements of {{mvar|X}} that map to {{mvar|y}}:

: <math>f^{-1}(y) = \left\{ x\in X : f(x) = y \right\} . </math>

The preimage of {{mvar|y}} can be thought of as the [[image (mathematics)|image]] of {{mvar|y}} under the (multivalued) full inverse of the function {{mvar|f}}.

The notion can be generalized to subsets of the range. Specifically, if {{mvar|S}} is any [[subset]] of {{mvar|Y}}, the preimage of {{mvar|S}}, denoted by <math>f^{-1}(S) </math>, is the set of all elements of {{mvar|X}} that map to {{mvar|S}}:

: <math>f^{-1}(S) = \left\{ x\in X : f(x) \in S \right\} . </math>

For example, take the function {{math|''f'': '''R''' → '''R'''; ''x'' ↦ ''x''<sup>2</sup>}}. This function is not invertible as it is not bijective, but preimages may be defined for subsets of the codomain, e.g.

: <math>f^{-1}(\left\{1,4,9,16\right\}) = \left\{-4,-3,-2,-1,1,2,3,4\right\}</math>.

The original notion and its generalization are related by the identity <math>f^{-1}(y) = f^{-1}(\{y\}),</math> The preimage of a single element {{math| ''y'' &isin; ''Y''}} – a [[singleton set]] {{math|{''y''} }} – is sometimes called the ''[[fiber (mathematics)|fiber]]'' of {{mvar|y}}. When {{mvar|Y}} is the set of real numbers, it is common to refer to {{math|''f''<sup> −1</sup>({''y''})}} as a ''[[level set]]''.

==See also==
* [[Lagrange inversion theorem]], gives the Taylor series expansion of the inverse function of an analytic function
* [[Integral of inverse functions]]
* [[Inverse Fourier transform]]
* [[Reversible computing]]

==Notes==
{{reflist|group="nb"|refs=
<ref group="nb" name="NB2">Not to be confused with numerical exponentiation such as taking the multiplicative inverse of a nonzero real number.</ref>
}}

==References==
{{reflist|refs=
<ref name="Hall_1909">{{cite book |title=Plane Trigonometry |author-first1=Arthur Graham |author-last1=Hall |author-first2=Fred Goodrich |author-last2=Frink |date=1909 |location=Ann Arbor, Michigan, USA |chapter=Article 14: Inverse trigonometric functions |publisher=[[Henry Holt & Company]] |publication-place=New York |pages=15–16 |chapter-url=https://archive.org/stream/planetrigonometr00hallrich#page/n30/mode/1up |access-date=2017-08-12 |quote=α&nbsp;= arcsin&nbsp;''m'' This notation is universally used in Europe and is fast gaining ground in this country. A less desirable symbol, α&nbsp;= sin{{sup|-1}}{{spaces|hair}}''m'', is still found in English and American texts. The notation α&nbsp;= inv sin ''m'' is perhaps better still on account of its general applicability. [...] A similar symbolic relation holds for the other [[trigonometric function]]s. 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.}}</ref>
<ref name="Korn_2000">{{cite book |title=Mathematical handbook for scientists and engineers: Definitions, theorems, and formulars for reference and review |url=https://archive.org/details/mathematicalhand00korn_849 |url-access=limited |author-first1=Grandino Arthur |author-last1=Korn |author-first2=Theresa M. |author-last2=Korn |author2-link= Theresa M. Korn |edition=3<!-- (based on 1968 edition by McGrawHill, Inc.) --> |date=2000 |orig-year=1961 |publisher=[[Dover Publications, Inc.]] |location=Mineola, New York, USA |chapter=21.2.-4. Inverse Trigonometric Functions |page=[https://archive.org/details/mathematicalhand00korn_849/page/n828 811] |isbn=978-0-486-41147-7}}</ref>
<ref name="Atlas_2009">{{cite book |title=An Atlas of Functions: with Equator, the Atlas Function Calculator |author-first1=Keith B. |author-last1=Oldham |author-first2=Jan C. |author-last2=Myland |author-first3=Jerome |author-last3=Spanier |publisher=[[Springer Science+Business Media, LLC]] |edition=2 |date=2009 |orig-year=1987 |isbn=978-0-387-48806-6 |doi=10.1007/978-0-387-48807-3 |lccn=2008937525}}</ref>
<ref name="Cajori_1929">{{cite book |author-first=Florian |author-last=Cajori |author-link=Florian Cajori |title=A History of Mathematical Notations |chapter=§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 |volume=2 |orig-year=March 1929 |publisher=[[Open court publishing company]] |location=Chicago, USA |date=1952 |edition=3rd corrected printing of 1929 issue, 2nd |pages=108, 176–179, 336, 346 |isbn=978-1-60206-714-1 |url=https://books.google.com/books?id=bT5suOONXlgC |access-date=2016-01-18 |quote=[...] §473. ''Iterated logarithms'' [...] We note here the symbolism used by [[Alfred Pringsheim|Pringsheim]] and [[Jules Molk|Molk]] in their joint ''Encyclopédie'' article: "<sup>2</sup>log<sub>''b''</sub> ''a'' = log<sub>''b''</sub> (log<sub>''b''</sub> ''a''), ..., <sup>''k''+1</sup>log<sub>''b''</sub> ''a'' = log<sub>''b''</sub> (<sup>''k''</sup>log<sub>''b''</sub> ''a'')."<!-- {{citeref|Pringsheim|Molk|1907|a}} --><!-- [10] --> [...] §533. ''[[John Frederick William Herschel|John Herschel]]'s notation for inverse functions,'' sin<sup>&minus;1</sup> ''x'', tan<sup>&minus;1</sup> ''x'', etc., was published by him in the ''[[Philosophical Transactions of London]]'', for the year 1813. He says ({{citeref|Herschel|1813|p.&nbsp;10|style=plain}}): "This notation cos.<sup>&minus;1</sup> ''e'' must not be understood to signify 1/cos.&nbsp;''e'', but what is usually written thus, arc (cos.=''e'')." He admits that some authors use cos.<sup>''m''</sup> ''A'' for (cos. ''A'')<sup>''m''</sup>, but he justifies his own notation by pointing out that since ''d''<sup>2</sup> ''x'', Δ<sup>3</sup> ''x'', Σ<sup>2</sup> ''x'' mean ''dd'' ''x'', ΔΔΔ ''x'', ΣΣ ''x'', we ought to write sin.<sup>2</sup> ''x'' for sin. sin. ''x'', log.<sup>3</sup> ''x'' for log. log. log. ''x''. Just as we write ''d''<sup>&minus;''n''</sup> V=∫<sup>''n''</sup> V, we may write similarly sin.<sup>&minus;1</sup> ''x''=arc (sin.=''x''), log.<sup>&minus;1</sup> ''x''.=c<sup>''x''</sup>. Some years later Herschel explained that in 1813 he used ''f''<sup>''n''</sup>(''x''), ''f''<sup>&minus;''n''</sup>(''x''), sin.<sup>&minus;1</sup> ''x'', etc., "as he then supposed for the first time. The work of a German Analyst, [[Hans Heinrich Bürmann|Burmann]], 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<sup>&minus;1</sup>, 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."{{citeref|Herschel|1820|a<!-- [4] -->}} [...] §535. ''Persistence of rival notations for inverse function.''&mdash; [...] The use of Herschel's notation underwent a slight change in [[Benjamin Peirce]]'s books, to remove the chief objection to them; Peirce wrote: "cos<sup>[&minus;1]</sup> ''x''," "log<sup>[&minus;1]</sup> ''x''."{{citeref|Peirce|1852|b<!-- [1] -->}} [...] §537. ''Powers of trigonometric functions.''&mdash;Three principal notations have been used to denote, say, the square of sin ''x'', namely, (sin ''x'')<sup>2</sup>, sin ''x''<sup>2</sup>, sin<sup>2</sup> ''x''. The prevailing notation at present is sin<sup>2</sup> ''x'', though the first is least likely to be misinterpreted. In the case of sin<sup>2</sup> ''x'' two interpretations suggest themselves; first, sin ''x'' · sin ''x''; second,{{citeref|Peano|1903|c<!-- [8] -->}} 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 log<sup>2</sup> ''x'', where log ''x'' · log ''x'' and log (log ''x'') are of frequent occurrence in analysis. [...] The notation sin<sup>''n''</sup> ''x'' for (sin ''x'')<sup>''n''</sup> has been widely used and is now the prevailing one. [...]}} (xviii+367+1 pages including 1 addenda page) (NB. ISBN and link for reprint of 2nd edition by Cosimo, Inc., New York, USA, 2013.)</ref>
<ref name="Herschel_1813">{{cite journal |author-first=John Frederick William |author-last=Herschel |author-link=John Frederick William Herschel |title=On a Remarkable Application of Cotes's Theorem |journal=[[Philosophical Transactions of the Royal Society of London]] |publisher=[[Royal Society of London]], printed by W. Bulmer and Co., Cleveland-Row, St. James's, sold by G. and W. Nicol, Pall-Mall |location=London |volume=103 |number=Part 1 |date=1813 |orig-year=1812-11-12 |jstor=107384 |pages=8–26 [10]|doi=10.1098/rstl.1813.0005 |s2cid=118124706 |doi-access=free }}</ref>
<ref name="Herschel_1820">{{cite book |author-first=John Frederick William |author-last=Herschel |author-link=John Frederick William Herschel |title=A Collection of Examples of the Applications of the Calculus of Finite Differences |chapter=Part III. Section I. Examples of the Direct Method of Differences |location=Cambridge, UK |publisher=Printed by J. Smith, sold by J. Deighton & sons |date=1820 |pages=1–13 [5–6] |chapter-url=https://books.google.com/books?id=PWcSAAAAIAAJ&pg=PA5 |access-date=2020-08-04 |url-status=live |archive-url=https://web.archive.org/web/20200804031020/https://books.google.de/books?hl=de&id=PWcSAAAAIAAJ&jtp=5 |archive-date=2020-08-04}} [https://archive.org/details/acollectionexam00lacrgoog] (NB. Inhere, Herschel refers to his {{citeref|Herschel|1813|1813 work|style=plain}} and mentions [[Hans Heinrich Bürmann]]'s older work.)</ref>
<ref name="Peirce_1852">{{cite book |author-first=Benjamin |author-last=Peirce |author-link=Benjamin Peirce |title=Curves, Functions and Forces |volume=I |edition=new |location=Boston, USA |date=1852 |page=203}}</ref>
<ref name="Peano_1903">{{cite book |author-first=Giuseppe |author-last=Peano |author-link=Giuseppe Peano |title=Formulaire mathématique |language=fr |volume=IV |date=1903 |page=229}}</ref>
<!-- <ref name="Pringsheim-Molk_1907">{{cite book |author-first1=Alfred |author-last1=Pringsheim |author-link1=Alfred Pringsheim |author-first2=Jules |author-last2=Molk |author-link2=Jules Molk |title=Encyclopédie des sciences mathématiques pures et appliquées |language=fr |id=Part I |volume=I |date=1907 |page=195}}</ref> -->
}}

== Bibliography ==
* {{cite book |author-first1=William |author-last1=Briggs |author-first2=Lyle |author-last2=Cochran |title=Calculus / Early Transcendentals Single Variable |date=2011 |publisher=[[Addison-Wesley]] |isbn=978-0-321-66414-3 |url-access=registration |url=https://archive.org/details/calculusearlytra0000brig}}
* {{cite book |author-first=Keith J. |author-last=Devlin |author-link=Keith J. Devlin |title=Sets, Functions, and Logic / An Introduction to Abstract Mathematics |edition=3 |publisher=[[Chapman & Hall]] / [[CRC Mathematics]] |date=2004 |isbn=978-1-58488-449-1}}
* {{cite book |author-first1=Peter |author-last1=Fletcher |author-first2=C. Wayne |author-last2=Patty |title=Foundations of Higher Mathematics |publisher=PWS-Kent |date=1988 |isbn=0-87150-164-3}}
* {{cite book |author-first=Steven R. |author-last=Lay |title=Analysis / With an Introduction to Proof |edition=4 |publisher=[[Pearson (publisher)|Pearson]] / [[Prentice Hall]] |date=2006 |isbn=978-0-13-148101-5 |url=https://books.google.com/books?id=k4k_AQAAIAAJ}}
* {{cite book |author-first1=Douglas |author-last1=Smith |author-first2=Maurice |author-last2=Eggen |author-first3=Richard |author-last3=St. Andre |title=A Transition to Advanced Mathematics |edition=6 |date=2006 |publisher=[[Thompson Brooks/Cole]] |isbn=978-0-534-39900-9}}
* {{cite book |author-first=George Brinton |author-last=Thomas Jr. |author-link=George Brinton Thomas Jr. |title=Calculus and Analytic Geometry Part 1: Functions of One Variable and Analytic Geometry |edition=Alternate |date=1972 |publisher=[[Addison-Wesley]] |ref={{SfnRef|Thomas|1972}}}}
* {{cite book |author-first=Robert S. |author-last=Wolf |title=Proof, Logic, and Conjecture / The Mathematician's Toolbox |publisher=[[W. H. Freeman and Co.]] |date=1998 |isbn=978-0-7167-3050-7}}

==Further reading==
* {{cite book |author-first1=John C. |author-last1=Amazigo |author-first2=Lester A. |author-last2=Rubenfeld |title=Advanced Calculus and its Applications to the Engineering and Physical Sciences |location=New York |publisher=Wiley |date=1980 |isbn=0-471-04934-4 |chapter=Implicit Functions; Jacobians; Inverse Functions |pages=[https://archive.org/details/advancedcalculus0000amaz/page/n114 103]–120 |url=https://archive.org/details/advancedcalculus0000amaz |url-access=registration}}
* {{cite book |author-first=Ken G. |last=Binmore |author-link=Ken Binmore |chapter=Inverse Functions |title=Calculus |location=New York |publisher=[[Cambridge University Press]] |date=1983 |isbn=0-521-28952-1 |pages=161–197}}
* {{cite book |author-last=Spivak |author-first=Michael |date=1994 |title=Calculus |publisher=Publish or Perish |edition=3 |isbn=0-914098-89-6}}
* {{cite book |author-last=Stewart |author-first=James |author-link=James Stewart (mathematician) |date=2002 |title=Calculus |publisher=[[Brooks Cole]] |edition=5 |isbn=978-0-534-39339-7 |url-access=registration |url=https://archive.org/details/calculus0000stew}}

==External links==
{{sister project links|d=y|b=Algebra/Functions#Inverse function|n=no|c=Category:Inverse function|v=no|voy=no|m=no|mw=no|wikt=inverse function|s=no|species=no|q=no}}
* {{springer|title=Inverse function|id=p/i052360}}

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[[Category:Basic concepts in set theory]]
[[Category:Inverse functions| ]]
[[Category:Unary operations]]