Editing Inverse function (section)

==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"/>