Inverse function: Difference between revisions

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{{short description|Mathematical concept}}

{{Distinguish|Multiplicativemultiplicative 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''⁻¹}}. Because {{mvar|f}} maps {{mvar|a}} to 3, the inverse {{math|''f''⁻¹}} 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''}}.

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===Notation===

While the notation {{math|''f''⁻¹(''x'')}} might be misunderstood,<ref name=":2" /> {{math|(''f''(''x''))⁻¹}} 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⁻¹(''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''))⁻¹}}.<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"/> Note that theThe expressions like {{math|sin⁻¹(''x'')}} can still be useful to distinguish the [[Multivalued_functionMultivalued function|multivalued]] inverse from the partial inverse: <math>\sin^{-1}(x) = \{(-1)^kn \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''⁻¹}} notation should be avoided.<ref name="Hall_1909"/><ref name="Atlas_2009"/>

== Examples ==

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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''⁻¹(''y'')}}

!scope="col" align="center" | Notes

|-

| align="center" | {{math|''x'' [[addition|+]] ''a''}}

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| align="center" | {{sfrac|1|{{mvar|y}}}} (i.e. {{math|''y''⁻¹}})

| {{math|''x'', ''y'' ≠ 0}}

|-

| align="center" | {{math|''x''²}}

| align="center" | <math>\sqrt y</math> (i.e. {{math|''y''^1/2}})

| {{math|''x'', ''y'' ≥ 0}} only

|-

| align="center" | {{math|[[cube (algebra)|''x''³]]}}

| align="center" | <math>\sqrt[3]y</math> (i.e. {{math|''y''^1/3}})

| no restriction on {{mvar|x}} and {{mvar|y}}

|-

| align="center" | {{math|''x''^''p''}}

| align="center" | <math>\sqrt[p]y</math> (i.e. {{math|''y''^1/''p''}})

| integer {{math|''p'' > 0}}; {{math|''x'', ''y'' ≥ 0}} if {{math|p}} is even; integer {{math|''p'' > 0}}

|-

| align="center" | {{math|2^''x''}}

| align="center" | {{math|[[binary logarithm|lb]] ''y''}}

| {{math|''y'' > 0}}

|-

| align="center" | {{math|[[e (mathematical constant)|''e'']]^''x''}}

| align="center" | {{math|[[natural logarithm|ln]] ''y''}}

| {{math|''y'' > 0}}

|-

| align="center" | {{math|10^''x''}}

| align="center" | {{math|[[decadic logarithm|log]] ''y''}}

| {{math|''y'' > 0}}

|-

| align="center" | {{math|''a''^''x''}}

| align="center" | {{math|[[logarithm|log]]_''a'' ''y''}}

| {{math|''y'' > 0}} and {{math|''a'' > 0}} and {{math|''a'' ≠ 1}}

|-

| align="center" | {{math|''x''[[e (mathematical constant)|''e'']]^''x''}}

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===Inverses and derivatives===

TheBy the [[inverse function theorem]] states that, a [[continuous function]] {{mvar|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>

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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'''^''n'' → '''R'''^''n''}} 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''⁻¹}} at {{math|''f''(''p'')}} is the [[matrix inverse]] of the Jacobian of {{mvar|f}} at {{mvar|p}}.

== Real-world examples ==

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: <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>

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

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''⁻¹(''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.]]

TheseThe 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>

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* <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 bookjournal|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{{Citation needed|date=February 2021}}

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

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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'')}}.<ref>{{CitationCite book |last=Loehr |first=Nicholas A. |url=https://books.google.com/books?id=mGUIEQAAQBAJ&pg=PA272 |title=An Introduction to Mathematical Proofs needed|date=February2019-11-20 2021|publisher=CRC Press |isbn=978-1-000-70962-9 |language=en}}</ref>

====Two-sided inverses====

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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}}.

SimilarlyThe 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>

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: <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''⁻¹({''y''})}} as a ''[[level set]]''.

==See also==

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* {{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}}

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==External links==

{{sisterlinkssister 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}}

{{springer|title=[https://www.turito.com/learn/math/inverse-functions Inverse function]|id=p/i052360}}

{{Authority control}}

[[Category:Basic concepts in set theory]]

[[Category:Inverse functions| ]]