Inverse function: Difference between revisions

Revision as of 22:29, 21 June 2024

Latest revision as of 09:30, 6 June 2025

(16 intermediate revisions by 13 users not shown)

Line 4:

[[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''}}.

Line 85:

| 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|''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''}}

Line 191:

===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>

Line 208:

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 ==

Line 238:

: <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>

Line 306 ⟶ 312:

: 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====

Line 320 ⟶ 326:

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>

Line 332 ⟶ 338:

: <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==