({{, (RefLink[InverseFunction, paclet:ref/InverseFunction])[f]
represents the inverse of the function f, defined so that (RefLink[InverseFunction, paclet:ref/InverseFunction])[f][y] gives the value of x for which f[x] is equal to y. }, {, (RefLink[InverseFunction, paclet:ref/InverseFunction])[f,n,tot]
represents the inverse with respect to the n(^th) argument when there are tot arguments in all. }})

OpenersNotesSection(Spacer1)Details

In (RefLink[OutputForm, paclet:ref/OutputForm]) and (RefLink[StandardForm, paclet:ref/StandardForm]), (RefLink[InverseFunction, paclet:ref/InverseFunction])[f] is printed as (f^((-1))).

As discussed in "Functions That Do Not Have Unique Values", many mathematical functions do not have unique inverses. In such cases, (RefLink[InverseFunction, paclet:ref/InverseFunction])[f] can represent only one of the possible inverses for f.

(RefLink[InverseFunction, paclet:ref/InverseFunction]) is generated by (RefLink[Solve, paclet:ref/Solve]) when the option (RefLink[InverseFunctions, paclet:ref/InverseFunctions]) is set to (RefLink[Automatic, paclet:ref/Automatic]) or (RefLink[True, paclet:ref/True]).

OpenersPrimaryExamplesSection(Spacer1)Examples  (15)

OpenersExampleSection0(Spacer1)Basic Examples  (3)

The "inverse function" of (RefLink[Sin, paclet:ref/Sin]) is (RefLink[ArcSin, paclet:ref/ArcSin]):

InverseFunction[Sin]

Inverse of a pure function:

InverseFunction[(a #+b)/(c#+d)&]

Symbolic inverse function:

InverseFunction[f][x]

Derivative of an inverse function:

D[%,x]

OpenersExampleSection1(Spacer1)Scope  (8)

Inverse of a one-to-one function:

InverseFunction[2#+3&]

When the function is not one-to-one, (RefLink[InverseFunction, paclet:ref/InverseFunction]) issues a message:

InverseFunction[#^2-3#+5&]

For functions with a named principal branch of the inverse, the message is not issued:

InverseFunction[Cos]

InverseFunction[Tan[Exp[#]]&]

Inverse function with respect to the second argument:

InverseFunction[#1^2+Log[#2]-1&,2,2]

Inverse of a function with a restricted domain:

InverseFunction[ConditionalExpression[#1^2+2,#1>0]&]

The domain of the inverse function is computed automatically:

InverseFunction[ConditionalExpression[Sin[#^2-1/2],0<#1<1]&]

Here a closed-form representation for the inverse function does not exist:

f=InverseFunction[Sin[#]-#&]

Evaluation of the inverse function at exact points yields exact numeric values:

f[2]

However, the inverse may not be unique:

Reduce[Sin[x]-x==2&&Abs[x]<5,x]

(RefLink[InverseFunction, paclet:ref/InverseFunction]) with respect to the first argument of a two-argument function:

g=InverseFunction[Function[{x,y},Beta[x,y]-x+y],1,2]

g[3,4]

Here a closed-form representation for the inverse function does not exist:

h=InverseFunction[(∑_k^∞)(k^(-#1 k))&]

Evaluation at an exact point does not find an exact numeric representation:

h[3]

Evaluation at an approximate point yields a numeric result:

h[3`50]

Automatic simplification of symbolic inverses:

f[InverseFunction[f][x]]

OpenersExampleSection2(Spacer1)Properties & Relations  (3)

For arbitrary function f and point x, f((f^((-1)))(f(x)))==f(x):

f=Log[Sin[#]+1]&;
g=InverseFunction[f]

{f[x],f[g[f[x]]]}

Note that neither (f^((-1)))(f(x))==x nor f((f^((-1)))(y))==y for arbitrary x and y:

{x,g[f[x]]}

%/.x->2Pi

{y,f[g[y]]}

%/.y->2I Pi

If solutions of f(x)==a exist, (f^((-1)))(a) gives a solution of f(x)==a:

f=Cos[#]+1&;
InverseFunction[f][3/2]

f[%]==3/2

Use (RefLink[Reduce, paclet:ref/Reduce]) to find all solutions of f(x)==a:

Reduce[f[x]==3/2,x]

Use (RefLink[FindInstance, paclet:ref/FindInstance]) to find a solution of f(x)==a:

FindInstance[f[x]==3/2,x]

For non-algebraic input, (RefLink[Solve, paclet:ref/Solve]) may use (RefLink[InverseFunction, paclet:ref/InverseFunction]) to represent solutions:

Solve[Erf[x]==a,x]

Solve[f[x]==a,x]

OpenersExampleSection3(Spacer1)Possible Issues  (1)

Equations (f^((-1)))(f(x))==x and f((f^((-1)))(y))==y may not hold for arbitrary x and y:

f=Log[Sin[#]+1]&;
g=InverseFunction[f]

{x,g[f[x]]}

%/.x->2Pi

{y,f[g[y]]}

%/.y->2I Pi

({{DocumentationFooterIconsRelatedFunction.png, ({{See Also}, {(RefLink[Solve, paclet:ref/Solve])  ▪  (RefLink[InverseSeries, paclet:ref/InverseSeries])  ▪  (RefLink[Composition, paclet:ref/Composition])  ▪  (RefLink[Derivative, paclet:ref/Derivative])  ▪  (RefLink[Identity, paclet:ref/Identity])}})}})

({{DocumentationFooterIconsRelatedTutorial.png, ({{Tutorials}, {▪(RefLinkPlain[Function Names as Expressions, paclet:tutorial/FunctionNamesAsExpressions])}, {▪(RefLinkPlain[Working with Operators, paclet:tutorial/WorkingWithOperators])}, {▪(RefLinkPlain[Equations in One Variable, paclet:tutorial/EquationsInOneVariable])}})}})

({{DocumentationFooterIconsRelatedGuide.png, ({{Related Guides}, {▪(RefLinkPlain[Inverse Functions, paclet:guide/InverseFunctions])}, {▪(RefLinkPlain[Function Composition & Operator Forms, paclet:guide/FunctionCompositionAndOperatorForms])}})}})

({{(Spacer1), ({{(Spacer1)Related Links}, {(WebLink[DocumentationFooterIconsRelatedLinks-NKS.png(Spacer1)NKS|Online, http://www.wolframscience.com/nksonline/index/search.cgi?SearchIndex=InverseFunction]) ((WebLink[A New Kind of Science, http://www.wolframscience.com/nksonline/]))}})}})