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InverseFunction
InverseFunction[
f
] represents the inverse of the function f, defined so that InverseFunction[
f
][
y
] gives the value of x for which f
[
x
] is equal to y.
For a function with several arguments, InverseFunction[
f
] represents the inverse with respect to the first argument. InverseFunction[
f
,
n
] represents the inverse with respect to the n
 argument. InverseFunction[
f
,
n
,
tot
] represents the inverse with respect to the n
 argument when there are tot arguments in all.
In OutputForm and StandardForm, InverseFunction[
f
] is printed as  .
As discussed in Section 3.2.7, many mathematical functions do not have unique inverses. In such cases, InverseFunction[
f
] can represent only one of the possible inverses for f.
Example: InverseFunction[Sin] .
InverseFunction is generated by Solve when the option InverseFunctions is set to Automatic or True.
See the Mathematica book: Section 2.2.1, Section 2.2.9, Section 3.4.5.
See also: Solve, InverseSeries, Composition, Derivative.
Further Examples
Here are two inverse functions evaluated at y.
In[1]:= 
Out[1]= 
In[2]:= 
Out[2]= 
Composing a function with its inverse in either order is the identity operation.
In[3]:= 
InverseFunction::ifun: Warning: Inverse functions are being used. Values may be lost for multivalued inverses.
Out[3]= 
In[4]:= 
InverseFunction::ifun: Warning: Inverse functions are being used. Values may be lost for multivalued inverses.
Out[4]= 
The inverse of a composition is another composition.
In[5]:= 
Out[5]= 
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