Given a function 
, its inverse 
is defined by
 |
(1)
|
Therefore, 
and 
are reflections about the line 
. In the Wolfram
Language, inverse functions are represented using InverseFunction[f].
As noted by Feynman (1997), the notation 
is unfortunate because it conflicts with the common
interpretation of a superscripted quantity as indicating a power,
i.e., 
.
It is therefore important to keep in mind that the symbols 
, 
, etc., refer to the inverse
sine, inverse cosine, etc., and not
to 
,
, etc.
A function 
admits an inverse function 
(i.e., "
is invertible") iff
it is bijective. However, inverse functions are commonly
defined for elementary functions that are multivalued
in the complex plane. In such cases, the inverse
relation holds on some subset of the complex plane but, over the whole plane, either
or both parts of the identity 
may fail to hold. A few examples
are illustrated above and in the following table. In the table 
and 
.
 |  |  |  |
 |  |  |  |
 |  |  |  |
 |  |  |  |
 |  |  |  |
 |  |  |  |
An additional counterintuitive property of inverse functions is that
![sqrt(z)sqrt(1/z)={-1 for I[z]=0 and R[z]<0; undefined for z=0; 1 otherwise,](/images/equations/InverseFunction/NumberedEquation2.svg) |
(2)
|
so the expected identity does not hold along the negative real axis.
