inverse function
Definition Suppose $f:X\to Y$ is a function^{} between sets $X$ and $Y$, and suppose ${f}^{1}:Y\to X$ is a mapping that satisfies
${f}^{1}\circ f$  $=$  ${\mathrm{id}}_{X},$  
$f\circ {f}^{1}$  $=$  ${\mathrm{id}}_{Y},$ 
where ${\mathrm{id}}_{A}$ denotes the identity function^{} on the set $A$. Then ${f}^{1}$ is called the inverse^{} of $f$, or the inverse function of $f$. If $f$ has an inverse near a point $x\in X$, then $f$ is invertible^{} near $x$. (That is, if there is a set $U$ containing $x$ such that the restriction^{} of $f$ to $U$ is invertible, then $f$ is invertible near $x$.) If $f$ is invertible near all $x\in X$, then $f$ is invertible.
Properties

1.
When an inverse function exists, it is unique.

2.
The inverse function and the inverse image of a set coincide in the following sense. Suppose ${f}^{1}(A)$ is the inverse image of a set $A\subset Y$ under a function $f:X\to Y$. If $f$ is a bijection, then ${f}^{1}(y)={f}^{1}(\{y\})$.

3.
The inverse function of a function $f:X\to Y$ exists if and only if $f$ is a bijection, that is, $f$ is an injection^{} and a surjection^{}.

4.
A linear mapping between vector spaces^{} is invertible if and only if the determinant^{} of the mapping is nonzero.

5.
For differentiable functions between Euclidean spaces, the inverse function theorem^{} gives a necessary and sufficient condition for the inverse to exist. This can be generalized to maps between Banach spaces which are differentiable^{} in the sense of Frechet.
Remarks
When $f$ is a linear mapping (for instance, a matrix), the term nonsingular is also used as a synonym for invertible.
Title  inverse function 
Canonical name  InverseFunction 
Date of creation  20130322 13:53:52 
Last modified on  20130322 13:53:52 
Owner  matte (1858) 
Last modified by  matte (1858) 
Numerical id  14 
Author  matte (1858) 
Entry type  Definition 
Classification  msc 0300 
Classification  msc 03E20 
Synonym  nonsingular function 
Synonym  nonsingular function 
Synonym  nonsingular 
Synonym  nonsingular 
Synonym  inverse 
Related topic  Function 
Defines  invertible function 
Defines  invertible 