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

	$\displaystyle f^{-1}\circ f$	$\displaystyle=$	$\displaystyle\operatorname{id}_{X},$
	$\displaystyle f\circ f^{-1}$	$\displaystyle=$	$\displaystyle\operatorname{id}_{Y},$

where $\operatorname{id}_{A}$ 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 non-singular is also used as a synonym for invertible.

Title	inverse function
Canonical name	InverseFunction
Date of creation	2013-03-22 13:53:52
Last modified on	2013-03-22 13:53:52
Owner	matte (1858)
Last modified by	matte (1858)
Numerical id	14
Author	matte (1858)
Entry type	Definition
Classification	msc 03-00
Classification	msc 03E20
Synonym	non-singular function
Synonym	nonsingular function
Synonym	non-singular
Synonym	nonsingular
Synonym	inverse
Related topic	Function
Defines	invertible function
Defines	invertible