inverse function

DefinitionSuppose $f:X\to Y$ 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 $f$ isinvertible 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 invertiblenear $x$.) 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 coincidein 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 onlyif $f$ is a bijection, that is, $f$ is an injection and a surjection.

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

5. For differentiable functions between Euclidean spaces, the inverse functiontheorem gives a necessary and sufficient condition for the inverse to exist.This can be generalized to maps between Banach spaces which are differentiablein the sense of Frechet.

Remarks

When $f$ is a linear mapping (for instance, a matrix), the term non-singular isalso 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