# 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. 1.

When an inverse function exists, it is unique.

2. 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. 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. 4.

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

5. 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