# 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.
4. 4.
5. 5.

## 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