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 \begin{eqnarray*} f^{-1}\circ f &=& \operatorname{id}_X, \\ f\circ f^{-1} &=& \operatorname{id}_Y, \end{eqnarray*}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