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# 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

## Mathematics Subject Classification

03-00*no label found*03E20

*no label found*

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