PlanetMath: inverse function

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inverse function

(Definition)

Definition Suppose $f:X\to Y$ is a function between sets and , 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

. Then $f^{-1}$ is called the inverse of

, or the inverse function of

. If

has an inverse near a point $x\in X$ , then

is invertible near

. (That is, if there is a set

containing

such that the restriction of

is invertible, then

is invertible near

.) If

is invertible near all $x\in X$ , then

is invertible.

When an inverse function exists, it is unique.
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 is a bijection, then $f^{-1}(y)=f^{-1}(\{y\})$ .
The inverse function of a function $f:X\to Y$ exists if and only if is a bijection, that is, is an injection and a surjection.
A linear mapping between vector spaces is invertible if and only if the determinant of the mapping is nonzero.
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.

When

is a linear mapping (for instance, a matrix), the term non-singular is also used as a synonym for invertible.

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"inverse function" is owned by matte. [ full author list (3) ]

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See Also: function

Other names:

non-singular function, nonsingular function, non-singular, nonsingular, inverse

Also defines:

invertible function, invertible

This object's parent.

Attachments:: derivative of inverse function (Theorem) by pahio
inverse of composition of functions (Theorem) by Wkbj79

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AMS MSC:	03-00 (Mathematical logic and foundations :: General reference works )
	03E20 (Mathematical logic and foundations :: Set theory :: Other classical set theory )

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