inverse function

Definition Suppose f:XY is a functionMathworldPlanetmath between sets X and Y, and suppose f-1:YX is a mapping that satisfies

f-1f = idX,
ff-1 = idY,

where idA denotes the identity functionMathworldPlanetmath on the set A. Then f-1 is called the inversePlanetmathPlanetmathPlanetmathPlanetmathPlanetmath of f, or the inverse function of f. If f has an inverse near a point xX, then f is invertiblePlanetmathPlanetmath near x. (That is, if there is a set U containing x such that the restrictionPlanetmathPlanetmathPlanetmath of f to U is invertible, then f is invertible near x.) If f is invertible near all xX, then f is invertible.


  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 AY under a function f:XY. If f is a bijection, then f-1(y)=f-1({y}).

  3. 3.

    The inverse function of a function f:XY exists if and only if f is a bijection, that is, f is an injectionMathworldPlanetmath and a surjectionMathworldPlanetmath.

  4. 4.

    A linear mapping between vector spacesMathworldPlanetmath is invertible if and only if the determinantDlmfMathworldPlanetmath of the mapping is nonzero.

  5. 5.

    For differentiable functions between Euclidean spaces, the inverse function theoremMathworldPlanetmath gives a necessary and sufficient condition for the inverse to exist. This can be generalized to maps between Banach spaces which are differentiableMathworldPlanetmath in the sense of Frechet.


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