The inverse function of a function f:D\to C is a function g:C \to D that does the opposite of f . A function has an inverse if and only if it is bijective. The inverse of a function f is denoted by f^{-1} (not to be confused with the reciprocal of f).

Given any two functions, f:D\to C and g:C\to D (notice the reversal of the domain and codomain), we say that f and g are inverses of each other, denoted f=g^{-1} and g=f^{-1} if:

  • f(g(x))=x for all x\in C
  • g(f(x))=x for all x\in D

A function that is not bijective can be "made" invertible by restricting the domain to that where the function is one-to-one and then restricting the codomain to its image on the domain restriction. For instance, the function f:\R\to \R defined by f(x)=x^2 is not bijective, and thus has no inverse, but restricting the domain of f to the interval [0,\infty) , we can obtain a function f\mid_{[0,\infty)}:[0,\infty)\to[0,\infty) defined by f\mid_{[0,\infty)}(x=x^2 , which is a bijective function.

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