FANDOM


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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