Knowlpedia

Inverse function

A function that undoes a bijective function

If f:A→Bf:A\to B is a , its inverse function is the function f−1:B→Af^{-1}:B\to A for which f−1(b)f^{-1}(b) is the unique a∈Aa\in A satisfying f(a)=bf(a)=b.

Equivalent characterization

Equivalently,

f−1∘f=idAandf∘f−1=idB,f^{-1}\circ f=\mathrm{id}_A\quad\text{and}\quad f\circ f^{-1}=\mathrm{id}_B,

where idA\mathrm{id}_A and idB\mathrm{id}_B are .

Remarks

The notation f−1f^{-1} is also used for the of a subset, an operation defined even when ff is not bijective.

Examples
  • The inverse of f:R→Rf:\mathbb R\to\mathbb R, f(x)=x3f(x)=x^3, is f−1(y)=y3f^{-1}(y)=\sqrt[3]{y}.
  • The inverse of f:Z→Zf:\mathbb Z\to\mathbb Z, f(n)=n+1f(n)=n+1, is f−1(m)=m−1f^{-1}(m)=m-1.