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Inverse function

For a bijection f:A→B, the unique function f^{-1}:B→A with f^{-1}(f(a))=a
Inverse function

Let f ⁣:A→Bf\colon A\to B be a . The inverse function of ff is the f−1 ⁣:B→Af^{-1}\colon B\to A defined by

f−1(b)= the unique a∈A such that f(a)=b. f^{-1}(b)=\text{ the unique }a\in A\text{ such that }f(a)=b.

It is characterized by the identities

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 ∘\circ is and id\mathrm{id} is the .

Examples:

  • If f ⁣:Z→Zf\colon\mathbb{Z}\to\mathbb{Z} is f(n)=n+1f(n)=n+1, then f−1(m)=m−1f^{-1}(m)=m-1.
  • If g ⁣:R→(0,∞)g\colon\mathbb{R}\to(0,\infty) is g(x)=exg(x)=e^x, then g−1(y)=ln⁡yg^{-1}(y)=\ln y.