If f:X->Y is a function then a function g:Y->X is an inverse of f iff fg=1Y and gf=1X. (Here 1X,1Y are the identity functions on X,Y.)

Notes

  • If an inverse of f exists it is unique and we usually denote it by f-1.
  • f has an inverse iff it is bijective.
  • The function f:R->R defined by f(x)=x+1 has an inverse defined by f-1(x)=x-1 but the same rule on the positive real numbers gives an injective (but not bijective) function h:R+->R+ which is not invertible (essentially because x -> x-1 is not well defined as a function R+->R+).