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+).