Let f be a one-to-one correspondence from A to B. The inverse function of f is the function that assigns to each element b of B an element a of A such that f(a)=b. We denote the inverse function f −1.

f −1(b) = a when f(a) = b.

The identity function on A is iA:AA where iA(a)=a for each aA. This is a one-to-one correspondence and so is invertible. It is, in fact, its own inverse.

The function f:ZZ such that f(a)=a−4 for each aZ is a one-to-one correspondence and f −1(a)=a+4.