Inverse Functions

Suppose f : A → B is a function whose domain is the set A and whose codomain is the set B. The function f is called invertible if there exists a function f ⁻¹ : B → A with the domain B and the codomain A such that

where x ∈ A, y ∈ B.

The function f ⁻¹ is then called the inverse of f.

Not all functions have an inverse. If a function f is not injective, different elements in its domain may have the same image:

In this case, the converse relation $f^{-1}$ is not a function because there are two preimages $x_1$ and $x_2$ for the element $y_1$ in the codomain $B.$ So, to have an inverse, the function must be injective.

If a function $f$ is not surjective, not all elements in the codomain have a preimage in the domain. In this case, the converse relation $f^{-1}$ is also not a function.

Thus, to have an inverse, the function must be surjective.

Recall that a function which is both injective and surjective is called bijective. Hence, to have an inverse, a function $f$ must be bijective. The converse is also true. If $f:A\to B$ is bijective, then it has an inverse function $f^{-1}.$