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 \({f^{-1}}.\)
