Question

Prove that if a function has an inverse function, then the inverse function is unique.

Short answer

The inverse function of a given function is unique.

Step-by-step solution

Definition of a Function and its Inverse

By definition, a function \(f: A \rightarrow B\) has an inverse \(f^{-1}: B \rightarrow A\) if and only if for every \(b \in B\), there exists a unique \(a \in A\) such that \(f(a) = b\). Similarly, \(f^{-1}(b) = a\) whenever \(f(a) = b\).

Assume two inverse functions

Now, let's assume that a function \(f\) has two inverse functions, denoted as \(g\) and \(h\). That means for every \(b \in B\), there exists \(a \in A\) such that \(f(a) = b\) and both \(g(b) = a\) and \(h(b) = a\).

Conclude the uniqueness of the inverse function

From Step 2, both \(g(b)\) and \(h(b)\) return the same result \(a\), meaning \(g = h\). Therefore, the inverse function of \(f\) is unique.