Question

Prove that if \(f\) has an inverse function, then \(\left(f^{-1}\right)^{-1}=f\).

Short answer

Based on the definition of the inverse function and properties of function composition, you can confirm that \(\left(f^{-1}\right)^{-1}=f\). This means that if you take the inverse of an inverse function, you get back the original function.

Step-by-step solution

Understand the definition of an inverse function

Before starting, it's important to know what an inverse function is. If \(f\) is a bijective function with an associated domain \(X\) and range \(Y\), then its inverse \(f^{-1}\) is the function that 'undoes' the action of \(f\). This means that for every \(y\) in \(Y\), there exists an \(x\) in \(X\) such that if \(f(x) = y\), then \(f^{-1}(y) = x\). We will use this property in the proof.

Start with the left-hand side

We want to prove that \(\left(f^{-1}\right)^{-1}=f\). So, let's start by considering \(\left(f^{-1}\right)^{-1}\). By definition of the inverse, \(\left(f^{-1}\right)^{-1}(y)\) should be the \(x\) such that \(f^{-1}(x) = y\).

Substitute and simplify

Next, we know that if \(f^{-1}(x) = y\), then \(f(y) = x\). So we can replace \(x\) with \(f(y)\) and find that \(\left(f^{-1}\right)^{-1}(y) = f(y)\).

Conclusion

Since \(\left(f^{-1}\right)^{-1}(y) = f(y)\), then we can assert that \(\left(f^{-1}\right)^{-1} = f\). This completes our proof.