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Chapter 1: Problem 136

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

Short Answer

Expert verified
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

01

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.
02

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 \(f^{-1}(x) = y\).
03

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 \(\left(f^{-1}\right)^{-1}(y) = f(y)\).
04

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.

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