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Chapter 5: Problem 95

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

Short Answer

Expert verified
\(\left(f^{-1}\right)^{-1}=f\), i.e., the inverse of the inverse of a function is the function itself, which is proved using the definition of inverse functions.

Step by step solution

01

Understanding Inverse Functions

Firstly, it is essential to understand the meaning and concept of an inverse function. By definition, for a function \(f: X \rightarrow Y\), an inverse function \(f^{-1}: Y \rightarrow X\) exists if the function \(f\) is bijective, meaning it is both injective (or one-to-one) and surjective (or onto). In simpler terms, if \(f(x) = y\), then \(f^{-1}(y) = x\).
02

Begin the proof

To prove that \(\left(f^{-1}\right)^{-1}=f\), we need to show that for any value of \(x\) in the domain, \((f^{-1})^{-1}(x) = f(x)\). Let's start by taking an arbitrary element \(x\) from the domain of \(f\).
03

Applying the Definition of an Inverse Function

Let \(y = f^{-1}(x)\). According to the definition of an inverse function, \(f(y) = x\). Replacing \(y\), we get \(f(f^{-1}(x)) = x\), which is the definition of an inverse function.
04

Final Conclusion

Since \(\left(f^{-1}\right)^{-1}(x) = f(x)\), we have successfully proven that \(\left(f^{-1}\right)^{-1}=f\). The assumption at the beginning of the proof that \(x\) was any element in the domain of \(f\) ensures the conclusion is valid for all such \(x\).

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