Question

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(f(x)=x^{n}\) where \(n\) is odd, then \(f^{-1}\) exists.

Short answer

The statement is true. The inverse function of \(f(x)=x^{n}\) does exist when \(n\) is an odd integer.

Step-by-step solution

Understand the Statement

The exercise states that if a function \(f(x) = x^{n}\) is given, with \(n\) being an odd integer, then the inverse function \(f^{-1}(x)\) exists. The question asks to determine if this statement is true, and if false, provide an explanation or example.

Check Goals for Inverse Existence

In order to have an inverse, a function must be both injective (one-to-one) and surjective (onto). The power function with an odd integer exponent is indeed both injective and surjective. This is because for every unique \(x\) value in the domain, there's a unique \(f(x)\) in the codomain, and also every element in the codomain has a corresponding \(x\) in the domain.

Evaluate Statement

Considering that the function \(f(x) = x^{n}\) with \(n\) being odd is both one-to-one and onto, an inverse function \(f^{-1}\) would indeed exist. This means the original statement is true.