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Chapter 1: Problem 10

Find \(f(g(x))\) and \(g(f(x))\) and determine whether each pair of functions \(f\) and \(g\) are inverses of each other. $$f(x)=\sqrt[3]{x-4} \text { and } g(x)=x^{3}+4$$

Short Answer

Expert verified
The functions \(f(x)=\sqrt[3]{x-4}\) and \(g(x)=x^{3}+4\) are inverses of each other, as \(f(g(x)) = x\) and \(g(f(x)) = x\).

Step by step solution

01

Calculate \(f(g(x))\)

First, substitute \(g(x)\) into \(f(x)\). That means, replace every \(x\) in \(f(x)\) with \(g(x)=x^{3}+4\). Therefore, \(f(g(x)) = \sqrt[3]{g(x) - 4} = \sqrt[3]{x^{3}+4 - 4}= \sqrt[3]{x^{3}}= x\).
02

Calculate \(g(f(x))\)

Now, substitute \(f(x)\) into \(g(x)\). Replace every \(x\) in \(g(x)\) with \(f(x)=\sqrt[3]{x-4}\), resulting in \(g(f(x)) = (f(x))^{3} + 4 = (\sqrt[3]{x - 4})^{3} + 4 = x-4+4 = x\).
03

Check if \(f\) and \(g\) are inverses of each other

For two functions to be inverses of each other, \(f(g(x))\) must equal \(x\) and \(g(f(x))\) must equal \(x\) as well. In this case, both \(f(g(x)) = x\) and \(g(f(x)) = x\) which indicates that \(f\) and \(g\) are inverses of each other.

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