Problem 10 - Precalculus Essentials

Question

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

Step-by-Step Solution

Answer

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 1: 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, \(f(g(x)) = \sqrt[3]{g(x) - 4} = \sqrt[3]{x^{3}+4 - 4}= \sqrt[3]{x^{3}}= x\).

Step 2: 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 \(g(f(x)) = (f(x))^{3} + 4 = (\sqrt[3]{x - 4})^{3} + 4 = x-4+4 = x\).

Step 3: 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.