Problem 10 - College Algebra 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 composite functions \(f(g(x))\) and \(g(f(x))\) both evaluate to \(x\). Therefore, the functions \(f(x)=\sqrt[3]{x-4}\) and \(g(x)=x^{3}+4\) are inverses of each other.

Step 1: Compute \(f(g(x))\)

To find \(f(g(x))\), substitute \(g(x)\) into \(f(x)\). That is, wherever there is \(x\) in \(f(x)\), replace it with \(g(x)\). So, \(f(g(x)) = f(x^{3}+4) = \sqrt[3]{x^{3}+4 - 4} = \sqrt[3]{x^{3}} = x\)

Step 2: Compute \(g(f(x))\)

Similarly, to find \(g(f(x))\), insert \(f(x)\) into \(g(x)\). So, \(g(f(x)) = g(\sqrt[3]{x - 4}) = (\sqrt[3]{x - 4})^{3} + 4 = x - 4 + 4 = x\)

Step 3: Check if \(f\) and \(g\) are inverses

Having found that both \(f(g(x)) = x\) and \(g(f(x)) = x\), it can be concluded that the functions \(f\) and \(g\) are indeed inverses of each other correct to the given exercise context. Inverses undo each other's operations - in this case, cubing and the cube-root. Hence, the conclusion.