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Chapter 2: 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
Both \(f(g(x)) = x\) and \(g(f(x)) = x\); therefore, the given functions \(f(x) = \sqrt[3]{x-4}\) and \(g(x) = x^{3}+4\) are inverses of each other.

Step by step solution

01

Compute \(f(g(x))\)

Firstly, we calculate the composition \(f(g(x))\). We replace \(x\) in the given function \(f(x) = \sqrt[3]{x-4}\) with \(g(x) = x^{3}+4\). So \(f(g(x)) = \sqrt[3]{(x^{3}+4)-4}\). This simplifies to \(f(g(x)) = \sqrt[3]{x^{3}} = x\).
02

Compute \(g(f(x))\)

We perform the same operation, this time computing \(g(f(x))\). We replace \(x\) in the function \(g(x) = x^{3}+4\) with \(f(x) = \sqrt[3]{x-4}\). Thus, \(g(f(x)) = (\sqrt[3]{x-4})^{3}+4 = x-4+4 = x\).
03

Conclude

Since both \(f(g(x))\) and \(g(f(x))\) simplify to \(x\), we can conclude that \(f\) and \(g\) are indeed inverse functions of each other. This aligns with the key property of inverse functions.

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