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Chapter 1: Problem 15

Verify that \(f\) and \(g\) are inverse functions. $$f(x)=x^{3}+5, \quad g(x)=\sqrt[3]{x-5}$$

Short Answer

Expert verified
The functions \(f(x)=x^{3}+5\) and \(g(x)=\sqrt[3]{x-5}\) are indeed inverse functions of each other since both \(f(g(x))\) and \(g(f(x))\) equal \(x\).

Step by step solution

01

Calculate f(g(x))

First, replace the \(x\) in \(f(x)\) by \(g(x)\), which yields: \(f(g(x))=f(\sqrt[3]{x-5})\). Then we replace \(x\) in \(f(x)=x^{3}+5\) with \(\sqrt[3]{x-5}\) to obtain \((\sqrt[3]{x-5})^{3}+5\) which simplifies to \(x\).
02

Calculate g(f(x))

Now, replace the \(x\) in \(g(x)\) by \(f(x)\), which yields: \(g(f(x))=g(x^{3}+5)\). Then we replace \(x\) in \(g(x)=\sqrt[3]{x-5}\) with \(x^{3}+5\) to obtain \(\sqrt[3]{(x^{3}+5)-5}\) which simplifies to \(x\).
03

Conclusion

Since \(f(g(x)) = x\) and \(g(f(x)) = x\), we can conclude that \(f\) and \(g\) are inverse functions of each other.

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