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Chapter 1: Problem 16

Verify that \(f\) and \(g\) are inverse functions. $$f(x)=\frac{x^{3}}{2}, \quad g(x)=\sqrt[3]{2 x}$$

Short Answer

Expert verified
Yes, the functions \(f(x)=\frac{x^{3}}{2}\) and \(g(x)=\sqrt[3]{2 x}\) are inverses of each other.

Step by step solution

01

Compute f(g(x))

Firstly compute the value of \(f(g(x))\). This is done by replacing each \(x\) in \(f(x)\) with \(g(x)\). Hence, \(f(g(x))=f(\sqrt[3]{2 x})=\frac{(\sqrt[3]{2 x})^{3}}{2}\). Simplifying this, you get \(f(g(x)) = x\).
02

Compute g(f(x))

Secondly, compute the value of \(g(f(x))\). This is done by replacing each \(x\) in \(g(x)\) with \(f(x)\). Hence, \(g(f(x))=g(\frac{x^{3}}{2})=\sqrt[3]{2 \cdot (\frac{x^{3}}{2})}\). Simplifying this, you get \(g(f(x)) = x\).
03

Conclude on the result

Since both \(f(g(x)) = x\) and \(g(f(x)) = x\), it's confirmed that functions \(f\) and \(g\) are indeed inverse functions.

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