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Chapter 1: Problem 11

Find the inverse function of \(f\) informally. Verify that \(f\left(f^{-1}(x)\right)=x\) and \(f^{-1}(f(x))=x\). $$f(x)=\sqrt[3]{x}$$

Short Answer

Expert verified
The inverse function of \(f(x) = \sqrt[3]{x}\) is \(f^{-1}(x) = x^3\). The relations \(f\left(f^{-1}(x)\right) = x\) and \(f^{-1}\left(f(x)\right) = x\) have been verified, which confirms \(f^{-1}(x) = x^3\) is indeed the inverse function of \(f(x) = \sqrt[3]{x}\).

Step by step solution

01

Find the inverse function

To find the inverse function \(f^{-1}(x)\) of \(f(x) = \sqrt[3]{x}\), swap the roles of \(x\) and \(y\) to get \(x = \sqrt[3]{y}\). Then, solve this equation for \(y\). Cubing both sides results in \(y = x^3\). So, \(f^{-1}(x) = x^3\).
02

Validate \(f\left(f^{-1}(x)\right) = x\)

Substitute \(f^{-1}(x)\) into the equation of \(f(x)\): \[f\left(f^{-1}(x)\right) = f(x^3) = \sqrt[3]{x^3} = x\]
03

Validate \(f^{-1}\left(f(x)\right) = x\)

Substitute \(f(x)\) into the equation of \(f^{-1}(x)\): \[f^{-1}\left(f(x)\right) = f^{-1}\left(\sqrt[3]{x}\right) = \left(\sqrt[3]{x}\right)^3 = x\]

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