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Chapter 1: Problem 88

Use the functions \(f(x)=\frac{1}{8} x-3\) and \(g(x)=x^{3}\) to find the indicated value or function. $$g^{-1} \circ f^{-1}$$

Short Answer

Expert verified
The composed function \(g^{-1} \circ f^{-1}\) equals \(\sqrt[3]{8x+24}\).

Step by step solution

01

Determine the inverses of the functions

The inverse of a function \(f(x)\) is found by switching \(x\) and \(y\) and solving for \(y\). It undoes the operations of the original function. For \(f(x)=\frac{1}{8} x-3\), exchange \(x\) and \(y\) to get \(x=\frac{1}{8} y-3\). Solving this for \(y\) gives the inverse, \(f^{-1}(x)=8x+24\). Now for \(g(x)=x^{3}\), after switching \(x\) and \(y\) you will get \(x=y^{3}\). Taking the cubic root of both sides will give \(g^{-1}(x)=\sqrt[3]{x}\).
02

Substitution and composition of functions

Composition of functions is a process of substituting one function into another. In this case, we substitute the inverse of \(f(x)\) which is \(f^{-1}(x)=8x+24\) into the inverse of \(g(x)\) which is \(g^{-1}(x)\). Therefore, \(g^{-1} \circ f^{-1} = g^{-1}(f^{-1}(x)) = g^{-1}(8x+24)\).
03

Simplification

After substitution, you need to simplify the expression. Here, you substitute \(8x+24\) into \(g^{-1}(x)=\sqrt[3]{x}\). Hence, \(g^{-1}(8x+24) = \sqrt[3]{8x+24}\). That is the final solution.

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