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Chapter 1: Problem 87

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

Short Answer

Expert verified
\((f \circ g)^{-1}(x)=\sqrt[3]{8(x+3)}

Step by step solution

01

Find the composition of functions \(f \circ g\)

First, it is necessary to find the composite function \(f \circ g(x)\). This means plugging the function \(g(x)\) into \(f(x)\). Therefore, \(f \circ g(x)=f(g(x))=f(x^{3})=\frac{1}{8} x^{3}-3\)
02

Find the inverse of the function \(f \circ g(x)\)

To find the inverse of a function is to switch \(x\) and \(y\) so that \(y=\frac{1}{8} x^{3}-3\) becomes \(x=\frac{1}{8} y^{3}-3\). The next step is to solve this equation for \(y\). Add 3 to both sides: \(x+3=\frac{1}{8} y^{3}\). Multiply by 8 to isolate the \(y^{3}\) term: \(8(x+3)=y^{3}\). Finally, take the cube root of both sides: \(y=\sqrt[3]{8(x+3)}\)
03

Simplify the Function

The goal here is to simplify the expression of the inverse function as much as possible. For this problem, the cubic root of \(8(x+3)\) doesn't simplify much further so the solution remains: \(y=\sqrt[3]{8(x+3)}\). This is equivalent to \( (f \circ g)^{-1}(x)=\sqrt[3]{8(x+3)}\)

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