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Chapter 5: Problem 76

Use the functions \(f(x)=\frac{1}{8} x-3\) and \(g(x)=x^{3}\) to find the given value. \(\left(g^{-1} \circ f^{-1}\right)(-3)\)

Short Answer

Expert verified
The result of \(\left(g^{-1} \circ f^{-1}\right)(-3)\) is 0.

Step by step solution

01

Find the Inverse of f(x) and g(x)

To find the inverse of \(f(x) = \frac{1}{8} x - 3\), one should first switch x and y to obtain \(x = \frac{1}{8} y - 3\). Solving for y gives the inverse function \(f^{-1}(x) = 8x + 24\). For \(g(x) = x^{3}\), switching x and y gives \(x = y^{3}\). Therefore, \(g^{-1}(x) = \sqrt[3]{x}\)since cubing this function gives back x.
02

Perform the Function Composition

\(g^{-1} \circ f^{-1}(x)\) means we need to substitute \(f^{-1}(x)\) into \(g^{-1}(x)\) equation. So instead of x in \(g^{-1}(x)\), we'll put \(f^{-1}(x)\) that means \(g^{-1} \circ f^{-1}(x) = \sqrt[3]{8x + 24}\)
03

Evaluate the Function Composition at -3

Substitute -3 into the function from step 2. This gives \(\sqrt[3]{8*-3 + 24} = \sqrt[3]{0}=0\)

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