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Chapter 1: Problem 84

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

Short Answer

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

Step by step solution

01

Finding the Inverse of Function f

To find the inverse of the function \(f(x)=\frac{1}{8} x-3\), interchange \(x\) and \(y\), then solve for \(y\). The equation becomes \(x=\frac{1}{8} y-3\), and solving for \(y\) gives \(f^{-1}(x) = 8x + 24\)
02

Finding the Inverse of Function g

For the function \(g(x)=x^{3}\), its inverse can be found by interchanging \(x\) and \(y\), then solving for \(y\). This procedure gives \(g^{-1}(x) = \sqrt[3]{x}\)
03

Calculate \(f^{-1}(-3)\)

Substitute \(x = -3\) into \(f^{-1}(x)\): \(f^{-1}(-3) = 8*(-3) + 24 = -24 + 24 = 0\).
04

Compose \(g^{-1}(f^{-1}(-3))\)

Now insert the result \(f^{-1}(-3) = 0\) into \(g^{-1}(x)\) to find \(g^{-1}(f^{-1}(-3)) = g^{-1}(0) = \sqrt[3]{0} = 0\).

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