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Chapter 1: Problem 83

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

Short Answer

Expert verified
The value of \( \left(f^{-1} \circ g^{-1}\right)(1) \) is 32.

Step by step solution

01

Find the inverse of function f(x)

The original function is \(f(x)=\frac{1}{8} x-3\). To find the inverse, switch the roles of x and y: \(x = \frac{1}{8}y - 3\). Solve for y to get the inverse function. Multiply both sides by 8 to get \(8x = y - 24\), then add 24 to both sides to obtain \(f^{-1}(x) = 8x + 24\).
02

Find the inverse of function g(x)

The original function is \(g(x) = x^3\). To find the inverse, switch x and y to get \(x = y^3\). Then take the cube root of both sides to obtain \(g^{-1}(x) = \sqrt[3]{x}\).
03

Find \(f^{-1} \circ g^{-1}\) (1)

The function \(f^{-1} \circ g^{-1}\) (x) is the composition of the inverses of \(f(x)\) and \(g(x)\) defined by \(f^{-1}( g^{-1}(x))\). We plug in 1 for x into \(g^{-1}\) to first evaluate \(g^{-1}(1) = \sqrt[3]{1} = 1\). Then plug this result into \(f^{-1}(x)\) to evaluate \(f^{-1}(1) = 8*1 + 24 = 32\).

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