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Chapter 1: Problem 105

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 \((f^{-1} \circ g^{-1})(1)\) is 32.

Step by step solution

01

Find the inverse of function \(g\)

Function \(g\) is given as \(g(x)=x^3\). To find its inverse, replace \(g(x)\) by \(y\): \(y = x^3\). The next step is to swap \(x\) and \(y\): \(x = y^3\). Finally, to solve for \(y\), take the cubed root of both sides: \(y = \sqrt[3]{x}\). So, \(g^{-1}(x) = \sqrt[3]{x}\).
02

Evaluate \(g^{-1}(1)\)

Now, we have to evaluate \(g^{-1}(1)\), or substitute \(1\) into \(g^{-1}(x)\): \(g^{-1}(1) = \sqrt[3]{1}\). The cubed root of 1 is 1, so \(g^{-1}(1) = 1\).
03

Find the inverse of function \(f\)

Function \(f\) is given as \(f(x)=\frac{1}{8} x-3\). To find its inverse, again, replace \(f(x)\) with \(y\): \(y = \frac{1}{8} x-3\). After swapping \(x\) and \(y\), we get \(x = \frac{1}{8} y-3\). Solving for \(y\) gives us \(y = 8x + 24\). So, \(f^{-1}(x) = 8x + 24\).
04

Evaluate \(f^{-1}(g^{-1}(1))\)

Finally, we evaluate \(f^{-1}(g^{-1}(1))\) by substituting the value of \(g^{-1}(1)\) into \(f^{-1}(x)\): \(f^{-1}(1) = 8*1 + 24 = 32\).

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Most popular questions from this chapter

An air traffic controller spots two planes flying at the same altitude. Their flight paths form a right angle at point \(P\). One plane is 150 miles from point \(P\) and is moving at 450 miles per hour. The other plane is 200 miles from point \(P\) and is moving at 450 miles per hour. Write the distance \(s\) between the planes as a function of time \(t.\)

Find the domain of the function.$$f(x)=\sqrt{100-x^{2}}$$.

Determine whether the function is even, odd, or neither (a) algebraically, (b) graphically by using a graphing utility to graph the function, and (c) numerically by using the table feature of the graphing utility to compare \(f(x)\) and \(f(-x)\) for several values of \(x\). $$h(x)=x^{5}-4 x^{3}$$

Determine whether the statement is true or false. Justify your answer. Given two functions \(f\) and \(g,\) you can calculate \((f \circ g)(x)\) if and only if the range of \(g\) is a subset of the domain of \(f\).

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