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Chapter 2: Problem 10

Find \(f(g(x))\) and \(g(f(x))\) and determine whether each pair of functions \(f\) and \(g\) are inverses of each other. $$f(x)=\sqrt[3]{x-4} \text { and } g(x)=x^{3}+4$$

Short Answer

Expert verified
The pair of functions \(f\) and \(g\) are inverses of each other as they satisfy the inverse property \(f(g(x)) = x\) and \(g(f(x)) = x\).

Step by step solution

01

Calculate f(g(x))

Apply function \(g(x)\) to function \(f\). So, calculating \(f(g(x))\) gives \(f(x^{3}+4)\), replacing \(x\) in \(f(x)\) by \(x^{3}+4\) will result in \(f(x^{3}+4) = \sqrt[3]{(x^{3}+4)-4} = \sqrt[3]{x^{3}}\). The cube root of \(x^{3}\) is \(x\). So, \(f(g(x)) = x\).
02

Calculate g(f(x))

Now, apply function \(f(x)\) to function \(g\). So, \(g(f(x))\) is calculated as \(g(\sqrt[3]{x-4})\), replacing \(x\) in \(g(x)\) by \(\sqrt[3]{x-4}\) gives \(g(\sqrt[3]{x-4}) = ((\sqrt[3]{x-4})^{3}+4) = x-4+4 = x\). So, \(g(f(x)) = x\).
03

Check the Conditions of Inverse

In order for functions \(f\) and \(g\) to be inverses of each other, they must satisfy the property that \(f(g(x)) = x\) and \(g(f(x)) = x\) for all \(x\) in the domain. Since they both yield \(x\), functions \(f(x)\) = \(\sqrt[3]{x-4}\) and \(g(x)\) = \(x^{3}+4\) are indeed inverses of each other.

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Most popular questions from this chapter

a. Use a graphing utility to graph \(f(x)=x^{2}+1\) b. Graph \(f(x)=x^{2}+1, g(x)=f(2 x), h(x)=f(3 x)\) and \(k(x)=f(4 x)\) in the same viewing rectangle. c. Describe the relationship among the graphs of \(f, g, h\) and \(k,\) with emphasis on different values of \(x\) for points on all four graphs that give the same \(y\) -coordinate. d. Generalize by describing the relationship between the graph of \(f\) and the graph of \(g,\) where \(g(x)=f(c x)\) for \(c>1\) e. Try out your generalization by sketching the graphs of \(f(c x)\) for \(c=1, c=2, c=3,\) and \(c=4\) for a function of your choice.

Group members should consult an almanac, newspaper, magazine, or the Internet to find data that lie approximately on or near a straight line. Working by hand or using a graphing utility, construct a scatter plot for the data. If working by hand, draw a line that approximately fits the data and then write its equation. If using a graphing utility, obtain the equation of the regression line. Then use the equation of the line to make a prediction about what might happen in the future. Are there circumstances that might affect the accuracy of this prediction? List some of these circumstances.

Prove that the equation of a line passing through \((a, 0)\) and \((0, b)(a \neq 0, b \neq 0)\) can be written in the form \(\frac{x}{a}+\frac{y}{b}=1 .\) Why is this called the intercept form of a line?

Begin by graphing the absolute value function, \(f(x)=|x| .\) Then use transformations of this graph to graph the given function. $$ g(x)=|x+3| $$

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