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Chapter 1: Problem 8

Find \(f(g(x))\) and \(g(f(x))\) and determine whether each pair of functions \(f\) and \(g\) are inverses of each other. $$f(x)=\frac{2}{x-5} \text { and } g(x)=\frac{2}{x}+5$$

Short Answer

Expert verified
The composite functions \(f(g(x))\) and \(g(f(x))\) are indeed \(x\). Hence, \(f(x)\) and \(g(x)\) are inverses of each other.

Step by step solution

01

Find \(f(g(x))\)

Substitute \(g(x)\) into \(f(x)\), this means wherever there's \(x\) in \(f(x)\), replace it with \({g(x)}\).\nSo, we get \(f(g(x))\) = \(f\left(\frac{2}{x} + 5\right)\) = \(\frac{2}{\frac{2}{x}+5-5}\) = \(\frac{2}{\frac{2}{x}}\) = \(x\).
02

Find \(g(f(x))\)

Similar to the above, replace \(x\) in \(g(x)\) with \(f(x)\) to get \(g(f(x))\). So we obtain \(g\left(\frac{2}{x-5}\right)\)=\(\frac{2}{\frac{2}{x-5}} + 5\) = \(x-5 + 5 = x\)
03

Determine the Inverseness

The functions \(f\) and \(g\) are inverses of one another if \(f(g(x)) = x\) and \(g(f(x)) = x\). As we have proven that both \(f(g(x))\) and \(g(f(x))\) equal \(x\), we can conclude that \(f\) and \(g\) are indeed inverses of each other.

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Most popular questions from this chapter

Will help you prepare for the material covered in the next section. Write an algebraic expression for the fare increase if a 200 dollars plane ticket is increased to \(x\) dollars.

$$\text { Solve and check: } \frac{x-1}{5}-\frac{x+3}{2}=1-\frac{x}{4}$$

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