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Chapter 2: 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
After substitution and simplification, if both \(f(g(x))\) and \(g(f(x))\) are equal to \(x\), then \(f(x)\) and \(g(x)\) are indeed inverses. Otherwise, they are not inverses.

Step by step solution

01

Finding \(f(g(x))\)

Firstly, substitute \(g(x)\) into \(f(x)\), then simplify the expression. That is, replace \(x\) in \(f(x)\) with \(g(x)\): \(f(g(x)) = f\left(\frac{2}{x} + 5\right) = \frac{2}{\left(\frac{2}{x} + 5\right) - 5}\). Simplify this to get it into standard form.
02

Finding \(g(f(x))\)

Now substitute \(f(x)\) into \(g(x)\), and simplify the expression. Replace \(x\) in \(g(x)\) with \(f(x)\): \(g(f(x)) = g\left(\frac{2}{x - 5}\right) = \frac{2}{\left(\frac{2}{x - 5}\right)} + 5\). Simplify this expression to get it in standard form.
03

Checking if \(f\) and \(g\) are Inverses

Now after simplifying both \(f(g(x))\) and \(g(f(x))\), check whether they are equal to \(x\). If they are, then \(f(x)\) and \(g(x)\) are inverses of each other. If not, they are not inverses.

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