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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
Yes, the given pair of functions \(f\) and \(g\) are inverses of each other as \(f(g(x)) = x\) and \(g(f(x)) = x\).

Step by step solution

01

Compute \(f(g(x))\)

To find the composite function \(f(g(x))\), substitute \(g(x)\) into function \(f\). Thus, \(f(g(x)) = f\left(\frac{2}{x}+5\right) = \frac{2}{\left(\frac{2}{x} + 5\right) - 5}.\n\nSimplify the expression by making the denominators same and then simplify the numerator to achieve \(f(g(x)) = x\).
02

Compute \(g(f(x))\)

To find the composite function \(g(f(x))\), substitute \(f(x)\) into function \(g\). Thus, \(g(f(x)) = g\left(\frac{2}{x-5}\right) = \frac{2}{\frac{2}{x-5}} + 5.\n\nSimplify the expression by making the denominators same and then simplify the numerator to achieve \(g(f(x)) = x\).
03

Identify function inverse

As both \(f(g(x)) = x\) and \(g(f(x)) = x\), we can conclude that the functions \(f\) and \(g\) are inverses of each other.

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