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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
The compositions of the functions are \(f(g(x)) = \frac{2}{x}\) and \(g(f(x)) = x + 5\). Thus, the functions \(f\) and \(g\) are not inverses of each other.

Step by step solution

01

Find Function Compositions

We first calculate the function compositions. For \(f(g(x))\), substitute \(g(x)\) into \(f(x)\): \(f(g(x)) = f(\frac{2}{x}+5) = \frac{2}{(\frac{2}{x}+5)-5}\). Simplifying the expression inside the bracket gives \(f(g(x)) = \frac{2}{x}\). For \(g(f(x))\), substitute \(f(x)\) into \(g(x)\): \(g(f(x)) = g(\frac{2}{x-5}) = \frac{2}{\frac{2}{x-5}}+5\). Simplifying this gives \(g(f(x)) = x+5\).
02

Check for Inverses

To be inverses, the calculations from the previous step need to yield \(f(g(x)) = x\) and \(g(f(x)) = x\). Clearly, \(f(g(x)) = \frac{2}{x}\) which is not equal to \(x\) for all \(x\) in the domain of \(g\). Thus these functions are not inverses of each other.

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