Problem 8 - College Algebra

Question

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

Step-by-Step Solution

Answer

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 1: 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 \(g(f(x)) = x+5\).

Step 2: 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.