Problem 8 - Precalculus Essentials

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

Step 1: 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 \(f(g(x))\) = \(f\left(\frac{2}{x} + 5\right)\) = \(\frac{2}{\frac{2}{x}+5-5}\) = \(\frac{2}{\frac{2}{x}}\) = \(x\).

Step 2: Find \(g(f(x))\)

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

Step 3: 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.