Problem 7 - 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. $$f(x)=\frac{3}{x-4} \text { and } g(x)=\frac{3}{x}+4$$

Step-by-Step Solution

Answer

The two functions \(f(x)=\frac{3}{x-4}\) and \(g(x)=\frac{3}{x}+4\) are inverse functions of each other.

Step 1: Find \(f(g(x))\)

Substitute \(g(x)\) into \(f(x)\), i.e. \(f(g(x))=f\left(\frac{3}{x}+4\right)=\frac{3}{\left(\frac{3}{x}+4\right)-4}\). Simplifying this results in \(x\).

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

Now substitute \(f(x)\) into \(g(x)\), i.e. \(g(f(x))=g\left(\frac{3}{x-4}\right)=\frac{3}{\left(\frac{3}{x-4}\right)}+4\). Simplifying this we also get \(x\).

Step 3: Determine if the functions are inverses

Since we found that both \(f(g(x))\) and \(g(f(x))\) are equal to \(x\), these two functions are indeed inverses of each other.