Problem 3 - 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. $$f(x)=3 x+8 \text { and } g(x)=\frac{x-8}{3}$$

Step-by-Step Solution

Answer

\(f(g(x)) = x\) and \(g(f(x)) = x\), hence \(f\) and \(g\) are inverses of each other.

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

Substitute \(g(x) = \frac{x-8}{3}\) into \(f(x) = 3x + 8\). That is, replace every 'x' in \(f(x)\) with \(\frac{x-8}{3}\). Then proceed to simplify the resulting expression.

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

Substitute \(f(x) = 3x + 8\) into \(g(x) = \frac{x-8}{3}\). That is, replace every 'x' in \(g(x)\) with \((3x + 8)\). Then proceed to simplify the resulting expression.

Step 3: Determine if \(f\) and \(g\) are inverses

Evaluate whether \(f(g(x)) = x\) and \(g(f(x)) = x\). If both equalities hold, then \(f\) and \(g\) are inverses of each other.