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

Step-by-Step Solution

Answer

The composite functions \(f(g(x))\) and \(g(f(x))\) both yield \(x\), indicating that \(f(x) = 3x+8\) and \(g(x) = (x - 8) / 3\) are inverse functions of each other.

Step 1: Compute the Composite Function \(f(g(x))\)

Replace \(x\) with the function \(g(x)\) in \(f(x)\). Hence, \(f(g(x)) = 3g(x) + 8 = 3(\frac{x-8}{3}) + 8 = x .\)

Step 2: Compute the Composite Function \(g(f(x))\)

Replace \(x\) with \(f(x)\) in \(g(x)\). Hence, \(g(f(x)) = \frac{f(x)-8}{3} = \frac{3x + 8 - 8}{3} = x.\)

Step 3: Check if f and g are Inverse Functions

By definition, if \(f(g(x)) = g(f(x)) = x\), then \(f\) and \(g\) are inverse functions. From Steps 1 and 2, we see that \(f(g(x)) = g(f(x)) = x\). Hence, \(f\) and \(g\) are indeed inverses of each other.