Problem 6 - College Algebra 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. $$f(x)=3 x-7 \text { and } g(x)=\frac{x+3}{7}$$

Step-by-Step Solution

Answer

The composite functions are \(f(g(x)) = x - \frac{45}{7}\) and \(g(f(x)) = x - \frac{4}{21}\). And the pair of functions \(f\) and \(g\) are not inverses of each other.

Step 1: Compute the composition f(g(x))

Insert \(g(x)\) into \(f(x)\). Here, \(g(x) = \frac{x+3}{7}\). So, \(f(g(x)) = f(\frac{x+3}{7}) = 3(\frac{x+3}{7})-7 = \frac{3x+9}{7} - 7 = x+ \frac{2}{7} - 7 = x - \frac{45}{7}\)

Step 2: Compute the composition g(f(x))

Insert \(f(x)\) into \(g(x)\). \(f(x) = 3x -7\). So, \(g(f(x)) = g(3x-7) = \frac{3x-7+3}{7} = \frac{3x-4}{7} = x - \frac{4}{21}\)

Step 3: Verify if f and g are inverses

Functions f and g are inverses of each other if and only if both \(f(g(x)) = x\) and \(g(f(x)) = x\) are satisfied. Here, neither \(f(g(x))\)-case nor \(g(f(x))\)-case result in \(x\), therefore, based on our results, functions f and g are not inverses of each other.