Determine if the functions \( f(x) = 3x + 1 \) and \( g(x) = \frac{x - 1}{3} \) are inverse functions.

To determine if the functions \( f(x) = 3x + 1 \) and \( g(x) = \frac{x - 1}{3} \) are inverse functions, we need to verify if \( f(g(x)) = x \) and \( g(f(x)) = x \). First, let's calculate \( f(g(x)) \): 1. Substitute \( g(x) \) into \( f(x) \): \[ f(g(x)) = f\left(\frac{x - 1}{3}\right) = 3\left(\frac{x - 1}{3}\right) + 1 \] Simplifying this: \[ = x - 1 + 1 = x \] Now, let's calculate \( g(f(x)) \): 1. Substitute \( f(x) \) into \( g(x) \): \[ g(f(x)) = g(3x + 1) = \frac{(3x + 1) - 1}{3} \] Simplifying this: \[ = \frac{3x}{3} = x \] Since both \( f(g(x)) = x \) and \( g(f(x)) = x \) hold true, we can conclude that \( f(x) \) and \( g(x) \) are indeed inverse functions of each other!