Watch the video and then solve the problem given below. Click here to watch the video. Find \( f(g(x)) \) and \( g(f(x)) \) and determine whether the pair of functions \( f \) and \( g \) are inverses of each other. \( f(x)=3 x+1 \) and \( g(x)=\frac{x-1}{3} \) a. \( f(g(x))=x \) (Simplify your answer.) b. \( g(f(x))=\square \) (Simplify your answer.)

To find \( f(g(x)) \) and \( g(f(x)) \), let's start with the first function composition. 1. First, calculate \( f(g(x)) \): \[ g(x) = \frac{x-1}{3} \] Now 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 gives: \[ f(g(x)) = x - 1 + 1 = x \] So, \( f(g(x)) = x \). 2. Next, calculate \( g(f(x)) \): \[ f(x) = 3x + 1 \] Now substitute \( f(x) \) into \( g(x) \): \[ g(f(x)) = g(3x+1) = \frac{(3x+1) - 1}{3} \] Simplifying this gives: \[ g(f(x)) = \frac{3x}{3} = x \] In summary: a. \( f(g(x)) = x \) b. \( g(f(x)) = x \) Since \( f(g(x)) = x \) and \( g(f(x)) = x \), the pair of functions \( f \) and \( g \) are indeed inverses of each other!