Find \( f(g(x)) \) and \( g(f(x)) \) and determine whether the pair of functions \( f \) and \( g \) are inverses of each other. \( f(x)=4 x+2 \) and \( g(x)=\frac{x-2}{4} \) a. \( f(g(x))=\square \) (Simplify your answer.)

To find \( f(g(x)) \), we first substitute \( g(x) \) into \( f(x) \): \[ g(x) = \frac{x-2}{4} \] Now, substituting \( g(x) \) into \( f(x) \): \[ f(g(x)) = f\left(\frac{x-2}{4}\right) = 4\left(\frac{x-2}{4}\right) + 2 \] When we simplify this: \[ f(g(x)) = (x-2) + 2 = x \] So, we have: \[ f(g(x)) = x \] Now, let’s find \( g(f(x)) \): First, we substitute \( f(x) \) into \( g(x) \): \[ f(x) = 4x + 2 \] Now substituting \( f(x) \) into \( g(x) \): \[ g(f(x)) = g(4x + 2) = \frac{(4x + 2) - 2}{4} \] Simplifying this gives: \[ g(f(x)) = \frac{4x}{4} = x \] So, we have: \[ g(f(x)) = x \] Since both \( f(g(x)) = x \) and \( g(f(x)) = x \), the functions \( f \) and \( g \) are indeed inverses of each other.