Find \( f(g(x)) \) and \( g(f(x)) \) and determine whether the pair of functions \( f \) and \( g \) are inverses of each other. \( f(x)=3 x \) and \( g(x)=\frac{x}{3} \) a. \( f(g(x))=\square \)

To find \( f(g(x)) \), we first substitute \( g(x) \) into \( f(x) \): \[ g(x) = \frac{x}{3} \] Now we calculate \( f(g(x)) \): \[ f(g(x)) = f\left(\frac{x}{3}\right) = 3\left(\frac{x}{3}\right) = x \] Now let's find \( g(f(x)) \): Substituting \( f(x) \) into \( g(x) \): \[ f(x) = 3x \] Now we calculate \( g(f(x)) \): \[ g(f(x)) = g(3x) = \frac{3x}{3} = x \] So, we have \( f(g(x)) = x \) and \( g(f(x)) = x \). Since both compositions result in \( x \), the functions \( f \) and \( g \) are indeed inverses of each other. Thus, the final outputs are: - \( f(g(x)) = x \) - \( g(f(x)) = x \)