Find \( f(g(x)) \) and \( g(f(x)) \) and determine whether the pair of functions \( f \) and \( g \) are inverses of each other. \( f(x)=9 x-5 \) and \( g(x)=\frac{x+9}{5} \) a. \( f(g(x))=\frac{9 x+56}{5} \) (Simplify your answer. Use integers or fractions for any numbers in the expression.) b. \( g(f(x))=\square \) (Simplify your answer. Use integers or fractions for any numbers in the expression.)

To determine \( f(g(x)) \) and \( g(f(x)) \), let’s start with each function. First, we calculate \( f(g(x)) \): 1. We know \( g(x) = \frac{x+9}{5} \). 2. Plugging this into \( f(x) \): \[ f(g(x)) = f\left(\frac{x+9}{5}\right) = 9\left(\frac{x+9}{5}\right) - 5 = \frac{9(x+9)}{5} - 5 = \frac{9x + 81}{5} - \frac{25}{5} = \frac{9x + 56}{5}. \] Thus, \( f(g(x)) = \frac{9x + 56}{5} \). Next, we calculate \( g(f(x)) \): 1. We know \( f(x) = 9x - 5 \). 2. Plugging this into \( g(x) \): \[ g(f(x)) = g(9x - 5) = \frac{(9x - 5) + 9}{5} = \frac{9x + 4}{5}. \] So, \( g(f(x)) = \frac{9x + 4}{5} \). Finally, we need to check if \( f \) and \( g \) are inverses. For functions to be inverses, it must be true that \( f(g(x)) = x \) and \( g(f(x)) = x \). - For \( f(g(x)) \): \[ f(g(x)) = \frac{9x + 56}{5} \quad \text{is not equal to } x. \] - For \( g(f(x)) \): \[ g(f(x)) = \frac{9x + 4}{5} \quad \text{is not equal to } x. \] Since neither condition is satisfied, \( f \) and \( g \) are not inverses of each other.