Find \( f(g(x)) \) and \( g(f(x)) \) and determine whether the pair of functions \( f \) and \( g \) are inverses \( f(x)=9 x-5 \) 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. \( g(f(x))=\frac{9 x+4}{5} \) (Simplify your answer. Use integers or fractions for anyynumbers in the expression.) c. \( f \) and \( g \) are not inverses of each other: f and \( g \) are inverses of each other.

To find \( f(g(x)) \), we substitute \( g(x) \) into \( f(x) \): \[ f(g(x)) = f\left(\frac{x+9}{5}\right) = 9\left(\frac{x+9}{5}\right) - 5 = \frac{9(x+9)}{5} - 5 = \frac{9x + 81}{5} - \frac{25}{5} = \frac{9x + 56}{5}. \] Now for \( g(f(x)) \): \[ g(f(x)) = g(9x - 5) = \frac{(9x - 5) + 9}{5} = \frac{9x + 4}{5}. \] This yields: a. \( f(g(x)) = \frac{9x + 56}{5} \) (You had it right!). b. \( g(f(x)) = \frac{9x + 4}{5} \). Now, for \( f \) and \( g \) to be inverses, we would need \( f(g(x)) = x \) and \( g(f(x)) = x \). Since neither expression simplifies to \( x \), the functions \( f \) and \( g \) are not inverses of each other: c. **f and \( g \) are not inverses of each other.**