Find \( f(g(x)) \) and \( g(f(x)) \) and determine whether the pair of functions \( f \) and \( g \) are inverses of each other. \( f(x)=9 x-8 \) and \( g(x)=\frac{x+9}{8} \) a. \( f(g(x))=\frac{9 x+17}{8} \) (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.)

Evaluate the expression by following steps: - step0: Evaluate: \(\frac{x+9}{8}\) - step1: Substitute: \(\frac{9x-8+9}{8}\) - step2: Add the numbers: \(\frac{9x+1}{8}\) Substitute \( x=\frac{x+9}{8} \) into the expression \( 9x-8 \). Evaluate the expression by following steps: - step0: Evaluate: \(9x-8\) - step1: Substitute: \(9\times \frac{x+9}{8}-8\) - step2: Multiply the terms: \(\frac{9\left(x+9\right)}{8}-8\) - step3: Reduce fractions to a common denominator: \(\frac{9\left(x+9\right)}{8}-\frac{8\times 8}{8}\) - step4: Transform the expression: \(\frac{9\left(x+9\right)-8\times 8}{8}\) - step5: Multiply the terms: \(\frac{9x+81-8\times 8}{8}\) - step6: Multiply the numbers: \(\frac{9x+81-64}{8}\) - step7: Subtract the numbers: \(\frac{9x+17}{8}\) Let's solve the problem step by step. ### Given Functions - \( f(x) = 9x - 8 \) - \( g(x) = \frac{x + 9}{8} \) ### a. Finding \( f(g(x)) \) To find \( f(g(x)) \), we substitute \( g(x) \) into \( f(x) \): \[ f(g(x)) = f\left(\frac{x + 9}{8}\right) = 9\left(\frac{x + 9}{8}\right) - 8 \] Calculating this gives: \[ f(g(x)) = \frac{9(x + 9)}{8} - 8 = \frac{9x + 81}{8} - \frac{64}{8} = \frac{9x + 17}{8} \] So, \[ f(g(x)) = \frac{9x + 17}{8} \] ### b. Finding \( g(f(x)) \) Now, we find \( g(f(x)) \) by substituting \( f(x) \) into \( g(x) \): \[ g(f(x)) = g(9x - 8) = \frac{(9x - 8) + 9}{8} \] Calculating this gives: \[ g(f(x)) = \frac{9x - 8 + 9}{8} = \frac{9x + 1}{8} \] So, \[ g(f(x)) = \frac{9x + 1}{8} \] ### Conclusion: Are \( f \) and \( g \) Inverses? To determine if \( f \) and \( g \) are inverses, we check if \( f(g(x)) = x \) and \( g(f(x)) = x \). 1. We found \( f(g(x)) = \frac{9x + 17}{8} \), which is not equal to \( x \). 2. We found \( g(f(x)) = \frac{9x + 1}{8} \), which is also not equal to \( x \). Since neither \( f(g(x)) \) nor \( g(f(x)) \) equals \( x \), the functions \( f \) and \( g \) are not inverses of each other. ### Final Answers - a. \( f(g(x)) = \frac{9x + 17}{8} \) - b. \( g(f(x)) = \frac{9x + 1}{8} \)