Find \( f(g(x)) \) and \( g(f(x)) \) and determine whether the pair of functions \( f \) and \( g \) are inverses of each other. \( f(x)=3 x-7 \) and \( g(x)=\frac{x+3}{7} \) a. \( f(g(x))=\square \) (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.) c. \( 0 \quad f \) and \( g \) are not inverses of each other. f and \( g \) are inverses of each other.

Evaluate the expression by following steps: - step0: Evaluate: \(\frac{\left(x+3\right)}{7}\) - step1: Substitute: \(\frac{\left(3x-7+3\right)}{7}\) - step2: Add the numbers: \(\frac{\left(3x-4\right)}{7}\) - step3: Remove the parentheses: \(\frac{3x-4}{7}\) Substitute \( x=(x+3)/7 \) into the expression \( 3x-7 \). Evaluate the expression by following steps: - step0: Evaluate: \(3x-7\) - step1: Substitute: \(3\times \frac{\left(x+3\right)}{7}-7\) - step2: Remove the parentheses: \(3\times \frac{x+3}{7}-7\) - step3: Multiply the terms: \(\frac{3\left(x+3\right)}{7}-7\) - step4: Reduce fractions to a common denominator: \(\frac{3\left(x+3\right)}{7}-\frac{7\times 7}{7}\) - step5: Transform the expression: \(\frac{3\left(x+3\right)-7\times 7}{7}\) - step6: Multiply the terms: \(\frac{3x+9-7\times 7}{7}\) - step7: Multiply the numbers: \(\frac{3x+9-49}{7}\) - step8: Subtract the numbers: \(\frac{3x-40}{7}\) Let's solve the problem step by step. ### a. Finding \( f(g(x)) \) We start with the function \( g(x) = \frac{x+3}{7} \) and substitute it into \( f(x) = 3x - 7 \): \[ f(g(x)) = f\left(\frac{x+3}{7}\right) = 3\left(\frac{x+3}{7}\right) - 7 \] Now, simplifying this expression: \[ f(g(x)) = \frac{3(x+3)}{7} - 7 = \frac{3x + 9}{7} - 7 \] To combine the terms, we convert \( 7 \) into a fraction with a denominator of \( 7 \): \[ f(g(x)) = \frac{3x + 9}{7} - \frac{49}{7} = \frac{3x + 9 - 49}{7} = \frac{3x - 40}{7} \] Thus, the answer is: \[ f(g(x)) = \frac{3x - 40}{7} \] ### b. Finding \( g(f(x)) \) Next, we substitute \( f(x) = 3x - 7 \) into \( g(x) \): \[ g(f(x)) = g(3x - 7) = \frac{(3x - 7) + 3}{7} \] Now, simplifying this expression: \[ g(f(x)) = \frac{3x - 7 + 3}{7} = \frac{3x - 4}{7} \] Thus, the answer is: \[ g(f(x)) = \frac{3x - 4}{7} \] ### c. Determining if \( f \) and \( g \) are inverses For \( f \) and \( g \) to be inverses of each other, we need to check if: 1. \( f(g(x)) = x \) 2. \( g(f(x)) = x \) From our calculations: - \( f(g(x)) = \frac{3x - 40}{7} \) - \( g(f(x)) = \frac{3x - 4}{7} \) Neither of these expressions equals \( x \). Therefore, we conclude that: \[ f \text{ and } g \text{ are not inverses of each other.} \] ### Final Answers a. \( f(g(x)) = \frac{3x - 40}{7} \) b. \( g(f(x)) = \frac{3x - 4}{7} \) c. \( 0 \quad f \text{ and } g \text{ are not inverses of each other.} \)