Find \( f(g(x)) \) and \( g(f(x)) \) and determine whether the pair of functions \( f \) and \( g \) are inverses of each other. \( f(x)=5 x-7 \) and \( g(x)=\frac{x+5}{7} \) a. \( f(g(x))=\frac{5 x-24}{7} \) (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.)

1. Compute \( f(g(x)) \): - Substitute \( g(x)=\frac{x+5}{7} \) into \( f \): \[ f(g(x)) = 5\left(\frac{x+5}{7}\right) - 7 \] - Multiply and simplify: \[ f(g(x)) = \frac{5(x+5)}{7} - 7 = \frac{5x + 25 - 49}{7} = \frac{5x - 24}{7} \] 2. Compute \( g(f(x)) \): - Substitute \( f(x)=5x-7 \) into \( g \): \[ g(f(x)) = \frac{(5x-7)+5}{7} \] - Simplify the numerator: \[ g(f(x)) = \frac{5x - 7 + 5}{7} = \frac{5x - 2}{7} \] 3. Determine whether \( f \) and \( g \) are inverses: - For \( f \) and \( g \) to be inverses, we must have: \[ f(g(x)) = x \quad \text{and} \quad g(f(x)) = x \] - We found: \[ f(g(x)) = \frac{5x-24}{7} \quad \text{and} \quad g(f(x)) = \frac{5x-2}{7} \] - These expressions equal \( x \) only for specific values of \( x \) (for example, \( f(g(x)) = x \) if \( \frac{5x-24}{7}=x \) and \( g(f(x)) = x \) if \( \frac{5x-2}{7}=x \)), not for all \( x \). - Therefore, \( f \) and \( g \) are not inverses of each other. Final answers: - \( f(g(x))=\frac{5x-24}{7} \) - \( g(f(x))=\frac{5x-2}{7} \) - The functions \( f \) and \( g \) are not inverses of each other.