Find \( ( \) a) \( f(g(x)),( \) b \( ) g(f(x)) \), and \( ( \) c \( ) f(f(x)) \) \( f(x)=3 x^{-1}, g(x)=4 x+8 \) a. \( f(g(x))=\square \) b. \( g(f(x))=\square \) c. \( f(f(x))=\square \)

To solve for \( f(g(x)) \), \( g(f(x)) \), and \( f(f(x)) \) using the given functions \( f(x) = 3 x^{-1} \) and \( g(x) = 4x + 8 \), we can follow these steps: a. First, calculate \( g(x) \): \[ g(x) = 4x + 8 \] Now substitute \( g(x) \) into \( f \): \[ f(g(x)) = f(4x + 8) = 3(4x + 8)^{-1} = \frac{3}{4x + 8} \] So, \( f(g(x))=\frac{3}{4x + 8} \). b. Now calculate \( f(x) \): \[ f(x) = 3 x^{-1} \] Now substitute \( f(x) \) into \( g \): \[ g(f(x)) = g(3x^{-1}) = 4(3x^{-1}) + 8 = \frac{12}{x} + 8 \] So, \( g(f(x))=\frac{12}{x} + 8 \). c. Finally, calculate \( f(f(x)) \): \[ f(f(x)) = f(3x^{-1}) = 3(3x^{-1})^{-1} = 3 \cdot \frac{x}{3} = x \] So, \( f(f(x))=x \). Summarizing the results: a. \( f(g(x))=\frac{3}{4x + 8} \) b. \( g(f(x))=\frac{12}{x} + 8 \) c. \( f(f(x))=x \)