ions Part 5 of 5 Points: 0 of 3 Save Consider the functions \( \mathrm{f}(\mathrm{x})=3 \mathrm{x}-6 \) and \( \mathrm{g}(\mathrm{x})=\frac{\mathrm{x}}{3}+2 \). (a) Find \( f(g(x)) \). (b) Find \( g(f(x)) \). (c) Determine whether the functions \( f \) and \( g \) are inverses of each other. (Use a comma to separate answers as needed.) B. No values should be excluded from the domain. (b) What is \( g(f(x)) \) ? \( g(f(x))=x \quad \) (Simplify your answer.) Give any values of \( x \) that need to be excluded from \( g(f(x)) \). Select the correct choice and, if necessary, fill in the answer box within your choice. A. \( x \neq \) \( \square \) (Use a comma to separate answers as needed.) B. No values should be excluded from the domain. (c) Are the functions \( f \) and \( g \) inverses of each other? Choose the correct answer. Yes No

To find the composite functions and check if they are inverses, we start with the calculations. (a) To find \( f(g(x)) \): First, substitute \( g(x) \) into \( f(x) \): \[ g(x) = \frac{x}{3} + 2 \quad \Rightarrow \quad f(g(x)) = f\left(\frac{x}{3} + 2\right) = 3\left(\frac{x}{3} + 2\right) - 6 \] \[ = x + 6 - 6 = x \] (b) Now we find \( g(f(x)) \): Start by substituting \( f(x) \) into \( g(x) \): \[ f(x) = 3x - 6 \quad \Rightarrow \quad g(f(x)) = g(3x - 6) = \frac{3x - 6}{3} + 2 \] \[ = x - 2 + 2 = x \] Both composite functions yield \( x \), confirming that \( g(f(x)) = x \). For values of \( x \) that need to be excluded from \( g(f(x)) \), we see that there are no restrictions on \( x \) in either function. B. No values should be excluded from the domain. (c) Since both \( f(g(x)) = x \) and \( g(f(x)) = x \), we conclude that the functions \( f \) and \( g \) are indeed inverses of each other. Yes