Glven the functions, \( f(x)=3 x-2 \) and \( g(x)=\frac{x+2}{3} \), complete parts 1 and 2 . 1. Find \( f(g(x)) \) and \( g(f(x)) \). Include your work in your final answer. 2. Use complete sentences to explain the relationship that exists between the composition of the functions, \( f(g(x)) \) and \( g(f(x)) \).

Certainly! Let's work through each part step by step. ### 1. Finding \( f(g(x)) \) and \( g(f(x)) \) **Given Functions:** - \( f(x) = 3x - 2 \) - \( g(x) = \frac{x + 2}{3} \) **a. Finding \( f(g(x)) \):** To find \( f(g(x)) \), we substitute \( g(x) \) into \( f(x) \). \[ f(g(x)) = f\left(\frac{x + 2}{3}\right) \] Now, replace \( x \) in \( f(x) \) with \( \frac{x + 2}{3} \): \[ f\left(\frac{x + 2}{3}\right) = 3\left(\frac{x + 2}{3}\right) - 2 \] Simplify the expression: \[ 3 \times \frac{x + 2}{3} = x + 2 \] So, \[ f(g(x)) = (x + 2) - 2 = x \] **b. Finding \( g(f(x)) \):** To find \( g(f(x)) \), we substitute \( f(x) \) into \( g(x) \). \[ g(f(x)) = g(3x - 2) \] Now, replace \( x \) in \( g(x) \) with \( 3x - 2 \): \[ g(3x - 2) = \frac{(3x - 2) + 2}{3} \] Simplify the expression: \[ (3x - 2) + 2 = 3x \] So, \[ g(f(x)) = \frac{3x}{3} = x \] **Summary:** \[ f(g(x)) = x \] \[ g(f(x)) = x \] ### 2. Explanation of the Relationship Between \( f(g(x)) \) and \( g(f(x)) \) Both compositions \( f(g(x)) \) and \( g(f(x)) \) result in the original input variable \( x \). This indicates that the functions \( f(x) \) and \( g(x) \) are **inverse functions** of each other. In other words, applying one function after the other returns the input value, effectively "undoing" each other's operations. Specifically: - When you apply \( g(x) \) first and then \( f(x) \), you get back to \( x \). - Similarly, applying \( f(x) \) first and then \( g(x) \) also brings you back to \( x \). This mutual inversion confirms that \( f \) and \( g \) are inverses, meaning they reverse each other's effects when composed.