Are the following functions inverses? \( \begin{array}{l}f(x)=4 x-2 \\ g(x)=\frac{x}{4}+2 \\ \text { No, they are not inverses } \\ \text { Yes, they are inverses }\end{array} \)

To determine if the functions \( f(x) = 4x - 2 \) and \( g(x) = \frac{x}{4} + 2 \) are inverses of each other, we need to check if the following conditions hold: 1. \( f(g(x)) = x \) 2. \( g(f(x)) = x \) Let's calculate \( f(g(x)) \) and \( g(f(x)) \) step by step. ### Step 1: Calculate \( f(g(x)) \) Substituting \( g(x) \) into \( f(x) \): \[ f(g(x)) = f\left(\frac{x}{4} + 2\right) \] Now, substitute \( \frac{x}{4} + 2 \) into \( f(x) \): \[ f\left(\frac{x}{4} + 2\right) = 4\left(\frac{x}{4} + 2\right) - 2 \] Now simplify: \[ = 4 \cdot \frac{x}{4} + 4 \cdot 2 - 2 \] \[ = x + 8 - 2 \] \[ = x + 6 \] ### Step 2: Calculate \( g(f(x)) \) Now, substitute \( f(x) \) into \( g(x) \): \[ g(f(x)) = g(4x - 2) \] Now, substitute \( 4x - 2 \) into \( g(x) \): \[ g(4x - 2) = \frac{4x - 2}{4} + 2 \] Now simplify: \[ = x - \frac{1}{2} + 2 \] \[ = x + \frac{3}{2} \] ### Conclusion We found: - \( f(g(x)) = x + 6 \) - \( g(f(x)) = x + \frac{3}{2} \) Since neither \( f(g(x)) \) nor \( g(f(x)) \) equals \( x \), the functions \( f(x) \) and \( g(x) \) are **not inverses** of each other. Thus, the answer is: **No, they are not inverses.**