Find \( f(g(x)) \) and \( g(f(x)) \) and determine whether the pair of functions \( f \) and \( g \) are inverses of each other \( f(x)=4 x+2 \) and \( g(x)=\frac{x-2}{4} \) a. \( f(g(x))=x \) (Simplify your answer) b. \( g(f(x))=\square \) (Simplify your answer)

Let's determine \( f(g(x)) \) and \( g(f(x)) \) for the given functions and verify whether \( f \) and \( g \) are inverses of each other. **Given Functions:** \[ f(x) = 4x + 2 \\ g(x) = \frac{x - 2}{4} \] ### a. Compute \( f(g(x)) \) \[ f(g(x)) = f\left(\frac{x - 2}{4}\right) \] \[ = 4 \left( \frac{x - 2}{4} \right) + 2 \] \[ = (x - 2) + 2 \] \[ = x \] So, \[ f(g(x)) = x \] ### b. Compute \( g(f(x)) \) \[ g(f(x)) = g(4x + 2) \] \[ = \frac{(4x + 2) - 2}{4} \] \[ = \frac{4x}{4} \] \[ = x \] So, \[ g(f(x)) = x \] ### Conclusion: Are \( f \) and \( g \) Inverses? Since both compositions satisfy: \[ f(g(x)) = x \quad \text{and} \quad g(f(x)) = x \] this means that \( f \) and \( g \) are indeed **inverse functions** of each other. **Answer Summary:** - **a.** \( f(g(x)) = x \) - **b.** \( g(f(x)) = x \) Since both compositions equal \( x \), \( f \) and \( g \) are inverses of one another.