Given \( f(x)=5 x^{2}-4 \) and \( g(x)=6-\frac{1}{2} x^{2} \), find the following expressions. \( \begin{array}{llll}\text { (a) }(f \circ g)(4) & \text { (b) }(g \circ f)(2) & \text { (c) }(f \circ f)(1) & \text { (d) }(g \circ g)(0)\end{array} \) \( \begin{array}{ll}\text { (a) }(f \circ g)(4)=\square & \text { (Simplify your answer.) }\end{array} \)

To solve the problem, we need to evaluate the compositions of the functions \( f \) and \( g \) at the specified points. Let's break it down step by step. ### Given Functions - \( f(x) = 5x^2 - 4 \) - \( g(x) = 6 - \frac{1}{2}x^2 \) ### (a) Calculate \( (f \circ g)(4) \) 1. **Evaluate \( g(4) \)**: \[ g(4) = 6 - \frac{1}{2}(4^2) = 6 - \frac{1}{2}(16) = 6 - 8 = -2 \] 2. **Now evaluate \( f(g(4)) = f(-2) \)**: \[ f(-2) = 5(-2)^2 - 4 = 5(4) - 4 = 20 - 4 = 16 \] Thus, \( (f \circ g)(4) = 16 \). ### (b) Calculate \( (g \circ f)(2) \) 1. **Evaluate \( f(2) \)**: \[ f(2) = 5(2^2) - 4 = 5(4) - 4 = 20 - 4 = 16 \] 2. **Now evaluate \( g(f(2)) = g(16) \)**: \[ g(16) = 6 - \frac{1}{2}(16^2) = 6 - \frac{1}{2}(256) = 6 - 128 = -122 \] Thus, \( (g \circ f)(2) = -122 \). ### (c) Calculate \( (f \circ f)(1) \) 1. **Evaluate \( f(1) \)**: \[ f(1) = 5(1^2) - 4 = 5(1) - 4 = 5 - 4 = 1 \] 2. **Now evaluate \( f(f(1)) = f(1) \)**: \[ f(1) = 1 \quad \text{(as calculated above)} \] Thus, \( (f \circ f)(1) = 1 \). ### (d) Calculate \( (g \circ g)(0) \) 1. **Evaluate \( g(0) \)**: \[ g(0) = 6 - \frac{1}{2}(0^2) = 6 - 0 = 6 \] 2. **Now evaluate \( g(g(0)) = g(6) \)**: \[ g(6) = 6 - \frac{1}{2}(6^2) = 6 - \frac{1}{2}(36) = 6 - 18 = -12 \] Thus, \( (g \circ g)(0) = -12 \). ### Summary of Results - \( (f \circ g)(4) = 16 \) - \( (g \circ f)(2) = -122 \) - \( (f \circ f)(1) = 1 \) - \( (g \circ g)(0) = -12 \) The final answers are: - (a) \( (f \circ g)(4) = 16 \) - (b) \( (g \circ f)(2) = -122 \) - (c) \( (f \circ f)(1) = 1 \) - (d) \( (g \circ g)(0) = -12 \)