Name \( \begin{array}{l}\text { Composite Functions Practice } \\ f(x)=-2 x^{2}, g(x)=\frac{1}{2} x+4, h(x)=3 x-5, j(x)=x^{2}+1 \\ \text { Use the equations above to find each composite function. } \\ \)\begin{tabular}{l|l|l|}\hline \text { 1. }\( j(g(x))= \) & \text { 2. }\( g(f(x))= \) & \text { 3. }\( f(h(x))= \)\end{tabular}\end{array}.

Sure! Let's find each of the composite functions step by step. ### Given Functions: - \( f(x) = -2x^{2} \) - \( g(x) = \frac{1}{2}x + 4 \) - \( h(x) = 3x - 5 \) - \( j(x) = x^{2} + 1 \) ### 1. \( j(g(x)) \) **Step 1:** Substitute \( g(x) \) into \( j(x) \): \[ j(g(x)) = j\left(\frac{1}{2}x + 4\right) = \left(\frac{1}{2}x + 4\right)^{2} + 1 \] **Step 2:** Expand the squared term: \[ \left(\frac{1}{2}x + 4\right)^{2} = \left(\frac{1}{2}x\right)^{2} + 2 \cdot \frac{1}{2}x \cdot 4 + 4^{2} = \frac{1}{4}x^{2} + 4x + 16 \] **Step 3:** Add 1: \[ j(g(x)) = \frac{1}{4}x^{2} + 4x + 16 + 1 = \frac{1}{4}x^{2} + 4x + 17 \] **Final Answer:** \[ j(g(x)) = \frac{1}{4}x^{2} + 4x + 17 \] --- ### 2. \( g(f(x)) \) **Step 1:** Substitute \( f(x) \) into \( g(x) \): \[ g(f(x)) = g(-2x^{2}) = \frac{1}{2}(-2x^{2}) + 4 \] **Step 2:** Simplify: \[ g(f(x)) = -x^{2} + 4 \] **Final Answer:** \[ g(f(x)) = -x^{2} + 4 \] --- ### 3. \( f(h(x)) \) **Step 1:** Substitute \( h(x) \) into \( f(x) \): \[ f(h(x)) = f(3x - 5) = -2(3x - 5)^{2} \] **Step 2:** Expand the squared term: \[ (3x - 5)^{2} = 9x^{2} - 30x + 25 \] **Step 3:** Multiply by -2: \[ f(h(x)) = -2(9x^{2} - 30x + 25) = -18x^{2} + 60x - 50 \] **Final Answer:** \[ f(h(x)) = -18x^{2} + 60x - 50 \] --- ### Summary of Composite Functions: | **1. \( j(g(x)) \)** | **2. \( g(f(x)) \)** | **3. \( f(h(x)) \)** | |----------------------|----------------------|-----------------------------| | \( \frac{1}{4}x^{2} + 4x + 17 \) | \( -x^{2} + 4 \) | \( -18x^{2} + 60x - 50 \) | Feel free to ask if you need further explanations or assistance!