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

Question

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

UpStudy AI · Accepted Answer

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!
1. $$ j(g(x)) = \frac{1}{4}x^{2} + 4x + 17 $$ 2. $$ g(f(x)) = -x^{2} + 4 $$ 3. $$ f(h(x)) = -18x^{2} + 60x - 50 $$

Name

1. 2. 3.

\end{array}.

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