2. If $$ f(x)=x^{2}+5 x-7 $$ and $$ g(x)=x+1 $$, find each of the following: $$ \begin{array}{l|l|l} & \text { a. } f(-3) \ =-3^{2}+5(-3)-7 & \begin{array}{l} \text { b. } g(f(-2)) \ =9-15-7 \ =9-2)=-2^{2}+5(-2)-7 \ \ =9 \end{array} & \begin{array}{l} \text { c. Find } k \text { so that } g(k+3)=2 g(2 k) \ f(-2)=4-10-7 \end{array} \ f(K+3)+1=((2 K)+1 \ f(-2)=-13 & K+4=4 K+2 \ g(-13)=-13+1 & \frac{5 k}{5}=\frac{6}{5} \quad K=\frac{6}{5} \ g(-13)=-12 & \end{array} $$

Question

UpStudy AI · Accepted Answer

Let's solve each part step by step.

### Given:
- $$ f(x) = x^{2} + 5x - 7 $$
- $$ g(x) = x + 1 $$

---

### **a. Compute $$ f(-3) $$**
$$
f(-3) = (-3)^{2} + 5(-3) - 7
$$
$$
= 9 - 15 - 7
$$
$$
= -13
$$

**Answer:** $$ f(-3) = -13 $$

---

### **b. Compute $$ g(f(-2)) $$**
First, find $$ f(-2) $$:
$$
f(-2) = (-2)^{2} + 5(-2) - 7
$$
$$
= 4 - 10 - 7
$$
$$
= -13
$$

Now, apply $$ g $$ to the result:
$$
g(f(-2)) = g(-13) = (-13) + 1 = -12
$$

**Answer:** $$ g(f(-2)) = -12 $$

---

### **c. Find $$ k $$ such that $$ g(k + 3) = 2 \cdot g(2k) $$**

Start by expressing both sides using the definition of $$ g(x) $$:
$$
g(k + 3) = (k + 3) + 1 = k + 4
$$
$$
2 \cdot g(2k) = 2 \cdot (2k + 1) = 4k + 2
$$

Set the two expressions equal to each other:
$$
k + 4 = 4k + 2
$$
$$
4 - 2 = 4k - k
$$
$$
2 = 3k
$$
$$
k = \frac{2}{3}
$$

**Answer:** $$ k = \frac{2}{3} $$

---

### **Summary of Answers:**
- **a.** $$ f(-3) = -13 $$
- **b.** $$ g(f(-2)) = -12 $$
- **c.** $$ k = \frac{2}{3} $$
**Answers:** - **a.** $$ f(-3) = -13 $$ - **b.** $$ g(f(-2)) = -12 $$ - **c.** $$ k = \frac{2}{3} $$

If and , find each of the following:

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