Evaluate $$ (g \circ f)(-2) $$ given $$ f(x)=3 x^{2}+2 x+1 $$ and $$ g(x)=x-5 $$.

Question

UpStudy AI · Accepted Answer

To evaluate $$ (g \circ f)(-2) $$, we need to find the value of $$ g(f(-2)) $$. This involves two steps:

1. **Calculate $$ f(-2) $$**
2. **Use the result from step 1 to find $$ g(f(-2)) $$**

Let's go through each step in detail.

---

### 1. Calculate $$ f(-2) $$

Given the function:
$$
f(x) = 3x^2 + 2x + 1
$$

Substitute $$ x = -2 $$ into the function:
$$
f(-2) = 3(-2)^2 + 2(-2) + 1
$$

Now, compute each term:
$$
(-2)^2 = 4 \
3 \times 4 = 12 \
2 \times (-2) = -4 \
$$

Add them up:
$$
f(-2) = 12 - 4 + 1 = 9
$$

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### 2. Calculate $$ g(f(-2)) $$ which is $$ g(9) $$

Given the function:
$$
g(x) = x - 5
$$

Substitute $$ x = 9 $$ into the function:
$$
g(9) = 9 - 5 = 4
$$

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### **Final Answer**

$$
(g \circ f)(-2) = 4
$$
$$ (g \circ f)(-2) = 4 $$

Evaluate \( (g \circ f)(-2) \) given \( f(x)=3 x^{2}+2 x+1 \) and \( g(x)=x-5 \).