QuestionLet $f(x)=3 x+2$ and $g(x)=5 x^{2}+2 x$ . Then $(f \circ g)(-3)=$ $\square$ $(f \circ g)(x)=$ $\square$

Studdy Solution

STEP 1

What is this asking? We're asked to find the value of a composite function $(f \circ g)$ at $x = -3$ , and also find the general formula for $(f \circ g)(x)$ . Watch out! Remember that $(f \circ g)(x)$ means $f(g(x))$ , not $g(f(x))$ !
It's $f$ of $g$ , like a delicious mathematical turducken!

STEP 2

1. Find $g(-3)$
2. Find $(f \circ g)(-3)$
3. Find the formula for $(f \circ g)(x)$

STEP 3

Let's kick things off by finding $g(-3)$ .
We know that $g(x) = 5x^2 + 2x$ , so we'll substitute $-3$ for $x$ :
$g(-3) = 5 \cdot (-3)^2 + 2 \cdot (-3)$

STEP 4

Now, let's carefully evaluate this expression.
Remember order of operations!
Exponents first:
$g(-3) = 5 \cdot 9 + 2 \cdot (-3)$

STEP 5

Next, we perform the multiplications:
$g(-3) = 45 - 6$

STEP 6

Finally, we subtract to get our result:
$g(-3) = 39$ So, $g(-3)$ is 39!

STEP 7

Now, we need to find $(f \circ g)(-3)$ , which means we're looking for $f(g(-3))$ .
We just found that $g(-3) = \textbf{39}$ , so we're really looking for $f(\textbf{39})$ .

STEP 8

We know that $f(x) = 3x + 2$ , so let's substitute $\textbf{39}$ for $x$ :
$f(39) = 3 \cdot 39 + 2$

STEP 9

Now, let's multiply:
$f(39) = 117 + 2$

STEP 10

Finally, we add to find our result:
$f(39) = 119$ So, $(f \circ g)(-3) = \textbf{119}$ !

STEP 11

To find the general formula for $(f \circ g)(x)$ , we need to find $f(g(x))$ .
We know that $g(x) = 5x^2 + 2x$ , so we're looking for $f(5x^2 + 2x)$ .

STEP 12

Since $f(x) = 3x + 2$ , we replace $x$ with $5x^2 + 2x$ in the formula for $f(x)$ :
$f(5x^2 + 2x) = 3 \cdot (5x^2 + 2x) + 2$

STEP 13

Now, we distribute the 3:
$f(5x^2 + 2x) = 15x^2 + 6x + 2$

STEP 14

So, the formula for $(f \circ g)(x)$ is $15x^2 + 6x + 2$ !

STEP 15

$(f \circ g)(-3) = \textbf{119}$ and $(f \circ g)(x) = 15x^2 + 6x + 2$ .

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