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Solved on Dec 06, 2023

Find the value of $(f \circ g)(3)$ where $f(x) = 7x^2 - 4x$ and $g(x) = 5x - 9$ .

STEP 1

Assumptions
1. The function $f(x)$ is defined as $f(x)=7x^2-4x$ .
2. The function $g(x)$ is defined as $g(x)=5x-9$ .
3. We need to find the value of the composition of $f$ and $g$ , denoted as $\left(f \circ g\right)(x)$ , at $x=3$ .

STEP 2

To find the composition $\left(f \circ g\right)(x)$ , we first need to evaluate $g(x)$ and then apply the result as the input for $f(x)$ .
$\left(f \circ g\right)(x) = f(g(x))$

STEP 3

Evaluate $g(x)$ at $x=3$ .
$g(3) = 5(3) - 9$

STEP 4

Calculate the value of $g(3)$ .
$g(3) = 15 - 9 = 6$

STEP 5

Now that we have $g(3) = 6$ , we will use this value as the input for the function $f(x)$ .
$f(g(3)) = f(6)$

STEP 6

Substitute $x=6$ into the function $f(x)$ .
$f(6) = 7(6)^2 - 4(6)$

STEP 7

Calculate the value of $f(6)$ .
$f(6) = 7(36) - 4(6)$

STEP 8

Continue the calculation.
$f(6) = 252 - 24$

STEP 9

Finalize the calculation to find $f(6)$ .
$f(6) = 228$

STEP 10

Thus, the value of $\left(f^{\circ} g\right)(3)$ is $228$ .
The correct answer is D) 228.

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