To determine [tex]\((f - g)(-3)\)[/tex], we need to follow a detailed, step-by-step process.
First, let's define the functions [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]:
[tex]\[
f(x) = 3x^2 + 6
\][/tex]
[tex]\[
g(x) = -5x + 2
\][/tex]
We need to find the value of [tex]\((f - g)(-3)\)[/tex]. This means we will evaluate both [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex] at [tex]\(x = -3\)[/tex], and then find the difference between these values.
1. Evaluate [tex]\(f(-3)\)[/tex]:
Substitute [tex]\(x = -3\)[/tex] into the function [tex]\(f(x)\)[/tex]:
[tex]\[
f(-3) = 3(-3)^2 + 6
\][/tex]
Calculate [tex]\((-3)^2\)[/tex]:
[tex]\[
(-3)^2 = 9
\][/tex]
Now, multiply by 3:
[tex]\[
3 \cdot 9 = 27
\][/tex]
Add 6:
[tex]\[
27 + 6 = 33
\][/tex]
Therefore:
[tex]\[
f(-3) = 33
\][/tex]
2. Evaluate [tex]\(g(-3)\)[/tex]:
Substitute [tex]\(x = -3\)[/tex] into the function [tex]\(g(x)\)[/tex]:
[tex]\[
g(-3) = -5(-3) + 2
\][/tex]
Calculate [tex]\(-5 \cdot -3\)[/tex]:
[tex]\[
-5 \cdot -3 = 15
\][/tex]
Add 2:
[tex]\[
15 + 2 = 17
\][/tex]
Therefore:
[tex]\[
g(-3) = 17
\][/tex]
3. Calculate [tex]\((f - g)(-3)\)[/tex]:
Subtract [tex]\(g(-3)\)[/tex] from [tex]\(f(-3)\)[/tex]:
[tex]\[
(f - g)(-3) = f(-3) - g(-3)
\][/tex]
Substitute the values we found:
[tex]\[
(f - g)(-3) = 33 - 17
\][/tex]
Perform the subtraction:
[tex]\[
33 - 17 = 16
\][/tex]
Thus:
[tex]\[
(f - g)(-3) = 16
\][/tex]
