To solve for [tex]\((fg)(-2)\)[/tex], we need to evaluate the functions [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex] at [tex]\(x = -2\)[/tex], and then find the product of the results.
1. Evaluating [tex]\(f(x)\)[/tex] at [tex]\(x = -2\)[/tex]:
The function [tex]\(f(x)\)[/tex] is defined as:
[tex]\[
f(x) = 8 - 10x
\][/tex]
Plugging in [tex]\(x = -2\)[/tex]:
[tex]\[
f(-2) = 8 - 10(-2)
\][/tex]
Calculate the expression inside the parentheses:
[tex]\[
f(-2) = 8 + 20
\][/tex]
Then sum the constants:
[tex]\[
f(-2) = 28
\][/tex]
2. Evaluating [tex]\(g(x)\)[/tex] at [tex]\(x = -2\)[/tex]:
The function [tex]\(g(x)\)[/tex] is defined as:
[tex]\[
g(x) = 5x + 4
\][/tex]
Plugging in [tex]\(x = -2\)[/tex]:
[tex]\[
g(-2) = 5(-2) + 4
\][/tex]
Calculate the expression inside the function:
[tex]\[
g(-2) = -10 + 4
\][/tex]
Then sum the constants:
[tex]\[
g(-2) = -6
\][/tex]
3. Calculating [tex]\((fg)(-2)\)[/tex]:
To find [tex]\((fg)(-2)\)[/tex], we multiply the values obtained from [tex]\(f(-2)\)[/tex] and [tex]\(g(-2)\)[/tex]:
[tex]\[
(fg)(-2) = f(-2) \cdot g(-2)
\][/tex]
Substituting the values we found:
[tex]\[
(fg)(-2) = 28 \cdot (-6)
\][/tex]
Perform the multiplication:
[tex]\[
(fg)(-2) = -168
\][/tex]
Therefore, the value of [tex]\((fg)(-2)\)[/tex] is [tex]\(\boxed{-168}\)[/tex].
