To solve for the value of [tex]\( f(g(-2)) \)[/tex] given the functions:
[tex]\[
\begin{array}{l}
f(x) = -\frac{1}{2} x^2 + 5 x \\
g(x) = x^2 + 2
\end{array}
\][/tex]
we'll follow these steps:
1. First, evaluate [tex]\( g(-2) \)[/tex].
[tex]\[
g(x) = x^2 + 2
\][/tex]
[tex]\[
g(-2) = (-2)^2 + 2 = 4 + 2 = 6
\][/tex]
So, [tex]\( g(-2) = 6 \)[/tex].
2. Next, with this result, we need to find [tex]\( f(g(-2)) \)[/tex], which is [tex]\( f(6) \)[/tex].
[tex]\[
f(x) = -\frac{1}{2} x^2 + 5 x
\][/tex]
[tex]\[
f(6) = -\frac{1}{2} (6)^2 + 5(6)
\][/tex]
Calculate [tex]\( 6^2 \)[/tex]:
[tex]\[
6^2 = 36
\][/tex]
Now substitute [tex]\( 36 \)[/tex] into the equation:
[tex]\[
f(6) = -\frac{1}{2} (36) + 5(6) = -18 + 30
\][/tex]
Finally, sum the results:
[tex]\[
-18 + 30 = 12
\][/tex]
So, [tex]\( f(g(-2)) = 12 \)[/tex].
Therefore, the correct answer is:
[tex]\[
\boxed{12}
\][/tex]
