To solve for [tex]\( f(g(-2)) \)[/tex] using the given functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex], follow these steps:
1. Evaluate [tex]\( g(-2) \)[/tex]:
The function [tex]\( g(x) \)[/tex] is given by:
[tex]\[
g(x) = x^2 + 2
\][/tex]
Substituting [tex]\( x = -2 \)[/tex] into the function:
[tex]\[
g(-2) = (-2)^2 + 2 = 4 + 2 = 6
\][/tex]
2. Evaluate [tex]\( f(g(-2)) \)[/tex]:
We now know that [tex]\( g(-2) = 6 \)[/tex]. Next, we need to find [tex]\( f(6) \)[/tex] using the function [tex]\( f(x) \)[/tex], which is given by:
[tex]\[
f(x) = -\frac{1}{2} x^2 + 5 x
\][/tex]
Substituting [tex]\( x = 6 \)[/tex] into the function:
[tex]\[
f(6) = -\frac{1}{2} (6)^2 + 5 \cdot 6
\][/tex]
Calculate [tex]\( 6^2 \)[/tex]:
[tex]\[
6^2 = 36
\][/tex]
Then:
[tex]\[
f(6) = -\frac{1}{2} \cdot 36 + 5 \cdot 6 = -18 + 30 = 12
\][/tex]
3. Conclusion:
Therefore, the value of [tex]\( f(g(-2)) \)[/tex] is:
[tex]\[
\boxed{12}
\][/tex]
Hence, the correct answer is:
C. 12
