To determine the value of [tex]\( g(f(-1)) \)[/tex], let’s follow these steps:
1. First, evaluate [tex]\( f(-1) \)[/tex]:
[tex]\[
f(x) = 5x^2 + 2
\][/tex]
Substitute [tex]\( x = -1 \)[/tex] into the function [tex]\( f(x) \)[/tex]:
[tex]\[
f(-1) = 5(-1)^2 + 2
\][/tex]
Simplify the equation:
[tex]\[
(-1)^2 = 1
\][/tex]
Therefore,
[tex]\[
f(-1) = 5 \cdot 1 + 2 = 5 + 2 = 7
\][/tex]
2. Next, evaluate [tex]\( g(f(-1)) \)[/tex] which is [tex]\( g(7) \)[/tex]:
[tex]\[
g(x) = x^2 - 1
\][/tex]
Substitute [tex]\( x = 7 \)[/tex] into the function [tex]\( g(x) \)[/tex]:
[tex]\[
g(7) = 7^2 - 1
\][/tex]
Simplify the equation:
[tex]\[
7^2 = 49
\][/tex]
Therefore,
[tex]\[
g(7) = 49 - 1 = 48
\][/tex]
So, the value of [tex]\( g(f(-1)) \)[/tex] is [tex]\( \boxed{48} \)[/tex]. The correct answer is [tex]\( \text{D. 48} \)[/tex].
