To solve the problem of finding the value of [tex]\( g(f(-1)) \)[/tex], follow these detailed steps:
1. Evaluate [tex]\( f(-1) \)[/tex]:
The function [tex]\( f(x) = 5x^2 + 2 \)[/tex] provides a way to calculate [tex]\( f \)[/tex] for any given [tex]\( x \)[/tex]. Substitute [tex]\( -1 \)[/tex] for [tex]\( x \)[/tex]:
[tex]\[
f(-1) = 5(-1)^2 + 2
\][/tex]
Calculate the square of [tex]\(-1\)[/tex]:
[tex]\[
(-1)^2 = 1
\][/tex]
Multiply by 5:
[tex]\[
5 \cdot 1 = 5
\][/tex]
Add 2:
[tex]\[
5 + 2 = 7
\][/tex]
Thus, [tex]\( f(-1) = 7 \)[/tex].
2. Evaluate [tex]\( g(7) \)[/tex]:
Now that [tex]\( f(-1) = 7 \)[/tex], we need to find [tex]\( g(7) \)[/tex] using the function [tex]\( g(x) = x^2 - 1 \)[/tex].
Substitute [tex]\( 7 \)[/tex] for [tex]\( x \)[/tex]:
[tex]\[
g(7) = 7^2 - 1
\][/tex]
Calculate the square of 7:
[tex]\[
7^2 = 49
\][/tex]
Subtract 1:
[tex]\[
49 - 1 = 48
\][/tex]
Therefore, [tex]\( g(7) = 48 \)[/tex].
Combining these results, the value of [tex]\( g(f(-1)) \)[/tex] is [tex]\( 48 \)[/tex].
Thus, the correct answer is:
[tex]\[
\boxed{48}
\][/tex]
