To evaluate [tex]\( f(g(2)) \)[/tex] given the functions [tex]\( f(x) = 3x^2 - 7 \)[/tex] and [tex]\( g(x) = x^3 + 1 \)[/tex], follow these steps:
1. Evaluate [tex]\( g(2) \)[/tex]:
[tex]\[
g(x) = x^3 + 1
\][/tex]
Substitute [tex]\( x = 2 \)[/tex]:
[tex]\[
g(2) = 2^3 + 1 = 8 + 1 = 9
\][/tex]
2. Now use the result of [tex]\( g(2) \)[/tex] to evaluate [tex]\( f(g(2)) \)[/tex]:
[tex]\[
f(x) = 3x^2 - 7
\][/tex]
Substitute [tex]\( x = g(2) \)[/tex] or [tex]\( x = 9 \)[/tex]:
[tex]\[
f(9) = 3(9^2) - 7
\][/tex]
Calculate [tex]\( 9^2 \)[/tex]:
[tex]\[
9^2 = 81
\][/tex]
Substitute back into the expression for [tex]\( f(9) \)[/tex]:
[tex]\[
f(9) = 3 \cdot 81 - 7
\][/tex]
Calculate the multiplication:
[tex]\[
3 \cdot 81 = 243
\][/tex]
Subtract 7:
[tex]\[
243 - 7 = 236
\][/tex]
So, the correct evaluation of [tex]\( f(g(2)) \)[/tex] is [tex]\( 236 \)[/tex].
Therefore, the answer is [tex]\( 236 \)[/tex].
