To determine the composition of the functions [tex]\( f \)[/tex] and [tex]\( g \)[/tex], denoted as [tex]\( f(g(x)) \)[/tex], follow these steps:
1. Identify the functions:
[tex]\[
f(x) = 3x - 6
\][/tex]
[tex]\[
g(x) = x^2 + 5x + 7
\][/tex]
2. Substitute [tex]\( g(x) \)[/tex] into [tex]\( f(x) \)[/tex]:
We need to evaluate [tex]\( f(g(x)) \)[/tex]. This means we substitute [tex]\( g(x) \)[/tex] into [tex]\( f \)[/tex]:
[tex]\[
f(g(x)) = f(x^2 + 5x + 7)
\][/tex]
3. Apply the function [tex]\( f \)[/tex] to [tex]\( g(x) \)[/tex]:
Recall that [tex]\( f(x) = 3x - 6 \)[/tex]. Substitute [tex]\( g(x) \)[/tex] into [tex]\( x \)[/tex] in [tex]\( f \)[/tex]:
[tex]\[
f(x^2 + 5x + 7) = 3(x^2 + 5x + 7) - 6
\][/tex]
4. Distribute and simplify the expression:
[tex]\[
= 3(x^2 + 5x + 7) - 6
\][/tex]
[tex]\[
= 3x^2 + 15x + 21 - 6
\][/tex]
5. Combine like terms:
[tex]\[
= 3x^2 + 15x + (21 - 6)
\][/tex]
[tex]\[
= 3x^2 + 15x + 15
\][/tex]
Therefore, the composition [tex]\( f(g(x)) \)[/tex] is:
[tex]\[
f(g(x)) = 3x^2 + 15x + 15
\][/tex]
So the correct answer from the given options is:
[tex]\[
\boxed{3x^2 + 15x + 15}
\][/tex]
