To find the value of [tex]\( g(f(x)) \)[/tex], we need to substitute the function [tex]\( f(x) \)[/tex] into the function [tex]\( g(x) \)[/tex]. Let's begin by detailing each function and how we can compose them.
Given:
[tex]\[
f(x) = \frac{1}{3} x^2 + 4
\][/tex]
[tex]\[
g(x) = 9x - 12
\][/tex]
Step-by-step procedure:
1. Determine [tex]\( f(x) \)[/tex]:
[tex]\[
f(x) = \frac{1}{3} x^2 + 4
\][/tex]
2. Substitute [tex]\( f(x) \)[/tex] into [tex]\( g(x) \)[/tex]:
[tex]\[
g(f(x)) = g\left(\frac{1}{3} x^2 + 4\right)
\][/tex]
3. Evaluate [tex]\( g(x) \)[/tex] at [tex]\( \left(\frac{1}{3} x^2 + 4\right) \)[/tex]:
[tex]\[
g\left(\frac{1}{3} x^2 + 4\right) = 9\left(\frac{1}{3} x^2 + 4\right) - 12
\][/tex]
4. Distribute the 9 inside the parentheses:
[tex]\[
9\left(\frac{1}{3} x^2 + 4\right) = 9 \cdot \frac{1}{3} x^2 + 9 \cdot 4
\][/tex]
5. Simplify the expression:
[tex]\[
9 \cdot \frac{1}{3} x^2 = 3x^2
\][/tex]
[tex]\[
9 \cdot 4 = 36
\][/tex]
6. Combine these results:
[tex]\[
3x^2 + 36 - 12
\][/tex]
7. Simplify further by combining constants:
[tex]\[
3x^2 + 36 - 12 = 3x^2 + 24
\][/tex]
Thus, the value of [tex]\( g(f(x)) \)[/tex] is:
[tex]\[
\boxed{3 x^2 + 24}
\][/tex]
Therefore, the correct answer is:
C. [tex]\( 3 x^2 + 24 \)[/tex]
