To solve for [tex]\((f \circ f)(x)\)[/tex], we need to find [tex]\(f(f(x))\)[/tex].
1. Start with the given function:
[tex]\[
f(x) = 2x^2
\][/tex]
2. To find [tex]\(f(f(x))\)[/tex], first evaluate [tex]\(f(x)\)[/tex]:
[tex]\[
f(x) = 2x^2
\][/tex]
3. Now substitute [tex]\(f(x)\)[/tex] back into the function [tex]\(f\)[/tex]:
[tex]\[
f(f(x)) = f(2x^2)
\][/tex]
4. Next, replace [tex]\(x\)[/tex] with [tex]\(2x^2\)[/tex] in the original function [tex]\(f\)[/tex]:
[tex]\[
f(2x^2) = 2(2x^2)^2
\][/tex]
5. Simplify the expression inside the parentheses:
[tex]\[
(2x^2)^2 = 4x^4
\][/tex]
6. Multiply by the coefficient from the function:
[tex]\[
2 \cdot 4x^4 = 8x^4
\][/tex]
Thus, the value of [tex]\((f \circ f)(x)\)[/tex] is:
[tex]\[
8x^4
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{C \; 8x^4}
\][/tex]
