To find the value of [tex]\( k(f(x)) \)[/tex], let's first understand what it means to apply the function [tex]\( k \)[/tex] to the output of the function [tex]\( f(x) \)[/tex].
Given the functions:
[tex]\[
f(x) = 3x^3 + 8x - 2
\][/tex]
[tex]\[
k(x) = 4x
\][/tex]
We need to find the composition [tex]\( k(f(x)) \)[/tex]. This means we substitute [tex]\( f(x) \)[/tex] into [tex]\( k(x) \)[/tex]:
[tex]\[
k(f(x)) = k(3x^3 + 8x - 2)
\][/tex]
The function [tex]\( k(x) \)[/tex] is defined as multiplying the input by 4. Hence, we replace the variable [tex]\( x \)[/tex] in [tex]\( k(x) \)[/tex] with the expression [tex]\( f(x) \)[/tex]:
[tex]\[
k(f(x)) = 4 \cdot (3x^3 + 8x - 2)
\][/tex]
Next, distribute the 4 to each term inside the parentheses:
[tex]\[
k(f(x)) = 4 \cdot 3x^3 + 4 \cdot 8x - 4 \cdot (-2)
\][/tex]
Simplify each term:
[tex]\[
4 \cdot 3x^3 = 12x^3
\][/tex]
[tex]\[
4 \cdot 8x = 32x
\][/tex]
[tex]\[
4 \cdot (-2) = -8
\][/tex]
Putting it all together, we get:
[tex]\[
k(f(x)) = 12x^3 + 32x - 8
\][/tex]
Thus, the correct answer is:
[tex]\[
\boxed{12x^3 + 32x - 8}
\][/tex]
So, the value of [tex]\( k(f(x)) \)[/tex] is:
[tex]\[
\boxed{D}
\][/tex]
