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Select the correct answer.

Consider these functions:

[tex]\[
\begin{array}{l}
f(x) = 3x^3 + 8x - 2 \\
k(x) = 4x
\end{array}
\][/tex]

What is the value of [tex]\( k(f(x)) \)[/tex]?

A. [tex]\( 192x^3 + 8x - 2 \)[/tex]
B. [tex]\( 192x^3 + 32x - 2 \)[/tex]
C. [tex]\( 12x^3 + 32x - 2 \)[/tex]
D. [tex]\( 12x^3 + 32x - 8 \)[/tex]



Answer :

GinnyAnswer
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]

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