Answer :

To solve the problem, we need to find the value of [tex]\( f(g(4)) \)[/tex] using the given functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex].

1. Find [tex]\( g(4) \)[/tex]:

The function [tex]\( g(x) \)[/tex] is given as [tex]\( g(x) = 2x \)[/tex]. To find [tex]\( g(4) \)[/tex], substitute 4 into the function:

[tex]\[
g(4) = 2 \times 4 = 8
\][/tex]

2. Use the result of [tex]\( g(4) \)[/tex] to find [tex]\( f(g(4)) \)[/tex], which is [tex]\( f(8) \)[/tex]:

The function [tex]\( f(x) \)[/tex] is given as [tex]\( f(x) = 3x^2 - 3x + 6 \)[/tex]. Substitute [tex]\( 8 \)[/tex] into the function:

[tex]\[
f(8) = 3(8)^2 - 3(8) + 6
\][/tex]

- First, calculate [tex]\( 8^2 \)[/tex] which is [tex]\( 64 \)[/tex].
- Then, calculate [tex]\( 3 \times 64 \)[/tex] which is [tex]\( 192 \)[/tex].
- Next, calculate [tex]\( 3 \times 8 \)[/tex] which is [tex]\( 24 \)[/tex].
- Now, substitute these into the equation:

[tex]\[
f(8) = 192 - 24 + 6
\][/tex]

- Calculate [tex]\( 192 - 24 = 168 \)[/tex].
- Finally, add 6: [tex]\( 168 + 6 = 174 \)[/tex].

Therefore, the value of [tex]\( f(g(4)) \)[/tex] is [tex]\( 174 \)[/tex].

The correct option is D) 174.

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