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Given the functions [tex]\( f(x) = 3x - 6 \)[/tex] and [tex]\( g(x) = x^2 + 5x + 7 \)[/tex], perform the indicated operation. When applicable, state the domain restriction.

[tex]\( f(g(x)) \)[/tex]

A. [tex]\( x^2 + 8x + 1 \)[/tex]

B. [tex]\( 3x^2 + 15x + 21 \)[/tex]

C. [tex]\( 3x^2 + 15x + 15 \)[/tex]

D. [tex]\( 3x^3 - x^2 - 9x - 42 \)[/tex]



Answer :

GinnyAnswer
To determine the composition of the functions [tex]\( f \)[/tex] and [tex]\( g \)[/tex], denoted as [tex]\( f(g(x)) \)[/tex], follow these steps:

1. Identify the functions:
[tex]\[ f(x) = 3x - 6 \][/tex]
[tex]\[ g(x) = x^2 + 5x + 7 \][/tex]

2. Substitute [tex]\( g(x) \)[/tex] into [tex]\( f(x) \)[/tex]:
We need to evaluate [tex]\( f(g(x)) \)[/tex]. This means we substitute [tex]\( g(x) \)[/tex] into [tex]\( f \)[/tex]:
[tex]\[ f(g(x)) = f(x^2 + 5x + 7) \][/tex]

3. Apply the function [tex]\( f \)[/tex] to [tex]\( g(x) \)[/tex]:
Recall that [tex]\( f(x) = 3x - 6 \)[/tex]. Substitute [tex]\( g(x) \)[/tex] into [tex]\( x \)[/tex] in [tex]\( f \)[/tex]:
[tex]\[ f(x^2 + 5x + 7) = 3(x^2 + 5x + 7) - 6 \][/tex]

4. Distribute and simplify the expression:
[tex]\[ = 3(x^2 + 5x + 7) - 6 \][/tex]
[tex]\[ = 3x^2 + 15x + 21 - 6 \][/tex]

5. Combine like terms:
[tex]\[ = 3x^2 + 15x + (21 - 6) \][/tex]
[tex]\[ = 3x^2 + 15x + 15 \][/tex]

Therefore, the composition [tex]\( f(g(x)) \)[/tex] is:
[tex]\[ f(g(x)) = 3x^2 + 15x + 15 \][/tex]

So the correct answer from the given options is:
[tex]\[ \boxed{3x^2 + 15x + 15} \][/tex]

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