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Consider functions [tex]\( f \)[/tex] and [tex]\( g \)[/tex]:
[tex]\[ f(x) = x^2 + 7x \][/tex]
[tex]\[ g(x) = 3x - 1 \][/tex]

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

A. [tex]\( 3x^2 + 7x - 1 \)[/tex]

B. [tex]\( 9x^2 + 15x - 6 \)[/tex]

C. [tex]\( 3x^2 + 21x + 1 \)[/tex]

D. [tex]\( 9x^2 + 21x - 1 \)[/tex]



Answer :

GinnyAnswer
To find [tex]\((f \circ g)(x)\)[/tex], we need to evaluate [tex]\(f(g(x))\)[/tex].

Given:
[tex]\[ f(x) = x^2 + 7x \][/tex]
[tex]\[ g(x) = 3x - 1 \][/tex]

Let's substitute [tex]\(g(x)\)[/tex] into [tex]\(f(x)\)[/tex]:

[tex]\[ f(g(x)) = f(3x - 1) \][/tex]

Substitute [tex]\(3x - 1\)[/tex] into [tex]\( f(x) \)[/tex]:
[tex]\[ f(3x - 1) = (3x - 1)^2 + 7(3x - 1) \][/tex]

Now, we expand and simplify this expression:

First, expand [tex]\((3x - 1)^2\)[/tex]:
[tex]\[ (3x - 1)^2 = 9x^2 - 6x + 1 \][/tex]

Next, expand [tex]\(7(3x - 1)\)[/tex]:
[tex]\[ 7(3x - 1) = 21x - 7 \][/tex]

Now add the results together:
[tex]\[ (3x - 1)^2 + 7(3x - 1) = 9x^2 - 6x + 1 + 21x - 7 \][/tex]

Combine like terms:
[tex]\[ 9x^2 + (21x - 6x) + (1 - 7) = 9x^2 + 15x - 6 \][/tex]

Thus, the value of [tex]\((f \circ g)(x)\)[/tex] is:
[tex]\[ 9x^2 + 15x - 6 \][/tex]

So the correct answer is:

B. [tex]\(9 x^2 + 15 x - 6\)[/tex]

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