To find [tex]\((f \circ g)(x)\)[/tex], we need to substitute [tex]\(g(x)\)[/tex] into [tex]\(f(x)\)[/tex]. Let's first identify the functions:
[tex]\[
f(x) = x^2 + 7x
\][/tex]
[tex]\[
g(x) = 3x - 1
\][/tex]
We want to find [tex]\(f(g(x))\)[/tex]. This means substituting [tex]\(3x - 1\)[/tex] into [tex]\(f(x)\)[/tex]:
[tex]\[
f(g(x)) = f(3x - 1)
\][/tex]
Now, plug [tex]\(3x - 1\)[/tex] in place of [tex]\(x\)[/tex] in [tex]\(f(x)\)[/tex]:
[tex]\[
f(3x - 1) = (3x - 1)^2 + 7(3x - 1)
\][/tex]
First, let's expand [tex]\((3x - 1)^2\)[/tex]:
[tex]\[
(3x - 1)^2 = 9x^2 - 6x + 1
\][/tex]
Next, let's distribute the 7 in [tex]\(7(3x - 1)\)[/tex]:
[tex]\[
7(3x - 1) = 21x - 7
\][/tex]
Now, add these two expressions together:
[tex]\[
(9x^2 - 6x + 1) + (21x - 7)
\][/tex]
Combine like terms:
[tex]\[
9x^2 - 6x + 1 + 21x - 7 = 9x^2 + 15x - 6
\][/tex]
Therefore, the value of [tex]\((f \circ g)(x)\)[/tex] is:
[tex]\[
9x^2 + 15x - 6
\][/tex]
So, the correct answer is:
D. [tex]\(9 x^2+15 x-6\)[/tex]
