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]
