To determine which expression is equivalent to [tex]\((f \circ g)(x)\)[/tex], we need to understand what the notation [tex]\((f \circ g)(x)\)[/tex] means. This notation represents the composition of the functions [tex]\(f\)[/tex] and [tex]\(g\)[/tex], where we first apply the function [tex]\(g\)[/tex] to [tex]\(x\)[/tex] and then apply the function [tex]\(f\)[/tex] to the result. Mathematically, this is written as:
[tex]\[
(f \circ g)(x) = f(g(x))
\][/tex]
Given the functions:
[tex]\[ f(x) = 3x + 2 \][/tex]
[tex]\[ g(x) = x^2 + 1 \][/tex]
We start by finding [tex]\(g(x)\)[/tex]:
[tex]\[ g(x) = x^2 + 1 \][/tex]
Next, we substitute [tex]\(g(x)\)[/tex] into [tex]\(f(x)\)[/tex]:
[tex]\[ f(g(x)) = f(x^2 + 1) \][/tex]
Substitute [tex]\(x^2 + 1\)[/tex] into [tex]\(f(x)\)[/tex]:
[tex]\[ f(x^2 + 1) = 3(x^2 + 1) + 2 \][/tex]
Simplify this expression step by step:
[tex]\[ 3(x^2 + 1) + 2 = 3x^2 + 3 \cdot 1 + 2 \][/tex]
[tex]\[ = 3x^2 + 3 + 2 \][/tex]
[tex]\[ = 3x^2 + 5 \][/tex]
So, the expression equivalent to [tex]\((f \circ g)(x)\)[/tex] is:
[tex]\[ 3(x^2 + 1) + 2 \][/tex]
Therefore, the correct option from the given choices is:
[tex]\[ 3(x^2 + 1) + 2 \][/tex]
[tex]\[ \boxed{3\left(x^2 + 1\right) + 2} \][/tex]
