To find the expression equivalent to [tex]\((f \circ g)(x)\)[/tex], we need to understand the composition of the functions [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]. This composition [tex]\((f \circ g)(x)\)[/tex] means we evaluate [tex]\(g(x)\)[/tex] first and then use its output as the input to [tex]\(f(x)\)[/tex].
Here are the steps:
1. Define the individual functions:
- [tex]\(f(x) = 3x + 2\)[/tex]
- [tex]\(g(x) = x^2 + 1\)[/tex]
2. Calculate [tex]\(g(x)\)[/tex]:
- [tex]\(g(x) = x^2 + 1\)[/tex]
3. Substitute [tex]\(g(x)\)[/tex] into [tex]\(f(x)\)[/tex]:
- We will find [tex]\(f(g(x))\)[/tex] which means substituting [tex]\(g(x) = x^2 + 1\)[/tex] into [tex]\(f(x)\)[/tex].
- So, we need to evaluate [tex]\(f(x^2 + 1)\)[/tex].
4. Substitute [tex]\(x^2 + 1\)[/tex] into [tex]\(f\)[/tex]:
- Recall that [tex]\(f(y) = 3y + 2\)[/tex]. Here, [tex]\(y = g(x) = x^2 + 1\)[/tex].
- Substitute [tex]\(y = x^2 + 1\)[/tex] into [tex]\(f(y)\)[/tex]:
[tex]\[
f(x^2 + 1) = 3(x^2 + 1) + 2
\][/tex]
5. Simplify the expression:
- Distribute the 3:
[tex]\[
3(x^2 + 1) + 2 = 3x^2 + 3 + 2
\][/tex]
- Combine like terms:
[tex]\[
3x^2 + 3 + 2 = 3x^2 + 5
\][/tex]
Therefore, the equivalent expression for [tex]\((f \circ g)(x)\)[/tex] is [tex]\(3x^2 + 5\)[/tex].
Among the given options, the correct choice is:
[tex]\[ \boxed{3(x^2 + 1) + 2} \][/tex]
