To find the expression equivalent to [tex]\((f \circ g)(x)\)[/tex], we need to compute the composition of the functions [tex]\(f\)[/tex] and [tex]\(g\)[/tex], which is defined 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]
Let's break down the computation step-by-step:
1. Find [tex]\(g(x)\)[/tex]:
[tex]\[
g(x) = x^2 + 1
\][/tex]
2. Substitute [tex]\(g(x)\)[/tex] into [tex]\(f(x)\)[/tex]:
We need to substitute [tex]\(g(x)\)[/tex] into [tex]\(f(x)\)[/tex], which means we will replace [tex]\(x\)[/tex] in [tex]\(f(x)\)[/tex] with [tex]\(g(x)\)[/tex].
[tex]\[
f(g(x)) = f(x^2 + 1)
\][/tex]
3. Evaluate [tex]\(f(x^2 + 1)\)[/tex]:
Using the definition of [tex]\(f(x)\)[/tex], we substitute [tex]\(x^2 + 1\)[/tex] into the function [tex]\(f\)[/tex]:
[tex]\[
f(x^2 + 1) = 3(x^2 + 1) + 2
\][/tex]
4. Simplify the expression:
Distribute the 3 and combine like terms:
[tex]\[
3(x^2 + 1) + 2 = 3x^2 + 3 + 2 = 3x^2 + 5
\][/tex]
Thus, the expression equivalent to [tex]\((f \circ g)(x)\)[/tex] is:
[tex]\[
\boxed{3x^2 + 5}
\][/tex]
