If [tex]f(x)=3x+2[/tex] and [tex]g(x)=x^2+1[/tex], which expression is equivalent to [tex](f \circ g)(x)[/tex]?

A. [tex](3x+2)(x^2+1)[/tex]

B. [tex]3x^2+1+2[/tex]

C. [tex](3x+2)^2+1[/tex]

D. [tex]3(x^2+1)+2[/tex]



Answer :

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]