To find the expression for [tex]\( (f \circ g)(x) \)[/tex], we need to apply the function [tex]\( g \)[/tex] first, and then apply the function [tex]\( f \)[/tex] to the result of [tex]\( g \)[/tex].
Given:
[tex]\[ f(x) = 3x + 2 \][/tex]
[tex]\[ g(x) = x^2 + 1 \][/tex]
1. First, evaluate [tex]\( g(x) \)[/tex]:
[tex]\[ g(x) = x^2 + 1 \][/tex]
2. Now, substitute [tex]\( g(x) \)[/tex] into [tex]\( f \)[/tex]:
[tex]\[ (f \circ g)(x) = f(g(x)) \][/tex]
The next step is to replace [tex]\( x \)[/tex] in [tex]\( f(x) = 3x + 2 \)[/tex] with [tex]\( g(x) \)[/tex]:
[tex]\[ (f \circ g)(x) = f(g(x)) = f(x^2 + 1) \][/tex]
3. Substitute [tex]\( x^2 + 1 \)[/tex] into [tex]\( f \)[/tex] in place of [tex]\( x \)[/tex]:
[tex]\[ f(x^2 + 1) = 3(x^2 + 1) + 2 \][/tex]
4. Distribute and simplify the expression:
[tex]\[ 3(x^2 + 1) + 2 = 3x^2 + 3 + 2 = 3x^2 + 5 \][/tex]
Therefore, the expression equivalent to [tex]\( (f \circ g)(x) \)[/tex] is:
[tex]\[ (f \circ g)(x) = 3x^2 + 5 \][/tex]
