To find [tex]\((f \circ g)(x)\)[/tex], which is the composition of the functions [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex], we need to follow these steps:
### Step-by-Step Solution:
1. Understand the composition notation:
[tex]\((f \circ g)(x)\)[/tex] means [tex]\(f(g(x))\)[/tex]. We need to evaluate [tex]\(f\)[/tex] at [tex]\(g(x)\)[/tex], which means substituting [tex]\(g(x)\)[/tex] into [tex]\(f(x)\)[/tex].
2. Identify the functions:
[tex]\[
f(x) = -3x + 4
\][/tex]
[tex]\[
g(x) = x^2 + 1
\][/tex]
3. Substitute [tex]\(g(x)\)[/tex] into [tex]\(f(x)\)[/tex]:
We replace [tex]\(x\)[/tex] in [tex]\(f(x)\)[/tex] with [tex]\(g(x)\)[/tex]:
[tex]\[
(f \circ g)(x) = f(g(x)) = f(x^2 + 1)
\][/tex]
4. Evaluate [tex]\(f(g(x))\)[/tex]:
Substitute [tex]\(x^2 + 1\)[/tex] into the function [tex]\(f(x)\)[/tex]:
[tex]\[
f(x^2 + 1) = -3(x^2 + 1) + 4
\][/tex]
5. Distribute and simplify:
[tex]\[
= -3(x^2) - 3(1) + 4
\][/tex]
[tex]\[
= -3x^2 - 3 + 4
\][/tex]
[tex]\[
= -3x^2 + 1
\][/tex]
So, the simplified form of [tex]\((f \circ g)(x)\)[/tex] is:
[tex]\[
(f \circ g)(x) = -3x^2 + 1
\][/tex]
Thus, the final answer is:
[tex]\[
(f \circ g)(x) = -3x^2 + 1
\][/tex]
