To find the combined function [tex]\((f+g)(x)\)[/tex] given [tex]\(f(x) = x^2 + 1\)[/tex] and [tex]\(g(x) = 5 - x\)[/tex], we need to sum the values of [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]. Let's break this down step-by-step:
1. Write down the functions:
- [tex]\( f(x) = x^2 + 1 \)[/tex]
- [tex]\( g(x) = 5 - x \)[/tex]
2. Determine the combined function (f+g)(x):
[tex]\[
(f+g)(x) = f(x) + g(x)
\][/tex]
3. Substitute the given functions into the combined function:
[tex]\[
(f+g)(x) = (x^2 + 1) + (5 - x)
\][/tex]
4. Combine like terms:
[tex]\[
(f+g)(x) = x^2 + 1 + 5 - x
\][/tex]
[tex]\[
(f+g)(x) = x^2 - x + 6
\][/tex]
Thus, the function [tex]\((f+g)(x)\)[/tex] is:
[tex]\[
x^2 - x + 6
\][/tex]
The correct answer from the given options is:
[tex]\[ x^2 - x + 6 \][/tex]
So, the answer is:
[tex]\[ x^2 - x + 6 \][/tex]
