To solve this problem, let's start by defining the functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] and then compute the sum of these functions to find [tex]\((f + g)(x)\)[/tex].
Given:
[tex]\[ f(x) = x^2 + 1 \][/tex]
[tex]\[ g(x) = 5 - x \][/tex]
We need to find [tex]\((f + g)(x)\)[/tex]. This is done by adding the expressions for [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] together:
[tex]\[ (f + g)(x) = f(x) + g(x) \][/tex]
Let's substitute [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] into the equation:
[tex]\[ (f + g)(x) = (x^2 + 1) + (5 - x) \][/tex]
Now, combine like terms:
1. Combine [tex]\( x^2 \)[/tex] terms:
[tex]\[ x^2 \][/tex]
2. Combine the constant terms:
[tex]\[ 1 + 5 = 6 \][/tex]
3. Combine the [tex]\(-x\)[/tex] term:
[tex]\[ -x \][/tex]
Putting it all together:
[tex]\[ (f + g)(x) = x^2 - x + 6 \][/tex]
So, the simplified expression for [tex]\((f + g)(x)\)[/tex] is [tex]\( x^2 - x + 6 \)[/tex].
Therefore, the correct answer from the given options is:
[tex]\[ \boxed{x^2 - x + 6} \][/tex]
