Let's work through the given functions step-by-step to find [tex]\((f+g)(x)\)[/tex] and evaluate it at [tex]\( x = -4 \)[/tex].
First, we are given the functions:
[tex]\[ f(x) = x^2 + 1 \][/tex]
[tex]\[ g(x) = 5 - x \][/tex]
### Step 1: Add the two functions
To find [tex]\((f+g)(x)\)[/tex], we need to add [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]:
[tex]\[ (f+g)(x) = f(x) + g(x) \][/tex]
Substitute the expressions for [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]:
[tex]\[ (f+g)(x) = (x^2 + 1) + (5 - x) \][/tex]
### Step 2: Simplify the resulting expression
Combine like terms:
[tex]\[ (f+g)(x) = x^2 + 1 + 5 - x \][/tex]
[tex]\[ (f+g)(x) = x^2 - x + 6 \][/tex]
So, the simplified form of [tex]\((f+g)(x)\)[/tex] is:
[tex]\[ (f+g)(x) = x^2 - x + 6 \][/tex]
### Step 3: Evaluate [tex]\((f+g)(x)\)[/tex] at [tex]\( x = -4 \)[/tex]
To evaluate the resulting function at [tex]\( x = -4 \)[/tex]:
Substitute [tex]\( x = -4 \)[/tex] into [tex]\((f+g)(x)\)[/tex]:
[tex]\[ (f+g)(-4) = (-4)^2 - (-4) + 6 \][/tex]
Calculate each term:
[tex]\[ (-4)^2 = 16 \][/tex]
[tex]\[ -(-4) = 4 \][/tex]
[tex]\[ 6 = 6 \][/tex]
Now, combine these values:
[tex]\[ (f+g)(-4) = 16 + 4 + 6 \][/tex]
[tex]\[ (f+g)(-4) = 26 \][/tex]
### Conclusion
The resulting function [tex]\((f+g)(x)\)[/tex] is:
[tex]\[ (f+g)(x) = x^2 - x + 6 \][/tex]
When evaluating this function at [tex]\( x = -4 \)[/tex], we get:
[tex]\[ (f+g)(-4) = 26 \][/tex]
