To find [tex]\((f-g)(x)\)[/tex], we need to subtract [tex]\(g(x)\)[/tex] from [tex]\(f(x)\)[/tex]. We start with the definitions of the functions [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]:
[tex]\[ f(x) = x^2 + 1 \][/tex]
[tex]\[ g(x) = 5 - x \][/tex]
We want to find:
[tex]\[ (f - g)(x) = f(x) - g(x) \][/tex]
Substituting in the given functions:
[tex]\[ (f - g)(x) = (x^2 + 1) - (5 - x) \][/tex]
Next, we distribute the subtraction across the terms inside the parentheses:
[tex]\[ (f - g)(x) = x^2 + 1 - 5 + x \][/tex]
Now, combine like terms:
[tex]\[ (f - g)(x) = x^2 + x + 1 - 5 \][/tex]
[tex]\[ (f - g)(x) = x^2 + x - 4 \][/tex]
So, the simplified expression for [tex]\((f - g)(x)\)[/tex] is:
[tex]\[ x^2 + x - 4 \][/tex]
Therefore, the correct answer from the given options is:
[tex]\[ \boxed{x^2 + x - 4} \][/tex]
