To solve the problem of performing the operation [tex]\((f - g)(x)\)[/tex] given the functions [tex]\( f(x) = x^2 + 1 \)[/tex] and [tex]\( g(x) = 5 - x \)[/tex], we should subtract [tex]\( g(x) \)[/tex] from [tex]\( f(x) \)[/tex].
Here's a detailed, step-by-step solution:
1. Write down the given functions:
[tex]\[
f(x) = x^2 + 1
\][/tex]
[tex]\[
g(x) = 5 - x
\][/tex]
2. Formulate the expression for [tex]\((f - g)(x)\)[/tex]:
[tex]\[
(f - g)(x) = f(x) - g(x)
\][/tex]
3. Substitute the given functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] into the expression:
[tex]\[
(f - g)(x) = (x^2 + 1) - (5 - x)
\][/tex]
4. Distribute the subtraction across the terms inside the parentheses:
[tex]\[
(f - g)(x) = x^2 + 1 - 5 + x
\][/tex]
5. Combine like terms:
[tex]\[
x^2 + x + (1 - 5)
\][/tex]
[tex]\[
x^2 + x - 4
\][/tex]
The expression for [tex]\((f - g)(x)\)[/tex] simplifies to [tex]\( x^2 + x - 4 \)[/tex].
Therefore, the correct answer is:
[tex]\[
\boxed{x^2 + x - 4}
\][/tex]
From the given options, the correct choice is:
[tex]\[
\text{a. } x^2 + x - 4
\][/tex]
