To solve the problem of finding [tex]\( (f-g)(x) \)[/tex], we first need to understand what the function [tex]\( (f-g)(x) \)[/tex] represents.
Given functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex], [tex]\( (f-g)(x) \)[/tex] is defined as the difference between [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]. Specifically:
[tex]\[
(f-g)(x) = f(x) - g(x)
\][/tex]
The expression we need to identify is:
[tex]\[
(f-g)(x) = x^2 + x - 4
\][/tex]
Now let's examine each of the given choices to see which one matches our expression.
1. [tex]\( x^2 + x - 4 \)[/tex]
2. [tex]\( x^2 + x + 4 \)[/tex]
3. [tex]\( x^2 - x + 6 \)[/tex]
4. [tex]\( x^2 + x + 6 \)[/tex]
The given expression is:
[tex]\[
x^2 + x - 4
\][/tex]
Comparing this expression with each of the choices:
1. [tex]\( x^2 + x - 4 \)[/tex] matches this expression exactly.
2. [tex]\( x^2 + x + 4 \)[/tex] has a [tex]\( +4 \)[/tex] instead of [tex]\( -4 \)[/tex], so it does not match.
3. [tex]\( x^2 - x + 6 \)[/tex] includes a [tex]\( -x \)[/tex] and [tex]\( +6 \)[/tex] which are different from our terms.
4. [tex]\( x^2 + x + 6 \)[/tex] includes an additional [tex]\( +6 \)[/tex] instead of the [tex]\( -4 \)[/tex] in our expression.
The correct option is:
[tex]\[
x^2 + x - 4
\][/tex]
