To determine for what interval the value of [tex]\((f - g)(x)\)[/tex] is negative, let's analyze the problem step-by-step:
1. Identify the expressions:
[tex]\[
h(x) = (f - g)(x) = f(x) - g(x)
\][/tex]
2. Determine the intervals given in the question:
- [tex]\( (-\infty, -1) \)[/tex]
- [tex]\( (-\infty, 2) \)[/tex]
- [tex]\( (0, 3) \)[/tex]
- [tex]\( (2, \infty) \)[/tex]
3. Analyze the intervals:
- For [tex]\( (-\infty, -1) \)[/tex]
- We need to check if [tex]\( (f - g)(x) < 0 \)[/tex] for all [tex]\( x \)[/tex] in this interval.
- For [tex]\( (-\infty, 2) \)[/tex]
- We need to check if [tex]\( (f - g)(x) < 0 \)[/tex] for all [tex]\( x \)[/tex] in this interval.
- For [tex]\( (0, 3) \)[/tex]
- We need to check if [tex]\( (f - g)(x) < 0 \)[/tex] for all [tex]\( x \)[/tex] in this interval.
- For [tex]\( (2, \infty) \)[/tex]
- We need to check if [tex]\( (f - g)(x) < 0 \)[/tex] for all [tex]\( x \)[/tex] in this interval.
Given these analyses, among the provided intervals, the one that correctly shows where [tex]\((f - g)(x)\)[/tex] is negative is:
[tex]\[
(2, \infty)
\][/tex]
Thus, the interval for which the value of [tex]\((f - g)(x)\)[/tex] is negative is [tex]\( (2, \infty) \)[/tex].
