To determine the interval where the value of [tex]\((f-g)(x)\)[/tex] is negative, we need to analyze the intervals provided and decide which one best fits this condition.
Analyzing the information, we have the following intervals to consider:
1. [tex]\((- \infty, -1)\)[/tex]
2. [tex]\((- \infty, 2)\)[/tex]
3. [tex]\((0, 3)\)[/tex]
4. [tex]\((2, \infty)\)[/tex]
We need to pick the interval where the difference [tex]\((f-g)(x)\)[/tex] remains negative throughout.
After considering the intervals, it is clear that:
- Interval [tex]\((- \infty, -1)\)[/tex] and [tex]\((f-g)(x)\)[/tex] is indeed negative.
- Interval [tex]\((- \infty, 2)\)[/tex] but this extends beyond [tex]\(-1\)[/tex] and isn't fitting for the exact negative condition expected.
- Interval [tex]\((0, 3)\)[/tex] is within a span where [tex]\((f-g)(x)\)[/tex] generally changes.
- Interval [tex]\((2, \infty)\)[/tex] is beyond any suitable span resulting in [tex]\((f-g)(x)\)[/tex] being purely negative.
Given all the intervals above, the correct interval ensuring [tex]\((f-g)(x)\)[/tex] is negative is:
[tex]\[
(-\infty, -1)
\][/tex]
Hence, the interval where [tex]\((f-g)(x)\)[/tex] is negative is [tex]\( (-\infty, -1) \)[/tex].
