To determine where [tex]\((f-g)(x)\)[/tex] is positive, we'll follow these steps:
1. Identify the values given:
- The [tex]\(x\)[/tex]-values are: [tex]\(-10, -7, -4, -1, 2\)[/tex]
- The corresponding values of [tex]\(f(x)\)[/tex] are: [tex]\(-3, 6, 15, 24, 33\)[/tex]
- The corresponding values of [tex]\(g(x)\)[/tex] are: [tex]\(9, 6, 3, 0, -3\)[/tex]
2. Calculate [tex]\((f-g)(x)\)[/tex]:
- At [tex]\(x = -10\)[/tex]: [tex]\(f(-10) - g(-10) = -3 - 9 = -12\)[/tex]
- At [tex]\(x = -7\)[/tex]: [tex]\(f(-7) - g(-7) = 6 - 6 = 0\)[/tex]
- At [tex]\(x = -4\)[/tex]: [tex]\(f(-4) - g(-4) = 15 - 3 = 12\)[/tex]
- At [tex]\(x = -1\)[/tex]: [tex]\(f(-1) - g(-1) = 24 - 0 = 24\)[/tex]
- At [tex]\(x = 2\)[/tex]: [tex]\(f(2) - g(2) = 33 - (-3) = 36\)[/tex]
So, the differences [tex]\((f-g)(x)\)[/tex] are:
- At [tex]\(x = -10\)[/tex]: [tex]\(-12\)[/tex]
- At [tex]\(x = -7\)[/tex]: [tex]\(0\)[/tex]
- At [tex]\(x = -4\)[/tex]: [tex]\(12\)[/tex]
- At [tex]\(x = -1\)[/tex]: [tex]\(24\)[/tex]
- At [tex]\(x = 2\)[/tex]: [tex]\(36\)[/tex]
3. Determine where [tex]\((f-g)(x)\)[/tex] is positive:
- We can see from the calculations that [tex]\((f-g)(x)\)[/tex] is positive at [tex]\(x = -4, -1,\text{ and } 2\)[/tex].
4. Find the range of [tex]\(x\)[/tex] values where [tex]\((f-g)(x)\)[/tex] is positive:
- For [tex]\((f-g)(x)\)[/tex] to be consistently positive, we observe the smallest [tex]\(x\)[/tex] value where it turns positive is [tex]\(-4\)[/tex] and it remains positive for all subsequent [tex]\(x\)[/tex] values.
- Since [tex]\((f-g)(x)\)[/tex] is zero at [tex]\(x = -7\)[/tex] and negative at [tex]\(x = -10\)[/tex], the positive interval must start just after [tex]\(-7\)[/tex].
Therefore, the interval where [tex]\((f-g)(x)\)[/tex] is positive is [tex]\((-7, \infty)\)[/tex].
