Let's solve the problem step-by-step.
1. List the given values from the tables:
- For [tex]\( x \)[/tex] values: [tex]\([-10, -7, -4, -1, 2]\)[/tex]
- For [tex]\( f(x) \)[/tex] values: [tex]\([-3, 6, 15, 24, 33]\)[/tex]
- For [tex]\( g(x) \)[/tex] values: [tex]\([9, 6, 3, 0, -3]\)[/tex]
2. Calculate [tex]\((f-g)(x)\)[/tex] for each corresponding [tex]\( x \)[/tex] value:
- When [tex]\( x = -10 \)[/tex]:
[tex]\[
(f-g)(-10) = f(-10) - g(-10) = -3 - 9 = -12
\][/tex]
- When [tex]\( x = -7 \)[/tex]:
[tex]\[
(f-g)(-7) = f(-7) - g(-7) = 6 - 6 = 0
\][/tex]
- When [tex]\( x = -4 \)[/tex]:
[tex]\[
(f-g)(-4) = f(-4) - g(-4) = 15 - 3 = 12
\][/tex]
- When [tex]\( x = -1 \)[/tex]:
[tex]\[
(f-g)(-1) = f(-1) - g(-1) = 24 - 0 = 24
\][/tex]
- When [tex]\( x = 2 \)[/tex]:
[tex]\[
(f-g)(2) = f(2) - g(2) = 33 - (-3) = 33 + 3 = 36
\][/tex]
Collect these results:
[tex]\[
(f - g)(x) = [-12, 0, 12, 24, 36]
\][/tex]
3. Identify where [tex]\((f - g)(x)\)[/tex] is positive:
- [tex]\((f - g)(-10) = -12\)[/tex] (not positive)
- [tex]\((f - g)(-7) = 0\)[/tex] (not positive)
- [tex]\((f - g)(-4) = 12\)[/tex] (positive)
- [tex]\((f - g)(-1) = 24\)[/tex] (positive)
- [tex]\((f - g)(2) = 36\)[/tex] (positive)
4. List the [tex]\( x \)[/tex] values where [tex]\((f - g)(x)\)[/tex] is positive:
[tex]\[
x = -4, -1, 2
\][/tex]
Therefore, [tex]\((f - g)(x)\)[/tex] is positive at [tex]\( x = -4, -1, \)[/tex] and [tex]\( 2 \)[/tex].
