Answer :
To compare the two functions [tex]\( f \)[/tex] and [tex]\( g \)[/tex] over the interval [tex]\([-1, 2]\)[/tex] and find the correct statement, let's go through the following steps:
1. Identify Changes in [tex]\( f \)[/tex]:
- From the table, the values of [tex]\( f(x) \)[/tex] are [tex]\(-22\)[/tex], [tex]\(-10\)[/tex], [tex]\(-4\)[/tex], and [tex]\(-1\)[/tex] for [tex]\( x = -1, 0, 1, 2 \)[/tex], respectively.
2. Check if [tex]\( f(x) \)[/tex] is Increasing:
- Compare consecutive values: [tex]\(-22 < -10 < -4 < -1\)[/tex].
- Since each value is larger than the previous one, function [tex]\( f(x) \)[/tex] is increasing.
3. Calculate Average Rate of Change for [tex]\( f \)[/tex]:
- Use the formula for the average rate of change between [tex]\( x = -1 \)[/tex] and [tex]\( x = 2 \)[/tex]:
[tex]\[
\text{Rate for } f = \frac{f(2) - f(-1)}{2 - (-1)} = \frac{-1 - (-22)}{3} = \frac{21}{3} = 7
\][/tex]
4. Check if [tex]\( f(x) \)[/tex] is Negative:
- All values given for [tex]\( f(x) \)[/tex] ([tex]\(-22\)[/tex], [tex]\(-10\)[/tex], [tex]\(-4\)[/tex], [tex]\(-1\)[/tex]) are negative.
5. Analyze Function [tex]\( g \)[/tex]:
- Function [tex]\( g(x) = -18\left(\frac{1}{3}\right)^x + 2 \)[/tex].
6. Calculate [tex]\( g(x) \)[/tex] for Interval [tex]\([-1, 2]\)[/tex]:
- Evaluate [tex]\( g(-1) \)[/tex], [tex]\( g(0) \)[/tex], [tex]\( g(1) \)[/tex], [tex]\( g(2) \)[/tex] generating:
- [tex]\( g(-1) = 2 + 18 \times 3 = 56 \)[/tex]
- [tex]\( g(0) = 2 - 18 = -16 \)[/tex]
- [tex]\( g(1) = 2 - 6 = -4 \)[/tex]
- [tex]\( g(2) = 2 - 2 = 0 \)[/tex]
7. Check if [tex]\( g(x) \)[/tex] is Increasing:
- Sequentially compare values: [tex]\( -16 < -4 < 0 \)[/tex].
- As values are increasing, function [tex]\( g(x) \)[/tex] is increasing.
8. Calculate Average Rate of Change for [tex]\( g \)[/tex]:
- From [tex]\( g(-1) \)[/tex] to [tex]\( g(2) \)[/tex]:
[tex]\[
\text{Rate for } g = \frac{g(2) - g(-1)}{2 - (-1)} = \frac{0 - 56}{3} = \frac{-56}{3} \approx 17.33
\][/tex]
9. Check if [tex]\( g(x) \)[/tex] is Negative:
- Check values: [tex]\( -16, -4, 0 \)[/tex]. Not all values are negative, as [tex]\( g(2) = 0 \)[/tex].
Conclusion:
- Both [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] are increasing over [tex]\([-1, 2]\)[/tex].
- [tex]\(\text{Rate for } g (17.33) > \text{Rate for } f (7)\)[/tex], meaning [tex]\( g \)[/tex] increases at a faster average rate.
- Only [tex]\( f(x) \)[/tex] is negative for the entire interval.
Based on this analysis, the correct statement is option D: "Only function [tex]\( f \)[/tex] is increasing, but both functions are negative."
1. Identify Changes in [tex]\( f \)[/tex]:
- From the table, the values of [tex]\( f(x) \)[/tex] are [tex]\(-22\)[/tex], [tex]\(-10\)[/tex], [tex]\(-4\)[/tex], and [tex]\(-1\)[/tex] for [tex]\( x = -1, 0, 1, 2 \)[/tex], respectively.
2. Check if [tex]\( f(x) \)[/tex] is Increasing:
- Compare consecutive values: [tex]\(-22 < -10 < -4 < -1\)[/tex].
- Since each value is larger than the previous one, function [tex]\( f(x) \)[/tex] is increasing.
3. Calculate Average Rate of Change for [tex]\( f \)[/tex]:
- Use the formula for the average rate of change between [tex]\( x = -1 \)[/tex] and [tex]\( x = 2 \)[/tex]:
[tex]\[
\text{Rate for } f = \frac{f(2) - f(-1)}{2 - (-1)} = \frac{-1 - (-22)}{3} = \frac{21}{3} = 7
\][/tex]
4. Check if [tex]\( f(x) \)[/tex] is Negative:
- All values given for [tex]\( f(x) \)[/tex] ([tex]\(-22\)[/tex], [tex]\(-10\)[/tex], [tex]\(-4\)[/tex], [tex]\(-1\)[/tex]) are negative.
5. Analyze Function [tex]\( g \)[/tex]:
- Function [tex]\( g(x) = -18\left(\frac{1}{3}\right)^x + 2 \)[/tex].
6. Calculate [tex]\( g(x) \)[/tex] for Interval [tex]\([-1, 2]\)[/tex]:
- Evaluate [tex]\( g(-1) \)[/tex], [tex]\( g(0) \)[/tex], [tex]\( g(1) \)[/tex], [tex]\( g(2) \)[/tex] generating:
- [tex]\( g(-1) = 2 + 18 \times 3 = 56 \)[/tex]
- [tex]\( g(0) = 2 - 18 = -16 \)[/tex]
- [tex]\( g(1) = 2 - 6 = -4 \)[/tex]
- [tex]\( g(2) = 2 - 2 = 0 \)[/tex]
7. Check if [tex]\( g(x) \)[/tex] is Increasing:
- Sequentially compare values: [tex]\( -16 < -4 < 0 \)[/tex].
- As values are increasing, function [tex]\( g(x) \)[/tex] is increasing.
8. Calculate Average Rate of Change for [tex]\( g \)[/tex]:
- From [tex]\( g(-1) \)[/tex] to [tex]\( g(2) \)[/tex]:
[tex]\[
\text{Rate for } g = \frac{g(2) - g(-1)}{2 - (-1)} = \frac{0 - 56}{3} = \frac{-56}{3} \approx 17.33
\][/tex]
9. Check if [tex]\( g(x) \)[/tex] is Negative:
- Check values: [tex]\( -16, -4, 0 \)[/tex]. Not all values are negative, as [tex]\( g(2) = 0 \)[/tex].
Conclusion:
- Both [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] are increasing over [tex]\([-1, 2]\)[/tex].
- [tex]\(\text{Rate for } g (17.33) > \text{Rate for } f (7)\)[/tex], meaning [tex]\( g \)[/tex] increases at a faster average rate.
- Only [tex]\( f(x) \)[/tex] is negative for the entire interval.
Based on this analysis, the correct statement is option D: "Only function [tex]\( f \)[/tex] is increasing, but both functions are negative."