Answer :
To solve this question, we need to compare the behaviors of both functions [tex]\( f \)[/tex] and [tex]\( g \)[/tex] on the interval [tex]\([-1, 2]\)[/tex] based on the given table for [tex]\( f \)[/tex] and the equation for [tex]\( g \)[/tex].
### Analyzing Function [tex]\( f \)[/tex]
From the table, we have the values of [tex]\( f(x) \)[/tex]:
- [tex]\( f(-1) = -22 \)[/tex]
- [tex]\( f(0) = -10 \)[/tex]
- [tex]\( f(1) = -4 \)[/tex]
- [tex]\( f(2) = -1 \)[/tex]
1. Is [tex]\( f(x) \)[/tex] increasing?
Check if each subsequent value is greater than the previous one:
- [tex]\( f(0) > f(-1) \)[/tex] because [tex]\(-10 > -22\)[/tex]
- [tex]\( f(1) > f(0) \)[/tex] because [tex]\(-4 > -10\)[/tex]
- [tex]\( f(2) > f(1) \)[/tex] because [tex]\(-1 > -4\)[/tex]
Since all these conditions are true, [tex]\( f(x) \)[/tex] is increasing on the interval [tex]\([-1, 2]\)[/tex].
2. Is [tex]\( f(x) \)[/tex] negative on this interval?
Checking each value:
- [tex]\( f(-1) = -22 < 0 \)[/tex]
- [tex]\( f(0) = -10 < 0 \)[/tex]
- [tex]\( f(1) = -4 < 0 \)[/tex]
- [tex]\( f(2) = -1 < 0 \)[/tex]
All values are negative, so function [tex]\( f(x) \)[/tex] is negative on this interval.
### Analyzing Function [tex]\( g \)[/tex]
Function [tex]\( g(x) \)[/tex] is given by the equation:
[tex]\[ g(x) = -18\left(\frac{1}{3}\right)^x + 2 \][/tex]
Calculate the values:
- [tex]\( g(-1) = -18 \cdot 3 + 2 = -54 + 2 = -52 \)[/tex]
- [tex]\( g(0) = -18 \cdot 1 + 2 = -18 + 2 = -16 \)[/tex]
- [tex]\( g(1) = -18 \cdot \frac{1}{3} + 2 = -6 + 2 = -4 \)[/tex]
- [tex]\( g(2) = -18 \cdot \frac{1}{9} + 2 = -2 + 2 = 0 \)[/tex]
1. Is [tex]\( g(x) \)[/tex] increasing?
Check if each subsequent value is greater than the previous one:
- [tex]\( g(0) > g(-1) \)[/tex] because [tex]\(-16 > -52\)[/tex]
- [tex]\( g(1) > g(0) \)[/tex] because [tex]\(-4 > -16\)[/tex]
- [tex]\( g(2) > g(1) \)[/tex] because [tex]\(0 > -4\)[/tex]
Since all conditions are true, [tex]\( g(x) \)[/tex] is increasing on the interval [tex]\([-1, 2]\)[/tex].
### Compare Average Rates of Change
Average rate of change for [tex]\( f(x) \)[/tex]:
[tex]\[ \text{Change in } f(x) = f(2) - f(-1) = -1 - (-22) = 21 \][/tex]
[tex]\[ \text{Average rate of change} = \frac{21}{2 - (-1)} = \frac{21}{3} = 7 \][/tex]
Average rate of change for [tex]\( g(x) \)[/tex]:
[tex]\[ \text{Change in } g(x) = g(2) - g(-1) = 0 - (-52) = 52 \][/tex]
[tex]\[ \text{Average rate of change} = \frac{52}{2 - (-1)} = \frac{52}{3} \approx 17.33 \][/tex]
### Conclusion
- Both [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] are increasing on the interval [tex]\([-1, 2]\)[/tex].
- Function [tex]\( f(x) \)[/tex] is negative on this interval, while [tex]\( g(x) \)[/tex] is not fully below zero.
- Function [tex]\( g(x) \)[/tex] increases at a faster average rate (approximately 17.33) compared to [tex]\( f(x)\)[/tex] (which increases at a rate of 7).
Based on these findings, no option in the standard multiple-choice answers directly describes the scenario correctly. However, if contextual paraphrasing of choices can be considered, the comparisons should note both functions are increasing, with [tex]\( g(x) \)[/tex] showing a more significant average rate increase.
### Analyzing Function [tex]\( f \)[/tex]
From the table, we have the values of [tex]\( f(x) \)[/tex]:
- [tex]\( f(-1) = -22 \)[/tex]
- [tex]\( f(0) = -10 \)[/tex]
- [tex]\( f(1) = -4 \)[/tex]
- [tex]\( f(2) = -1 \)[/tex]
1. Is [tex]\( f(x) \)[/tex] increasing?
Check if each subsequent value is greater than the previous one:
- [tex]\( f(0) > f(-1) \)[/tex] because [tex]\(-10 > -22\)[/tex]
- [tex]\( f(1) > f(0) \)[/tex] because [tex]\(-4 > -10\)[/tex]
- [tex]\( f(2) > f(1) \)[/tex] because [tex]\(-1 > -4\)[/tex]
Since all these conditions are true, [tex]\( f(x) \)[/tex] is increasing on the interval [tex]\([-1, 2]\)[/tex].
2. Is [tex]\( f(x) \)[/tex] negative on this interval?
Checking each value:
- [tex]\( f(-1) = -22 < 0 \)[/tex]
- [tex]\( f(0) = -10 < 0 \)[/tex]
- [tex]\( f(1) = -4 < 0 \)[/tex]
- [tex]\( f(2) = -1 < 0 \)[/tex]
All values are negative, so function [tex]\( f(x) \)[/tex] is negative on this interval.
### Analyzing Function [tex]\( g \)[/tex]
Function [tex]\( g(x) \)[/tex] is given by the equation:
[tex]\[ g(x) = -18\left(\frac{1}{3}\right)^x + 2 \][/tex]
Calculate the values:
- [tex]\( g(-1) = -18 \cdot 3 + 2 = -54 + 2 = -52 \)[/tex]
- [tex]\( g(0) = -18 \cdot 1 + 2 = -18 + 2 = -16 \)[/tex]
- [tex]\( g(1) = -18 \cdot \frac{1}{3} + 2 = -6 + 2 = -4 \)[/tex]
- [tex]\( g(2) = -18 \cdot \frac{1}{9} + 2 = -2 + 2 = 0 \)[/tex]
1. Is [tex]\( g(x) \)[/tex] increasing?
Check if each subsequent value is greater than the previous one:
- [tex]\( g(0) > g(-1) \)[/tex] because [tex]\(-16 > -52\)[/tex]
- [tex]\( g(1) > g(0) \)[/tex] because [tex]\(-4 > -16\)[/tex]
- [tex]\( g(2) > g(1) \)[/tex] because [tex]\(0 > -4\)[/tex]
Since all conditions are true, [tex]\( g(x) \)[/tex] is increasing on the interval [tex]\([-1, 2]\)[/tex].
### Compare Average Rates of Change
Average rate of change for [tex]\( f(x) \)[/tex]:
[tex]\[ \text{Change in } f(x) = f(2) - f(-1) = -1 - (-22) = 21 \][/tex]
[tex]\[ \text{Average rate of change} = \frac{21}{2 - (-1)} = \frac{21}{3} = 7 \][/tex]
Average rate of change for [tex]\( g(x) \)[/tex]:
[tex]\[ \text{Change in } g(x) = g(2) - g(-1) = 0 - (-52) = 52 \][/tex]
[tex]\[ \text{Average rate of change} = \frac{52}{2 - (-1)} = \frac{52}{3} \approx 17.33 \][/tex]
### Conclusion
- Both [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] are increasing on the interval [tex]\([-1, 2]\)[/tex].
- Function [tex]\( f(x) \)[/tex] is negative on this interval, while [tex]\( g(x) \)[/tex] is not fully below zero.
- Function [tex]\( g(x) \)[/tex] increases at a faster average rate (approximately 17.33) compared to [tex]\( f(x)\)[/tex] (which increases at a rate of 7).
Based on these findings, no option in the standard multiple-choice answers directly describes the scenario correctly. However, if contextual paraphrasing of choices can be considered, the comparisons should note both functions are increasing, with [tex]\( g(x) \)[/tex] showing a more significant average rate increase.