Answer :
To determine which statement correctly compares the functions [tex]\( f \)[/tex] and [tex]\( g \)[/tex] on the interval [tex]\([-1, 2]\)[/tex], we'll analyze the behavior of both functions over this interval.
Step 1: Examine Function [tex]\( f(x) \)[/tex]
The given values of [tex]\( f(x) \)[/tex] corresponding to [tex]\( x = -1 \)[/tex] to [tex]\( x = 2 \)[/tex] are:
- [tex]\( f(-1) = -22 \)[/tex]
- [tex]\( f(0) = -10 \)[/tex]
- [tex]\( f(1) = -4 \)[/tex]
- [tex]\( f(2) = -1 \)[/tex]
Determine if [tex]\( f(x) \)[/tex] is increasing:
Check if the function values form an increasing sequence:
- From [tex]\( f(-1) = -22 \)[/tex] to [tex]\( f(0) = -10 \)[/tex], [tex]\( -22 < -10 \)[/tex].
- From [tex]\( f(0) = -10 \)[/tex] to [tex]\( f(1) = -4 \)[/tex], [tex]\( -10 < -4 \)[/tex].
- From [tex]\( f(1) = -4 \)[/tex] to [tex]\( f(2) = -1 \)[/tex], [tex]\( -4 < -1 \)[/tex].
Since [tex]\( f(x) \)[/tex] is strictly increasing from [tex]\( x = -1 \)[/tex] to [tex]\( x = 2 \)[/tex], function [tex]\( f(x) \)[/tex] is increasing.
Determine if [tex]\( f(x) \)[/tex] is negative:
All values of [tex]\( f(x) \)[/tex] provided are negative, so [tex]\( f(x) \)[/tex] is negative over this interval.
Average rate of change for [tex]\( f(x) \)[/tex]:
This is calculated using the endpoints of the interval:
[tex]\[
\text{Average rate of change of } f = \frac{f(2) - f(-1)}{2 - (-1)} = \frac{-1 - (-22)}{2 - (-1)} = \frac{21}{3} = 7
\][/tex]
Step 2: Examine Function [tex]\( g(x) = -18\left(\frac{1}{3}\right)^x + 2 \)[/tex]
Calculate [tex]\( g(x) \)[/tex] at each point in the interval [tex]\([-1, 2]\)[/tex]:
- [tex]\( g(-1) = -18 \times 3 + 2 = -54 + 2 = -52 \)[/tex]
- [tex]\( g(0) = -18 \times 1 + 2 = -18 + 2 = -16 \)[/tex]
- [tex]\( g(1) = -18 \times \frac{1}{3} + 2 = -6 + 2 = -4 \)[/tex]
- [tex]\( g(2) = -18 \times \frac{1}{9} + 2 = -2 + 2 = 0 \)[/tex]
Determine if [tex]\( g(x) \)[/tex] is increasing:
Check if the function values increase:
- From [tex]\( g(-1) = -52 \)[/tex] to [tex]\( g(0) = -16 \)[/tex], [tex]\( -52 < -16 \)[/tex].
- From [tex]\( g(0) = -16 \)[/tex] to [tex]\( g(1) = -4 \)[/tex], [tex]\( -16 < -4 \)[/tex].
- From [tex]\( g(1) = -4 \)[/tex] to [tex]\( g(2) = 0 \)[/tex], [tex]\( -4 < 0 \)[/tex].
Function [tex]\( g(x) \)[/tex] is also increasing from [tex]\( x = -1 \)[/tex] to [tex]\( x = 2 \)[/tex].
Determine if [tex]\( g(x) \)[/tex] is negative:
Not all values are negative (e.g., [tex]\( g(2) = 0 \)[/tex]), so [tex]\( g(x) \)[/tex] is not consistently negative over this interval.
Average rate of change for [tex]\( g(x) \)[/tex]:
[tex]\[
\text{Average rate of change of } g = \frac{g(2) - g(-1)}{2 - (-1)} = \frac{0 - (-52)}{3} = \frac{52}{3} \approx 17.33
\][/tex]
Conclusion:
- Both functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] are increasing on the interval [tex]\([-1, 2]\)[/tex].
- The average rate of change of [tex]\( f(x) \)[/tex] is 7, and for [tex]\( g(x) \)[/tex] it is approximately 17.33.
Thus, the correct statement is:
D. Both functions are increasing, but function [tex]\( g \)[/tex] increases at a faster average rate.
Step 1: Examine Function [tex]\( f(x) \)[/tex]
The given values of [tex]\( f(x) \)[/tex] corresponding to [tex]\( x = -1 \)[/tex] to [tex]\( x = 2 \)[/tex] are:
- [tex]\( f(-1) = -22 \)[/tex]
- [tex]\( f(0) = -10 \)[/tex]
- [tex]\( f(1) = -4 \)[/tex]
- [tex]\( f(2) = -1 \)[/tex]
Determine if [tex]\( f(x) \)[/tex] is increasing:
Check if the function values form an increasing sequence:
- From [tex]\( f(-1) = -22 \)[/tex] to [tex]\( f(0) = -10 \)[/tex], [tex]\( -22 < -10 \)[/tex].
- From [tex]\( f(0) = -10 \)[/tex] to [tex]\( f(1) = -4 \)[/tex], [tex]\( -10 < -4 \)[/tex].
- From [tex]\( f(1) = -4 \)[/tex] to [tex]\( f(2) = -1 \)[/tex], [tex]\( -4 < -1 \)[/tex].
Since [tex]\( f(x) \)[/tex] is strictly increasing from [tex]\( x = -1 \)[/tex] to [tex]\( x = 2 \)[/tex], function [tex]\( f(x) \)[/tex] is increasing.
Determine if [tex]\( f(x) \)[/tex] is negative:
All values of [tex]\( f(x) \)[/tex] provided are negative, so [tex]\( f(x) \)[/tex] is negative over this interval.
Average rate of change for [tex]\( f(x) \)[/tex]:
This is calculated using the endpoints of the interval:
[tex]\[
\text{Average rate of change of } f = \frac{f(2) - f(-1)}{2 - (-1)} = \frac{-1 - (-22)}{2 - (-1)} = \frac{21}{3} = 7
\][/tex]
Step 2: Examine Function [tex]\( g(x) = -18\left(\frac{1}{3}\right)^x + 2 \)[/tex]
Calculate [tex]\( g(x) \)[/tex] at each point in the interval [tex]\([-1, 2]\)[/tex]:
- [tex]\( g(-1) = -18 \times 3 + 2 = -54 + 2 = -52 \)[/tex]
- [tex]\( g(0) = -18 \times 1 + 2 = -18 + 2 = -16 \)[/tex]
- [tex]\( g(1) = -18 \times \frac{1}{3} + 2 = -6 + 2 = -4 \)[/tex]
- [tex]\( g(2) = -18 \times \frac{1}{9} + 2 = -2 + 2 = 0 \)[/tex]
Determine if [tex]\( g(x) \)[/tex] is increasing:
Check if the function values increase:
- From [tex]\( g(-1) = -52 \)[/tex] to [tex]\( g(0) = -16 \)[/tex], [tex]\( -52 < -16 \)[/tex].
- From [tex]\( g(0) = -16 \)[/tex] to [tex]\( g(1) = -4 \)[/tex], [tex]\( -16 < -4 \)[/tex].
- From [tex]\( g(1) = -4 \)[/tex] to [tex]\( g(2) = 0 \)[/tex], [tex]\( -4 < 0 \)[/tex].
Function [tex]\( g(x) \)[/tex] is also increasing from [tex]\( x = -1 \)[/tex] to [tex]\( x = 2 \)[/tex].
Determine if [tex]\( g(x) \)[/tex] is negative:
Not all values are negative (e.g., [tex]\( g(2) = 0 \)[/tex]), so [tex]\( g(x) \)[/tex] is not consistently negative over this interval.
Average rate of change for [tex]\( g(x) \)[/tex]:
[tex]\[
\text{Average rate of change of } g = \frac{g(2) - g(-1)}{2 - (-1)} = \frac{0 - (-52)}{3} = \frac{52}{3} \approx 17.33
\][/tex]
Conclusion:
- Both functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] are increasing on the interval [tex]\([-1, 2]\)[/tex].
- The average rate of change of [tex]\( f(x) \)[/tex] is 7, and for [tex]\( g(x) \)[/tex] it is approximately 17.33.
Thus, the correct statement is:
D. Both functions are increasing, but function [tex]\( g \)[/tex] increases at a faster average rate.