Answer :
To compare the two functions [tex]\( f \)[/tex] and [tex]\( g \)[/tex] on the interval [tex]\([-1,2]\)[/tex], let's follow these steps:
1. Evaluate Function [tex]\( f \)[/tex]:
- From the table of values, the given [tex]\( f(x) \)[/tex] values for [tex]\( x = -1, 0, 1, 2 \)[/tex] are:
- [tex]\( f(-1) = -22 \)[/tex]
- [tex]\( f(0) = -10 \)[/tex]
- [tex]\( f(1) = -4 \)[/tex]
- [tex]\( f(2) = -1 \)[/tex]
- We can see that the values of [tex]\( f(x) \)[/tex] consistently increase from [tex]\(-22\)[/tex] to [tex]\(-1\)[/tex] as [tex]\( x \)[/tex] moves from [tex]\(-1\)[/tex] to [tex]\( 2\)[/tex]. Hence, function [tex]\( f \)[/tex] is increasing on the interval [tex]\([-1, 2]\)[/tex].
2. Evaluate Function [tex]\( g \)[/tex]:
- The function [tex]\( g(x) = -18 \left(\frac{1}{3}\right)^x + 2 \)[/tex] needs to be evaluated at the same points:
- [tex]\( g(-1) = -18\left(\frac{1}{3}\right)^{-1} + 2 = -18 \times 3 + 2 = -52 \)[/tex]
- [tex]\( g(0) = -18\left(\frac{1}{3}\right)^{0} + 2 = -18 + 2 = -16 \)[/tex]
- [tex]\( g(1) = -18\left(\frac{1}{3}\right)^{1} + 2 = -6 + 2 = -4 \)[/tex]
- [tex]\( g(2) = -18\left(\frac{1}{3}\right)^{2} + 2 = -2 + 2 = 0 \)[/tex]
- Observing that the values of [tex]\( g(x) \)[/tex] are increasing from [tex]\(-52\)[/tex] to [tex]\(0\)[/tex], we can conclude that function [tex]\( g \)[/tex] is also increasing on the interval [tex]\([-1, 2]\)[/tex].
3. Compare Average Rates of Change:
- The average rate of change for [tex]\( f \)[/tex] from [tex]\( x = -1 \)[/tex] to [tex]\( x = 2 \)[/tex] is calculated as:
[tex]\[
\text{Rate for } f = \frac{f(2) - f(-1)}{2 - (-1)} = \frac{-1 - (-22)}{3} = \frac{21}{3} = 7
\][/tex]
- Similarly, the average rate of change for [tex]\( g \)[/tex] from [tex]\( x = -1 \)[/tex] to [tex]\( x = 2 \)[/tex]:
[tex]\[
\text{Rate for } g = \frac{g(2) - g(-1)}{2 - (-1)} = \frac{0 - (-52)}{3} = \frac{52}{3} \approx 17.33
\][/tex]
4. Make the Comparison:
- Both [tex]\( f \)[/tex] and [tex]\( g \)[/tex] are increasing over the interval [tex]\([-1, 2]\)[/tex].
- However, function [tex]\( g \)[/tex] has a larger average rate of change (approximately 17.33) compared to function [tex]\( f \)[/tex] (which has a rate of 7). Hence, [tex]\( g \)[/tex] increases at a faster average rate than [tex]\( f \)[/tex].
Therefore, the correct statement that compares the two functions is:
C. Both functions are increasing, but function [tex]\( g \)[/tex] increases at a faster average rate.
1. Evaluate Function [tex]\( f \)[/tex]:
- From the table of values, the given [tex]\( f(x) \)[/tex] values for [tex]\( x = -1, 0, 1, 2 \)[/tex] are:
- [tex]\( f(-1) = -22 \)[/tex]
- [tex]\( f(0) = -10 \)[/tex]
- [tex]\( f(1) = -4 \)[/tex]
- [tex]\( f(2) = -1 \)[/tex]
- We can see that the values of [tex]\( f(x) \)[/tex] consistently increase from [tex]\(-22\)[/tex] to [tex]\(-1\)[/tex] as [tex]\( x \)[/tex] moves from [tex]\(-1\)[/tex] to [tex]\( 2\)[/tex]. Hence, function [tex]\( f \)[/tex] is increasing on the interval [tex]\([-1, 2]\)[/tex].
2. Evaluate Function [tex]\( g \)[/tex]:
- The function [tex]\( g(x) = -18 \left(\frac{1}{3}\right)^x + 2 \)[/tex] needs to be evaluated at the same points:
- [tex]\( g(-1) = -18\left(\frac{1}{3}\right)^{-1} + 2 = -18 \times 3 + 2 = -52 \)[/tex]
- [tex]\( g(0) = -18\left(\frac{1}{3}\right)^{0} + 2 = -18 + 2 = -16 \)[/tex]
- [tex]\( g(1) = -18\left(\frac{1}{3}\right)^{1} + 2 = -6 + 2 = -4 \)[/tex]
- [tex]\( g(2) = -18\left(\frac{1}{3}\right)^{2} + 2 = -2 + 2 = 0 \)[/tex]
- Observing that the values of [tex]\( g(x) \)[/tex] are increasing from [tex]\(-52\)[/tex] to [tex]\(0\)[/tex], we can conclude that function [tex]\( g \)[/tex] is also increasing on the interval [tex]\([-1, 2]\)[/tex].
3. Compare Average Rates of Change:
- The average rate of change for [tex]\( f \)[/tex] from [tex]\( x = -1 \)[/tex] to [tex]\( x = 2 \)[/tex] is calculated as:
[tex]\[
\text{Rate for } f = \frac{f(2) - f(-1)}{2 - (-1)} = \frac{-1 - (-22)}{3} = \frac{21}{3} = 7
\][/tex]
- Similarly, the average rate of change for [tex]\( g \)[/tex] from [tex]\( x = -1 \)[/tex] to [tex]\( x = 2 \)[/tex]:
[tex]\[
\text{Rate for } g = \frac{g(2) - g(-1)}{2 - (-1)} = \frac{0 - (-52)}{3} = \frac{52}{3} \approx 17.33
\][/tex]
4. Make the Comparison:
- Both [tex]\( f \)[/tex] and [tex]\( g \)[/tex] are increasing over the interval [tex]\([-1, 2]\)[/tex].
- However, function [tex]\( g \)[/tex] has a larger average rate of change (approximately 17.33) compared to function [tex]\( f \)[/tex] (which has a rate of 7). Hence, [tex]\( g \)[/tex] increases at a faster average rate than [tex]\( f \)[/tex].
Therefore, the correct statement that compares the two functions is:
C. Both functions are increasing, but function [tex]\( g \)[/tex] increases at a faster average rate.