Answer :
To compare the two functions [tex]\( f \)[/tex] and [tex]\( g \)[/tex] on the interval [tex]\([-1, 2]\)[/tex], let's break down the steps to find which option is correct:
1. Identify Function Values:
- For function [tex]\( f \)[/tex]:
- The table provides values for [tex]\( f(x) \)[/tex]: [tex]\( f(-1) = -22 \)[/tex], [tex]\( f(0) = -10 \)[/tex], [tex]\( f(1) = -4 \)[/tex], [tex]\( f(2) = -1 \)[/tex].
2. Check if Function [tex]\( f \)[/tex] is Increasing:
- Calculate the changes as [tex]\( x \)[/tex] increases:
- From [tex]\( x = -1 \)[/tex] to [tex]\( x = 0 \)[/tex], [tex]\( f \)[/tex] changes from [tex]\(-22\)[/tex] to [tex]\(-10\)[/tex]: The change is [tex]\( +12 \)[/tex].
- From [tex]\( x = 0 \)[/tex] to [tex]\( x = 1 \)[/tex], [tex]\( f \)[/tex] changes from [tex]\(-10\)[/tex] to [tex]\(-4\)[/tex]: The change is [tex]\( +6 \)[/tex].
- From [tex]\( x = 1 \)[/tex] to [tex]\( x = 2 \)[/tex], [tex]\( f \)[/tex] changes from [tex]\(-4\)[/tex] to [tex]\(-1\)[/tex]: The change is [tex]\( +3 \)[/tex].
- All these changes are positive, so function [tex]\( f \)[/tex] is increasing.
3. Identify Function [tex]\( g \)[/tex] Values and Changes:
- Equation for function [tex]\( g \)[/tex] is [tex]\( g(x) = -18\left(\frac{1}{3}\right)^x + 2 \)[/tex].
- Calculate [tex]\( g(x) \)[/tex] values for [tex]\( x = -1, 0, 1, 2 \)[/tex]:
- [tex]\( g(-1) = -18\left(\frac{1}{3}\right)^{-1} + 2 = 36 + 2 = 38 \)[/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]
- Calculate changes for [tex]\( g(x) \)[/tex]:
- From [tex]\( x = -1 \)[/tex] to [tex]\( x = 0 \)[/tex]: Change is [tex]\(-16 - 38 = -54\)[/tex].
- From [tex]\( x = 0 \)[/tex] to [tex]\( x = 1 \)[/tex]: Change is [tex]\(-4 - (-16) = 12\)[/tex].
- From [tex]\( x = 1 \)[/tex] to [tex]\( x = 2 \)[/tex]: Change is [tex]\(0 - (-4) = 4\)[/tex].
4. Determine Function [tex]\( g \)[/tex] Behavior:
- Just like with function [tex]\( f \)[/tex], calculate changes over the interval:
- Since both earlier changes are positive for [tex]\( x = 0 \)[/tex] to [tex]\( x = 2 \)[/tex], function [tex]\( g \)[/tex] is increasing.
5. Average Rate of Increase:
- Find average rate of increase for both functions:
- Average rate for [tex]\( f \)[/tex] is calculated as [tex]\((12 + 6 + 3)/3 = 7\)[/tex].
- Average rate for [tex]\( g \)[/tex] is [tex]\((12 + 4)/2 = 8\)[/tex].
6. Compare the Functions:
- Both functions are increasing over the interval.
- Function [tex]\( g \)[/tex] increases at a faster average rate than function [tex]\( f \)[/tex].
7. Conclusion:
- Since both functions are increasing and [tex]\( g(x) \)[/tex] increases at a faster average rate on the interval, the correct statement is:
D. Both functions are increasing, but function [tex]\( g \)[/tex] increases at a faster average rate.
1. Identify Function Values:
- For function [tex]\( f \)[/tex]:
- The table provides values for [tex]\( f(x) \)[/tex]: [tex]\( f(-1) = -22 \)[/tex], [tex]\( f(0) = -10 \)[/tex], [tex]\( f(1) = -4 \)[/tex], [tex]\( f(2) = -1 \)[/tex].
2. Check if Function [tex]\( f \)[/tex] is Increasing:
- Calculate the changes as [tex]\( x \)[/tex] increases:
- From [tex]\( x = -1 \)[/tex] to [tex]\( x = 0 \)[/tex], [tex]\( f \)[/tex] changes from [tex]\(-22\)[/tex] to [tex]\(-10\)[/tex]: The change is [tex]\( +12 \)[/tex].
- From [tex]\( x = 0 \)[/tex] to [tex]\( x = 1 \)[/tex], [tex]\( f \)[/tex] changes from [tex]\(-10\)[/tex] to [tex]\(-4\)[/tex]: The change is [tex]\( +6 \)[/tex].
- From [tex]\( x = 1 \)[/tex] to [tex]\( x = 2 \)[/tex], [tex]\( f \)[/tex] changes from [tex]\(-4\)[/tex] to [tex]\(-1\)[/tex]: The change is [tex]\( +3 \)[/tex].
- All these changes are positive, so function [tex]\( f \)[/tex] is increasing.
3. Identify Function [tex]\( g \)[/tex] Values and Changes:
- Equation for function [tex]\( g \)[/tex] is [tex]\( g(x) = -18\left(\frac{1}{3}\right)^x + 2 \)[/tex].
- Calculate [tex]\( g(x) \)[/tex] values for [tex]\( x = -1, 0, 1, 2 \)[/tex]:
- [tex]\( g(-1) = -18\left(\frac{1}{3}\right)^{-1} + 2 = 36 + 2 = 38 \)[/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]
- Calculate changes for [tex]\( g(x) \)[/tex]:
- From [tex]\( x = -1 \)[/tex] to [tex]\( x = 0 \)[/tex]: Change is [tex]\(-16 - 38 = -54\)[/tex].
- From [tex]\( x = 0 \)[/tex] to [tex]\( x = 1 \)[/tex]: Change is [tex]\(-4 - (-16) = 12\)[/tex].
- From [tex]\( x = 1 \)[/tex] to [tex]\( x = 2 \)[/tex]: Change is [tex]\(0 - (-4) = 4\)[/tex].
4. Determine Function [tex]\( g \)[/tex] Behavior:
- Just like with function [tex]\( f \)[/tex], calculate changes over the interval:
- Since both earlier changes are positive for [tex]\( x = 0 \)[/tex] to [tex]\( x = 2 \)[/tex], function [tex]\( g \)[/tex] is increasing.
5. Average Rate of Increase:
- Find average rate of increase for both functions:
- Average rate for [tex]\( f \)[/tex] is calculated as [tex]\((12 + 6 + 3)/3 = 7\)[/tex].
- Average rate for [tex]\( g \)[/tex] is [tex]\((12 + 4)/2 = 8\)[/tex].
6. Compare the Functions:
- Both functions are increasing over the interval.
- Function [tex]\( g \)[/tex] increases at a faster average rate than function [tex]\( f \)[/tex].
7. Conclusion:
- Since both functions are increasing and [tex]\( g(x) \)[/tex] increases at a faster average rate on the interval, the correct statement is:
D. Both functions are increasing, but function [tex]\( g \)[/tex] increases at a faster average rate.
