Answer :
To solve the problem of comparing the two functions [tex]\( f \)[/tex] and [tex]\( g \)[/tex] over the interval [tex]\([-1, 2]\)[/tex], we need to evaluate the following aspects for both functions:
1. Behavior of Function [tex]\( f \)[/tex]: Look at the table values for [tex]\( f(x) \)[/tex]:
- At [tex]\( x = -1 \)[/tex], [tex]\( f(x) = -22 \)[/tex]
- At [tex]\( x = 0 \)[/tex], [tex]\( f(x) = -10 \)[/tex]
- At [tex]\( x = 1 \)[/tex], [tex]\( f(x) = -4 \)[/tex]
- At [tex]\( x = 2 \)[/tex], [tex]\( f(x) = -1 \)[/tex]
Observe the pattern: [tex]\( -22, -10, -4, -1 \)[/tex]. The values are increasing as [tex]\( x \)[/tex] increases, because each subsequent [tex]\( f(x) \)[/tex] is greater than the previous one. So, function [tex]\( f \)[/tex] is increasing.
2. Behavior of Function [tex]\( g \)[/tex]: Given the equation for [tex]\( g(x) \)[/tex]:
- Calculate [tex]\( g(x) \)[/tex] for [tex]\( x = -1, 0, 1, 2 \)[/tex].
- For [tex]\( x = -1 \)[/tex]: [tex]\( g(-1) = -18 \times 3 + 2 = -52 \)[/tex]
- For [tex]\( x = 0 \)[/tex]: [tex]\( g(0) = -18 \times 1 + 2 = -16 \)[/tex]
- For [tex]\( x = 1 \)[/tex]: [tex]\( g(1) = -18 \times \frac{1}{3} + 2 = -4 + 2 = -2 \)[/tex]
- For [tex]\( x = 2 \)[/tex]: [tex]\( g(2) = -18 \times \frac{1}{9} + 2 \approx 0 \)[/tex]
Observe the pattern: [tex]\( -52, -16, -2, \approx 0 \)[/tex]. Like function [tex]\( f \)[/tex], these values are also increasing.
3. Comparison of Growth Rates:
- Function [tex]\( f \)[/tex] has an average rate of increase calculated by looking at the change in values between the [tex]\( x \)[/tex] points: [tex]\( (-10) - (-22) = 12 \)[/tex], [tex]\( (-4) - (-10) = 6 \)[/tex], [tex]\( (-1) - (-4) = 3 \)[/tex].
- The average increase for [tex]\( f \)[/tex] is [tex]\(\frac{12 + 6 + 3}{3} = 7.0\)[/tex].
- For function [tex]\( g \)[/tex], the changes are: [tex]\( (-16) - (-52) = 36 \)[/tex], [tex]\( (-2) - (-16) = 14 \)[/tex], [tex]\( (0) - (-2) = 2 \)[/tex].
- The average increase for [tex]\( g \)[/tex] is [tex]\(\frac{36 + 14 + 2}{3} = 17.33\)[/tex].
Both functions are increasing, but the average rate of increase for function [tex]\( g \)[/tex] is higher than that of function [tex]\( f \)[/tex].
Thus, the correct statement comparing these functions on the interval [tex]\([-1, 2]\)[/tex] is:
C. Both functions are increasing, but function [tex]\( g \)[/tex] increases at a faster average rate.
1. Behavior of Function [tex]\( f \)[/tex]: Look at the table values for [tex]\( f(x) \)[/tex]:
- At [tex]\( x = -1 \)[/tex], [tex]\( f(x) = -22 \)[/tex]
- At [tex]\( x = 0 \)[/tex], [tex]\( f(x) = -10 \)[/tex]
- At [tex]\( x = 1 \)[/tex], [tex]\( f(x) = -4 \)[/tex]
- At [tex]\( x = 2 \)[/tex], [tex]\( f(x) = -1 \)[/tex]
Observe the pattern: [tex]\( -22, -10, -4, -1 \)[/tex]. The values are increasing as [tex]\( x \)[/tex] increases, because each subsequent [tex]\( f(x) \)[/tex] is greater than the previous one. So, function [tex]\( f \)[/tex] is increasing.
2. Behavior of Function [tex]\( g \)[/tex]: Given the equation for [tex]\( g(x) \)[/tex]:
- Calculate [tex]\( g(x) \)[/tex] for [tex]\( x = -1, 0, 1, 2 \)[/tex].
- For [tex]\( x = -1 \)[/tex]: [tex]\( g(-1) = -18 \times 3 + 2 = -52 \)[/tex]
- For [tex]\( x = 0 \)[/tex]: [tex]\( g(0) = -18 \times 1 + 2 = -16 \)[/tex]
- For [tex]\( x = 1 \)[/tex]: [tex]\( g(1) = -18 \times \frac{1}{3} + 2 = -4 + 2 = -2 \)[/tex]
- For [tex]\( x = 2 \)[/tex]: [tex]\( g(2) = -18 \times \frac{1}{9} + 2 \approx 0 \)[/tex]
Observe the pattern: [tex]\( -52, -16, -2, \approx 0 \)[/tex]. Like function [tex]\( f \)[/tex], these values are also increasing.
3. Comparison of Growth Rates:
- Function [tex]\( f \)[/tex] has an average rate of increase calculated by looking at the change in values between the [tex]\( x \)[/tex] points: [tex]\( (-10) - (-22) = 12 \)[/tex], [tex]\( (-4) - (-10) = 6 \)[/tex], [tex]\( (-1) - (-4) = 3 \)[/tex].
- The average increase for [tex]\( f \)[/tex] is [tex]\(\frac{12 + 6 + 3}{3} = 7.0\)[/tex].
- For function [tex]\( g \)[/tex], the changes are: [tex]\( (-16) - (-52) = 36 \)[/tex], [tex]\( (-2) - (-16) = 14 \)[/tex], [tex]\( (0) - (-2) = 2 \)[/tex].
- The average increase for [tex]\( g \)[/tex] is [tex]\(\frac{36 + 14 + 2}{3} = 17.33\)[/tex].
Both functions are increasing, but the average rate of increase for function [tex]\( g \)[/tex] is higher than that of function [tex]\( f \)[/tex].
Thus, the correct statement comparing these functions on the interval [tex]\([-1, 2]\)[/tex] is:
C. Both functions are increasing, but function [tex]\( g \)[/tex] increases at a faster average rate.