Answer :
To compare the functions [tex]\(f\)[/tex] and [tex]\(g\)[/tex] on the interval [tex]\([-1, 2]\)[/tex], we need to look at both their behavior (whether they are increasing or decreasing) and their rates of change.
1. Analyze Function [tex]\(f\)[/tex]:
- From the table, the values for function [tex]\(f\)[/tex] over the interval [tex]\([-1, 2]\)[/tex] are: [tex]\(-22, -10, -4, -1\)[/tex].
- These values are increasing because each subsequent value is larger than the previous one. Therefore, function [tex]\(f\)[/tex] is increasing on the interval [tex]\([-1, 2]\)[/tex].
2. Analyze Function [tex]\(g\)[/tex]:
- The equation for function [tex]\(g\)[/tex] is [tex]\(g(x) = -18\left(\frac{1}{3}\right)^x + 2\)[/tex].
- On evaluating [tex]\(g(x)\)[/tex] for [tex]\( x = -1, 0, 1, 2\)[/tex], we determine that [tex]\(g\)[/tex] is also increasing over the interval [tex]\([-1, 2]\)[/tex].
3. Compare the Average Rates of Change:
- The rate of change is essentially the slope when considering the changes over an interval.
- For both functions, calculate the average rate of change over the interval [tex]\([-1, 2]\)[/tex]:
- Function [tex]\(f\)[/tex] has a steady increase in values, showing it's consistently increasing.
- Function [tex]\(g\)[/tex] is based on an exponential decay modified by translations, but across this specific interval, it is increasing too.
4. Comparison:
- Once both functions are confirmed to be increasing, compare their rates of increase.
- Function [tex]\(g\)[/tex] increases at a faster average rate than function [tex]\(f\)[/tex].
Based on this analysis, the correct statement is:
A. Both functions are increasing, but function [tex]\(g\)[/tex] increases at a faster average rate.
1. Analyze Function [tex]\(f\)[/tex]:
- From the table, the values for function [tex]\(f\)[/tex] over the interval [tex]\([-1, 2]\)[/tex] are: [tex]\(-22, -10, -4, -1\)[/tex].
- These values are increasing because each subsequent value is larger than the previous one. Therefore, function [tex]\(f\)[/tex] is increasing on the interval [tex]\([-1, 2]\)[/tex].
2. Analyze Function [tex]\(g\)[/tex]:
- The equation for function [tex]\(g\)[/tex] is [tex]\(g(x) = -18\left(\frac{1}{3}\right)^x + 2\)[/tex].
- On evaluating [tex]\(g(x)\)[/tex] for [tex]\( x = -1, 0, 1, 2\)[/tex], we determine that [tex]\(g\)[/tex] is also increasing over the interval [tex]\([-1, 2]\)[/tex].
3. Compare the Average Rates of Change:
- The rate of change is essentially the slope when considering the changes over an interval.
- For both functions, calculate the average rate of change over the interval [tex]\([-1, 2]\)[/tex]:
- Function [tex]\(f\)[/tex] has a steady increase in values, showing it's consistently increasing.
- Function [tex]\(g\)[/tex] is based on an exponential decay modified by translations, but across this specific interval, it is increasing too.
4. Comparison:
- Once both functions are confirmed to be increasing, compare their rates of increase.
- Function [tex]\(g\)[/tex] increases at a faster average rate than function [tex]\(f\)[/tex].
Based on this analysis, the correct statement is:
A. Both functions are increasing, but function [tex]\(g\)[/tex] increases at a faster average rate.