Answer :
Let's compare the two functions, [tex]\( f \)[/tex] and [tex]\( g \)[/tex], over the interval [tex]\([-1, 2]\)[/tex], based on the information provided:
### Function [tex]\( f \)[/tex]
The values provided for [tex]\( f(x) \)[/tex] at specific points are:
- [tex]\( f(-1) = -22 \)[/tex]
- [tex]\( f(0) = -10 \)[/tex]
- [tex]\( f(1) = -4 \)[/tex]
- [tex]\( f(2) = -1 \)[/tex]
#### Analyzing [tex]\( f \)[/tex]
- Checking if [tex]\( f \)[/tex] is increasing:
- [tex]\( f(-1) = -22 \)[/tex] and [tex]\( f(0) = -10 \)[/tex]. Here, [tex]\(-22 < -10\)[/tex], so [tex]\( f(x) \)[/tex] is increasing from -1 to 0.
- [tex]\( f(0) = -10 \)[/tex] and [tex]\( f(1) = -4 \)[/tex]. Here, [tex]\(-10 < -4\)[/tex], so [tex]\( f(x) \)[/tex] continues to increase from 0 to 1.
- [tex]\( f(1) = -4 \)[/tex] and [tex]\( f(2) = -1 \)[/tex]. Here, [tex]\(-4 < -1\)[/tex], so [tex]\( f(x) \)[/tex] continues to increase from 1 to 2.
- Since [tex]\( f(x) \)[/tex] increases at each step, it is increasing over the whole interval [tex]\([-1, 2]\)[/tex].
- Checking if [tex]\( f \)[/tex] is negative:
- All given values [tex]\(-22, -10, -4\)[/tex], and [tex]\(-1\)[/tex] are negative, indicating that [tex]\( f(x) \)[/tex] is negative through the entire interval [tex]\([-1, 2]\)[/tex].
### Function [tex]\( g \)[/tex]
The function [tex]\( g(x) = -18 \left(\frac{1}{3}\right)^x + 2 \)[/tex] needs analyzing at [tex]\( x = -1, 0, 1, 2 \)[/tex].
1. [tex]\( g(-1) = -18 \times \frac{3}{1} + 2 = -54 + 2 = -52 \)[/tex]
2. [tex]\( g(0) = -18 \times 1 + 2 = -18 + 2 = -16 \)[/tex]
3. [tex]\( g(1) = -18 \times \frac{1}{3} + 2 = -6 + 2 = -4 \)[/tex]
4. [tex]\( g(2) = -18 \times \frac{1}{9} + 2 = -2 + 2 = 0 \)[/tex]
#### Analyzing [tex]\( g \)[/tex]
- Checking if [tex]\( g \)[/tex] is increasing:
- [tex]\( g(-1) = -52 \)[/tex] and [tex]\( g(0) = -16 \)[/tex]. Here, [tex]\(-52 < -16\)[/tex], so [tex]\( g(x) \)[/tex] is increasing from -1 to 0.
- [tex]\( g(0) = -16 \)[/tex] and [tex]\( g(1) = -4 \)[/tex]. Here, [tex]\(-16 < -4\)[/tex], so [tex]\( g(x) \)[/tex] continues to increase from 0 to 1.
- [tex]\( g(1) = -4 \)[/tex] and [tex]\( g(2) = 0 \)[/tex]. Here, [tex]\(-4 < 0\)[/tex], so [tex]\( g(x) \)[/tex] continues to increase from 1 to 2.
- Since [tex]\( g(x) \)[/tex] increases at each step, it is increasing over the whole interval [tex]\([-1, 2]\)[/tex].
- Checking if [tex]\( g \)[/tex] is negative:
- The values [tex]\(-52, -16, -4\)[/tex] at [tex]\( x = -1, 0, 1 \)[/tex] are negative, but the value at [tex]\( x = 2 \)[/tex] is 0, which is not negative. So, [tex]\( g(x) \)[/tex] is not entirely negative over the interval.
### Conclusion
Both functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] are increasing over the interval [tex]\([-1, 2]\)[/tex]. Now let's look at the average rate of increase from [tex]\( x = -1 \)[/tex] to [tex]\( x = 2 \)[/tex]:
- For [tex]\( f(x) \)[/tex], the change is [tex]\( f(2) - f(-1) = -1 - (-22) = 21 \)[/tex].
- For [tex]\( g(x) \)[/tex], the change is [tex]\( g(2) - g(-1) = 0 - (-52) = 52 \)[/tex].
Since [tex]\( g(x) \)[/tex] increases by 52 compared to [tex]\( f(x)\)[/tex]'s increase of 21, function [tex]\( g \)[/tex] increases at a faster rate.
Given all this analysis, the correct statement is:
C. Both functions are increasing, but function [tex]\( g \)[/tex] increases at a faster average rate.
### Function [tex]\( f \)[/tex]
The values provided for [tex]\( f(x) \)[/tex] at specific points are:
- [tex]\( f(-1) = -22 \)[/tex]
- [tex]\( f(0) = -10 \)[/tex]
- [tex]\( f(1) = -4 \)[/tex]
- [tex]\( f(2) = -1 \)[/tex]
#### Analyzing [tex]\( f \)[/tex]
- Checking if [tex]\( f \)[/tex] is increasing:
- [tex]\( f(-1) = -22 \)[/tex] and [tex]\( f(0) = -10 \)[/tex]. Here, [tex]\(-22 < -10\)[/tex], so [tex]\( f(x) \)[/tex] is increasing from -1 to 0.
- [tex]\( f(0) = -10 \)[/tex] and [tex]\( f(1) = -4 \)[/tex]. Here, [tex]\(-10 < -4\)[/tex], so [tex]\( f(x) \)[/tex] continues to increase from 0 to 1.
- [tex]\( f(1) = -4 \)[/tex] and [tex]\( f(2) = -1 \)[/tex]. Here, [tex]\(-4 < -1\)[/tex], so [tex]\( f(x) \)[/tex] continues to increase from 1 to 2.
- Since [tex]\( f(x) \)[/tex] increases at each step, it is increasing over the whole interval [tex]\([-1, 2]\)[/tex].
- Checking if [tex]\( f \)[/tex] is negative:
- All given values [tex]\(-22, -10, -4\)[/tex], and [tex]\(-1\)[/tex] are negative, indicating that [tex]\( f(x) \)[/tex] is negative through the entire interval [tex]\([-1, 2]\)[/tex].
### Function [tex]\( g \)[/tex]
The function [tex]\( g(x) = -18 \left(\frac{1}{3}\right)^x + 2 \)[/tex] needs analyzing at [tex]\( x = -1, 0, 1, 2 \)[/tex].
1. [tex]\( g(-1) = -18 \times \frac{3}{1} + 2 = -54 + 2 = -52 \)[/tex]
2. [tex]\( g(0) = -18 \times 1 + 2 = -18 + 2 = -16 \)[/tex]
3. [tex]\( g(1) = -18 \times \frac{1}{3} + 2 = -6 + 2 = -4 \)[/tex]
4. [tex]\( g(2) = -18 \times \frac{1}{9} + 2 = -2 + 2 = 0 \)[/tex]
#### Analyzing [tex]\( g \)[/tex]
- Checking if [tex]\( g \)[/tex] is increasing:
- [tex]\( g(-1) = -52 \)[/tex] and [tex]\( g(0) = -16 \)[/tex]. Here, [tex]\(-52 < -16\)[/tex], so [tex]\( g(x) \)[/tex] is increasing from -1 to 0.
- [tex]\( g(0) = -16 \)[/tex] and [tex]\( g(1) = -4 \)[/tex]. Here, [tex]\(-16 < -4\)[/tex], so [tex]\( g(x) \)[/tex] continues to increase from 0 to 1.
- [tex]\( g(1) = -4 \)[/tex] and [tex]\( g(2) = 0 \)[/tex]. Here, [tex]\(-4 < 0\)[/tex], so [tex]\( g(x) \)[/tex] continues to increase from 1 to 2.
- Since [tex]\( g(x) \)[/tex] increases at each step, it is increasing over the whole interval [tex]\([-1, 2]\)[/tex].
- Checking if [tex]\( g \)[/tex] is negative:
- The values [tex]\(-52, -16, -4\)[/tex] at [tex]\( x = -1, 0, 1 \)[/tex] are negative, but the value at [tex]\( x = 2 \)[/tex] is 0, which is not negative. So, [tex]\( g(x) \)[/tex] is not entirely negative over the interval.
### Conclusion
Both functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] are increasing over the interval [tex]\([-1, 2]\)[/tex]. Now let's look at the average rate of increase from [tex]\( x = -1 \)[/tex] to [tex]\( x = 2 \)[/tex]:
- For [tex]\( f(x) \)[/tex], the change is [tex]\( f(2) - f(-1) = -1 - (-22) = 21 \)[/tex].
- For [tex]\( g(x) \)[/tex], the change is [tex]\( g(2) - g(-1) = 0 - (-52) = 52 \)[/tex].
Since [tex]\( g(x) \)[/tex] increases by 52 compared to [tex]\( f(x)\)[/tex]'s increase of 21, function [tex]\( g \)[/tex] increases at a faster rate.
Given all this analysis, the correct statement is:
C. Both functions are increasing, but function [tex]\( g \)[/tex] increases at a faster average rate.