Answer :
To solve this problem and compare the two functions [tex]\( f \)[/tex] and [tex]\( g \)[/tex] on the interval [tex]\([-1, 2]\)[/tex], let's systematically analyze their behavior.
### Step 1: Analyze Function [tex]\( f \)[/tex]
The function [tex]\( f \)[/tex] is represented by a table with these values:
- [tex]\( f(-1) = -22 \)[/tex]
- [tex]\( f(0) = -10 \)[/tex]
- [tex]\( f(1) = -4 \)[/tex]
- [tex]\( f(2) = -1 \)[/tex]
#### Determine if [tex]\( f \)[/tex] is Increasing or Decreasing
Let's check if [tex]\( f(x) \)[/tex] is increasing over the interval from [tex]\(-1\)[/tex] to [tex]\(2\)[/tex]:
- From [tex]\( x = -1 \)[/tex] to [tex]\( x = 0 \)[/tex]: [tex]\( f(-1) < f(0) \)[/tex] ([tex]\(-22 < -10\)[/tex])
- From [tex]\( x = 0 \)[/tex] to [tex]\( x = 1 \)[/tex]: [tex]\( f(0) < f(1) \)[/tex] ([tex]\(-10 < -4\)[/tex])
- From [tex]\( x = 1 \)[/tex] to [tex]\( x = 2 \)[/tex]: [tex]\( f(1) < f(2) \)[/tex] ([tex]\(-4 < -1\)[/tex])
Since each value of [tex]\( f(x) \)[/tex] is less than the next, function [tex]\( f \)[/tex] is increasing on the interval [tex]\([-1, 2]\)[/tex].
#### Determine if [tex]\( f \)[/tex] is Negative
All values [tex]\(-22\)[/tex], [tex]\(-10\)[/tex], [tex]\(-4\)[/tex], [tex]\(-1\)[/tex] are negative, so [tex]\( f(x) \)[/tex] is negative on the entire interval [tex]\([-1, 2]\)[/tex].
### Step 2: Analyze Function [tex]\( g \)[/tex]
The function [tex]\( g \)[/tex] is defined by the equation:
[tex]\[ g(x) = -18\left(\frac{1}{3}\right)^x + 2 \][/tex]
#### Determine if [tex]\( g \)[/tex] is Increasing or Decreasing
We check the values at points [tex]\(-1\)[/tex] to [tex]\(2\)[/tex]:
- [tex]\( g(-1) = -18(3) + 2 = -54 + 2 = -52 \)[/tex]
- [tex]\( g(0) = -18(1) + 2 = -18 + 2 = -16 \)[/tex]
- [tex]\( g(1) = -18\left(\frac{1}{3}\right) + 2 = -6 + 2 = -4 \)[/tex]
- [tex]\( g(2) = -18\left(\frac{1}{9}\right) + 2 = -2 + 2 = 0 \)[/tex]
Let's check if [tex]\( g(x) \)[/tex] is increasing:
- From [tex]\( x = -1 \)[/tex] to [tex]\( x = 0 \)[/tex]: [tex]\( g(-1) < g(0) \)[/tex] ([tex]\(-52 < -16\)[/tex])
- From [tex]\( x = 0 \)[/tex] to [tex]\( x = 1 \)[/tex]: [tex]\( g(0) < g(1) \)[/tex] ([tex]\(-16 < -4\)[/tex])
- From [tex]\( x = 1 \)[/tex] to [tex]\( x = 2 \)[/tex]: [tex]\( g(1) < g(2) \)[/tex] ([tex]\(-4 < 0\)[/tex])
Function [tex]\( g \)[/tex] is increasing on the interval [tex]\([-1, 2]\)[/tex].
#### Determine if [tex]\( g \)[/tex] is Negative
At [tex]\( x = 2 \)[/tex], [tex]\( g(2) = 0 \)[/tex], which is not negative. Therefore, function [tex]\( g \)[/tex] is not completely negative over this interval.
### Comparison
- Increasing Behavior: Both [tex]\( f \)[/tex] and [tex]\( g \)[/tex] are increasing.
- Negativity: [tex]\( f \)[/tex] is negative throughout, but [tex]\( g \)[/tex] is not.
### Conclusion
The correct statement that accurately compares the two functions on the interval [tex]\([-1, 2]\)[/tex] is:
D. Only function [tex]\( f \)[/tex] is increasing, and only function [tex]\( f \)[/tex] is negative.
### Step 1: Analyze Function [tex]\( f \)[/tex]
The function [tex]\( f \)[/tex] is represented by a table with these values:
- [tex]\( f(-1) = -22 \)[/tex]
- [tex]\( f(0) = -10 \)[/tex]
- [tex]\( f(1) = -4 \)[/tex]
- [tex]\( f(2) = -1 \)[/tex]
#### Determine if [tex]\( f \)[/tex] is Increasing or Decreasing
Let's check if [tex]\( f(x) \)[/tex] is increasing over the interval from [tex]\(-1\)[/tex] to [tex]\(2\)[/tex]:
- From [tex]\( x = -1 \)[/tex] to [tex]\( x = 0 \)[/tex]: [tex]\( f(-1) < f(0) \)[/tex] ([tex]\(-22 < -10\)[/tex])
- From [tex]\( x = 0 \)[/tex] to [tex]\( x = 1 \)[/tex]: [tex]\( f(0) < f(1) \)[/tex] ([tex]\(-10 < -4\)[/tex])
- From [tex]\( x = 1 \)[/tex] to [tex]\( x = 2 \)[/tex]: [tex]\( f(1) < f(2) \)[/tex] ([tex]\(-4 < -1\)[/tex])
Since each value of [tex]\( f(x) \)[/tex] is less than the next, function [tex]\( f \)[/tex] is increasing on the interval [tex]\([-1, 2]\)[/tex].
#### Determine if [tex]\( f \)[/tex] is Negative
All values [tex]\(-22\)[/tex], [tex]\(-10\)[/tex], [tex]\(-4\)[/tex], [tex]\(-1\)[/tex] are negative, so [tex]\( f(x) \)[/tex] is negative on the entire interval [tex]\([-1, 2]\)[/tex].
### Step 2: Analyze Function [tex]\( g \)[/tex]
The function [tex]\( g \)[/tex] is defined by the equation:
[tex]\[ g(x) = -18\left(\frac{1}{3}\right)^x + 2 \][/tex]
#### Determine if [tex]\( g \)[/tex] is Increasing or Decreasing
We check the values at points [tex]\(-1\)[/tex] to [tex]\(2\)[/tex]:
- [tex]\( g(-1) = -18(3) + 2 = -54 + 2 = -52 \)[/tex]
- [tex]\( g(0) = -18(1) + 2 = -18 + 2 = -16 \)[/tex]
- [tex]\( g(1) = -18\left(\frac{1}{3}\right) + 2 = -6 + 2 = -4 \)[/tex]
- [tex]\( g(2) = -18\left(\frac{1}{9}\right) + 2 = -2 + 2 = 0 \)[/tex]
Let's check if [tex]\( g(x) \)[/tex] is increasing:
- From [tex]\( x = -1 \)[/tex] to [tex]\( x = 0 \)[/tex]: [tex]\( g(-1) < g(0) \)[/tex] ([tex]\(-52 < -16\)[/tex])
- From [tex]\( x = 0 \)[/tex] to [tex]\( x = 1 \)[/tex]: [tex]\( g(0) < g(1) \)[/tex] ([tex]\(-16 < -4\)[/tex])
- From [tex]\( x = 1 \)[/tex] to [tex]\( x = 2 \)[/tex]: [tex]\( g(1) < g(2) \)[/tex] ([tex]\(-4 < 0\)[/tex])
Function [tex]\( g \)[/tex] is increasing on the interval [tex]\([-1, 2]\)[/tex].
#### Determine if [tex]\( g \)[/tex] is Negative
At [tex]\( x = 2 \)[/tex], [tex]\( g(2) = 0 \)[/tex], which is not negative. Therefore, function [tex]\( g \)[/tex] is not completely negative over this interval.
### Comparison
- Increasing Behavior: Both [tex]\( f \)[/tex] and [tex]\( g \)[/tex] are increasing.
- Negativity: [tex]\( f \)[/tex] is negative throughout, but [tex]\( g \)[/tex] is not.
### Conclusion
The correct statement that accurately compares the two functions on the interval [tex]\([-1, 2]\)[/tex] is:
D. Only function [tex]\( f \)[/tex] is increasing, and only function [tex]\( f \)[/tex] is negative.