Answer :
To solve this question, we need to compare the two functions, [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex], on the interval [tex]\([-1, 2]\)[/tex].
### Step 1: Examine Function [tex]\( f(x) \)[/tex]
The table provides the values of function [tex]\( f \)[/tex]:
- When [tex]\( x = -1 \)[/tex], [tex]\( f(x) = -22 \)[/tex]
- When [tex]\( x = 0 \)[/tex], [tex]\( f(x) = -10 \)[/tex]
- When [tex]\( x = 1 \)[/tex], [tex]\( f(x) = -4 \)[/tex]
- When [tex]\( x = 2 \)[/tex], [tex]\( f(x) = -1 \)[/tex]
By observing these values, we can see that [tex]\( f(x) \)[/tex] is increasing as [tex]\( x \)[/tex] increases from [tex]\(-1\)[/tex] to [tex]\(2\)[/tex]:
- From [tex]\(-22 \to -10\)[/tex]
- From [tex]\(-10 \to -4\)[/tex]
- From [tex]\(-4 \to -1\)[/tex]
All the values of [tex]\( f(x) \)[/tex] in this interval are negative.
### Step 2: Examine Function [tex]\( g(x) \)[/tex]
Function [tex]\( g(x) \)[/tex] is given by the equation:
[tex]\[ g(x) = -18\left(\frac{1}{3}\right)^x + 2 \][/tex]
We need to calculate [tex]\( g(x) \)[/tex] at each point within the interval [tex]\([-1, 2]\)[/tex]:
- [tex]\( g(-1) = -18 \left(\frac{1}{3}\right)^{-1} + 2 = -18 \cdot 3 + 2 = -54 + 2 = -52 \)[/tex]
- [tex]\( g(0) = -18 \left(\frac{1}{3}\right)^{0} + 2 = -18 \cdot 1 + 2 = -16 \)[/tex]
- [tex]\( g(1) = -18 \left(\frac{1}{3}\right)^{1} + 2 = -18 \cdot \frac{1}{3} + 2 = -6 + 2 = -4 \)[/tex]
- [tex]\( g(2) = -18 \left(\frac{1}{3}\right)^{2} + 2 = -18 \cdot \frac{1}{9} + 2 = -2 + 2 = 0 \)[/tex]
From these values, we note that [tex]\( g(x) \)[/tex] is increasing in the interval [tex]\([-1, 2]\)[/tex]:
- From [tex]\(-52 \to -16\)[/tex]
- From [tex]\(-16 \to -4\)[/tex]
- From [tex]\(-4 \to 0\)[/tex]
However, only the value of [tex]\( g(x) \)[/tex] at [tex]\( x = 2\)[/tex] is non-negative. Until then, the function remains negative.
### Step 3: Compare the Functions
Both functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] are increasing over this interval. However, function [tex]\( g(x) \)[/tex] becomes non-negative at the endpoint, while function [tex]\( f(x) \)[/tex] remains negative throughout the interval from [tex]\(-1\)[/tex] to [tex]\(2 \)[/tex].
### Conclusion
The correct statement about the functions on the interval [tex]\([-1, 2]\)[/tex] is:
D. Both functions are increasing, but function [tex]\( g \)[/tex] increases at a faster average rate.
### Step 1: Examine Function [tex]\( f(x) \)[/tex]
The table provides the values of function [tex]\( f \)[/tex]:
- When [tex]\( x = -1 \)[/tex], [tex]\( f(x) = -22 \)[/tex]
- When [tex]\( x = 0 \)[/tex], [tex]\( f(x) = -10 \)[/tex]
- When [tex]\( x = 1 \)[/tex], [tex]\( f(x) = -4 \)[/tex]
- When [tex]\( x = 2 \)[/tex], [tex]\( f(x) = -1 \)[/tex]
By observing these values, we can see that [tex]\( f(x) \)[/tex] is increasing as [tex]\( x \)[/tex] increases from [tex]\(-1\)[/tex] to [tex]\(2\)[/tex]:
- From [tex]\(-22 \to -10\)[/tex]
- From [tex]\(-10 \to -4\)[/tex]
- From [tex]\(-4 \to -1\)[/tex]
All the values of [tex]\( f(x) \)[/tex] in this interval are negative.
### Step 2: Examine Function [tex]\( g(x) \)[/tex]
Function [tex]\( g(x) \)[/tex] is given by the equation:
[tex]\[ g(x) = -18\left(\frac{1}{3}\right)^x + 2 \][/tex]
We need to calculate [tex]\( g(x) \)[/tex] at each point within the interval [tex]\([-1, 2]\)[/tex]:
- [tex]\( g(-1) = -18 \left(\frac{1}{3}\right)^{-1} + 2 = -18 \cdot 3 + 2 = -54 + 2 = -52 \)[/tex]
- [tex]\( g(0) = -18 \left(\frac{1}{3}\right)^{0} + 2 = -18 \cdot 1 + 2 = -16 \)[/tex]
- [tex]\( g(1) = -18 \left(\frac{1}{3}\right)^{1} + 2 = -18 \cdot \frac{1}{3} + 2 = -6 + 2 = -4 \)[/tex]
- [tex]\( g(2) = -18 \left(\frac{1}{3}\right)^{2} + 2 = -18 \cdot \frac{1}{9} + 2 = -2 + 2 = 0 \)[/tex]
From these values, we note that [tex]\( g(x) \)[/tex] is increasing in the interval [tex]\([-1, 2]\)[/tex]:
- From [tex]\(-52 \to -16\)[/tex]
- From [tex]\(-16 \to -4\)[/tex]
- From [tex]\(-4 \to 0\)[/tex]
However, only the value of [tex]\( g(x) \)[/tex] at [tex]\( x = 2\)[/tex] is non-negative. Until then, the function remains negative.
### Step 3: Compare the Functions
Both functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] are increasing over this interval. However, function [tex]\( g(x) \)[/tex] becomes non-negative at the endpoint, while function [tex]\( f(x) \)[/tex] remains negative throughout the interval from [tex]\(-1\)[/tex] to [tex]\(2 \)[/tex].
### Conclusion
The correct statement about the functions on the interval [tex]\([-1, 2]\)[/tex] is:
D. Both functions are increasing, but function [tex]\( g \)[/tex] increases at a faster average rate.