Answer :
To solve this problem, we need to compare two functions, [tex]\( f \)[/tex] and [tex]\( g \)[/tex], on the interval [tex]\([-1, 2]\)[/tex]. Here’s how we approach this:
### Step 1: Analyze Function [tex]\( f \)[/tex]
The function [tex]\( f \)[/tex] is represented by a table of values:
[tex]\[
\begin{array}{cccccc}
x & -2 & -1 & 0 & 1 & 2 \\
f(x) & -46 & -22 & -10 & -4 & -1
\end{array}
\][/tex]
Increasing Behavior:
- Look at the values of [tex]\( f(x) \)[/tex] as [tex]\( x \)[/tex] moves from [tex]\(-1\)[/tex] to [tex]\(2\)[/tex].
- From the table:
- [tex]\( f(-1) = -22 \)[/tex]
- [tex]\( f(0) = -10 \)[/tex]
- [tex]\( f(1) = -4 \)[/tex]
- [tex]\( f(2) = -1 \)[/tex]
As [tex]\( x \)[/tex] increases from [tex]\(-1\)[/tex] to 2, [tex]\( f(x) \)[/tex] is consistently increasing because each consecutive value is greater than the previous one.
Sign:
- All values [tex]\( f(x) \)[/tex] at [tex]\( x = -1, 0, 1, 2 \)[/tex] are negative: [tex]\(-22, -10, -4, -1\)[/tex].
### Step 2: Analyze Function [tex]\( g \)[/tex]
The function [tex]\( g \)[/tex] is given by the equation:
[tex]\[ g(x) = -18 \left(\frac{1}{3}\right)^x + 2 \][/tex]
We calculate the values for [tex]\( g(x) \)[/tex] at [tex]\( x = -1, 0, 1, 2 \)[/tex]:
1. [tex]\( g(-1) = -18 \left(\frac{1}{3}\right)^{-1} + 2 = -18 \times 3 + 2 = -54 + 2 = -52 \)[/tex]
2. [tex]\( g(0) = -18 \left(\frac{1}{3}\right)^{0} + 2 = -18 \times 1 + 2 = -18 + 2 = -16 \)[/tex]
3. [tex]\( g(1) = -18 \left(\frac{1}{3}\right)^{1} + 2 = -18 \times \frac{1}{3} + 2 = -6 + 2 = -4 \)[/tex]
4. [tex]\( g(2) = -18 \left(\frac{1}{3}\right)^{2} + 2 = -18 \times \frac{1}{9} + 2 = -2 + 2 = 0 \)[/tex]
Increasing Behavior:
- As [tex]\( x \)[/tex] increases, [tex]\( g(-1) = -52 \)[/tex] to [tex]\( g(2) = 0 \)[/tex], the values of [tex]\( g(x) \)[/tex] increase.
Sign:
- Examine the values:
- [tex]\( g(-1) = -52\)[/tex]
- [tex]\( g(0) = -16\)[/tex]
- [tex]\( g(1) = -4\)[/tex]
- [tex]\( g(2) = 0\)[/tex]
The values are negative except [tex]\( g(2) = 0 \)[/tex], so function [tex]\( g \)[/tex] is not entirely negative.
### Step 3: Compare Average Rate of Change
The average rate of change for a function on an interval [tex]\([a, b]\)[/tex] is given by:
[tex]\[
\text{Average Rate of Change} = \frac{\text{Change in } y}{\text{Change in } x}
\][/tex]
Function [tex]\( f \)[/tex] on [tex]\([-1, 2]\)[/tex]:
- Change in [tex]\( f(x) = f(2) - f(-1) = -1 - (-22) = 21 \)[/tex]
- Change in [tex]\( x = 2 - (-1) = 3 \)[/tex]
- Average rate = [tex]\(\frac{21}{3} = 7\)[/tex]
Function [tex]\( g \)[/tex] on [tex]\([-1, 2]\)[/tex]:
- Change in [tex]\( g(x) = g(2) - g(-1) = 0 - (-52) = 52 \)[/tex]
- Change in [tex]\( x = 2 - (-1) = 3 \)[/tex]
- Average rate = [tex]\(\frac{52}{3} \approx 17.33\)[/tex]
### Conclusion
Both functions [tex]\( f \)[/tex] and [tex]\( g \)[/tex] are increasing on the interval [tex]\([-1, 2]\)[/tex]. However, function [tex]\( g \)[/tex] increases at a faster average rate compared to function [tex]\( f \)[/tex].
Thus, the correct statement is:
- Both functions are increasing, but function g increases at a faster average rate.
### Step 1: Analyze Function [tex]\( f \)[/tex]
The function [tex]\( f \)[/tex] is represented by a table of values:
[tex]\[
\begin{array}{cccccc}
x & -2 & -1 & 0 & 1 & 2 \\
f(x) & -46 & -22 & -10 & -4 & -1
\end{array}
\][/tex]
Increasing Behavior:
- Look at the values of [tex]\( f(x) \)[/tex] as [tex]\( x \)[/tex] moves from [tex]\(-1\)[/tex] to [tex]\(2\)[/tex].
- From the table:
- [tex]\( f(-1) = -22 \)[/tex]
- [tex]\( f(0) = -10 \)[/tex]
- [tex]\( f(1) = -4 \)[/tex]
- [tex]\( f(2) = -1 \)[/tex]
As [tex]\( x \)[/tex] increases from [tex]\(-1\)[/tex] to 2, [tex]\( f(x) \)[/tex] is consistently increasing because each consecutive value is greater than the previous one.
Sign:
- All values [tex]\( f(x) \)[/tex] at [tex]\( x = -1, 0, 1, 2 \)[/tex] are negative: [tex]\(-22, -10, -4, -1\)[/tex].
### Step 2: Analyze Function [tex]\( g \)[/tex]
The function [tex]\( g \)[/tex] is given by the equation:
[tex]\[ g(x) = -18 \left(\frac{1}{3}\right)^x + 2 \][/tex]
We calculate the values for [tex]\( g(x) \)[/tex] at [tex]\( x = -1, 0, 1, 2 \)[/tex]:
1. [tex]\( g(-1) = -18 \left(\frac{1}{3}\right)^{-1} + 2 = -18 \times 3 + 2 = -54 + 2 = -52 \)[/tex]
2. [tex]\( g(0) = -18 \left(\frac{1}{3}\right)^{0} + 2 = -18 \times 1 + 2 = -18 + 2 = -16 \)[/tex]
3. [tex]\( g(1) = -18 \left(\frac{1}{3}\right)^{1} + 2 = -18 \times \frac{1}{3} + 2 = -6 + 2 = -4 \)[/tex]
4. [tex]\( g(2) = -18 \left(\frac{1}{3}\right)^{2} + 2 = -18 \times \frac{1}{9} + 2 = -2 + 2 = 0 \)[/tex]
Increasing Behavior:
- As [tex]\( x \)[/tex] increases, [tex]\( g(-1) = -52 \)[/tex] to [tex]\( g(2) = 0 \)[/tex], the values of [tex]\( g(x) \)[/tex] increase.
Sign:
- Examine the values:
- [tex]\( g(-1) = -52\)[/tex]
- [tex]\( g(0) = -16\)[/tex]
- [tex]\( g(1) = -4\)[/tex]
- [tex]\( g(2) = 0\)[/tex]
The values are negative except [tex]\( g(2) = 0 \)[/tex], so function [tex]\( g \)[/tex] is not entirely negative.
### Step 3: Compare Average Rate of Change
The average rate of change for a function on an interval [tex]\([a, b]\)[/tex] is given by:
[tex]\[
\text{Average Rate of Change} = \frac{\text{Change in } y}{\text{Change in } x}
\][/tex]
Function [tex]\( f \)[/tex] on [tex]\([-1, 2]\)[/tex]:
- Change in [tex]\( f(x) = f(2) - f(-1) = -1 - (-22) = 21 \)[/tex]
- Change in [tex]\( x = 2 - (-1) = 3 \)[/tex]
- Average rate = [tex]\(\frac{21}{3} = 7\)[/tex]
Function [tex]\( g \)[/tex] on [tex]\([-1, 2]\)[/tex]:
- Change in [tex]\( g(x) = g(2) - g(-1) = 0 - (-52) = 52 \)[/tex]
- Change in [tex]\( x = 2 - (-1) = 3 \)[/tex]
- Average rate = [tex]\(\frac{52}{3} \approx 17.33\)[/tex]
### Conclusion
Both functions [tex]\( f \)[/tex] and [tex]\( g \)[/tex] are increasing on the interval [tex]\([-1, 2]\)[/tex]. However, function [tex]\( g \)[/tex] increases at a faster average rate compared to function [tex]\( f \)[/tex].
Thus, the correct statement is:
- Both functions are increasing, but function g increases at a faster average rate.
