Answer :
To solve this problem, we are comparing two functions, [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex], over the interval [tex]\([-1, 2]\)[/tex]. Let's go through each step to see which statement correctly compares the functions.
### Step 1: Analyze Function [tex]\(f(x)\)[/tex]
Function [tex]\(f(x)\)[/tex] is represented by a table:
[tex]\[
\begin{array}{|c|c|c|c|c|}
\hline
x & -1 & 0 & 1 & 2 \\
\hline
f(x) & -22 & -10 & -4 & -1 \\
\hline
\end{array}
\][/tex]
- Calculate Differences:
- From [tex]\(x = -1\)[/tex] to [tex]\(x = 0\)[/tex]: [tex]\(-10 - (-22) = 12\)[/tex]
- From [tex]\(x = 0\)[/tex] to [tex]\(x = 1\)[/tex]: [tex]\(-4 - (-10) = 6\)[/tex]
- From [tex]\(x = 1\)[/tex] to [tex]\(x = 2\)[/tex]: [tex]\(-1 - (-4) = 3\)[/tex]
Since all these differences are positive, function [tex]\(f(x)\)[/tex] is increasing on the interval [tex]\([-1, 2]\)[/tex].
- Check if [tex]\(f(x)\)[/tex] is Negative:
- All values [tex]\(-22, -10, -4, -1\)[/tex] are less than zero, so function [tex]\(f(x)\)[/tex] is negative over the interval [tex]\([-1, 2]\)[/tex].
### Step 2: Analyze 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].
- Calculate [tex]\(g(x)\)[/tex] Values:
- [tex]\( g(-1) = -18\left(\frac{1}{3}\right)^{-1} + 2 = -18 \times 3 + 2 = -54 + 2 = -52 \)[/tex]
- [tex]\( g(0) = -18\left(\frac{1}{3}\right)^{0} + 2 = -18 \times 1 + 2 = -16 \)[/tex]
- [tex]\( g(1) = -18\left(\frac{1}{3}\right)^{1} + 2 = -18 \times \frac{1}{3} + 2 = -6 + 2 = -4 \)[/tex]
- [tex]\( g(2) = -18\left(\frac{1}{3}\right)^{2} + 2 = -18 \times \frac{1}{9} + 2 = -2 + 2 = 0 \)[/tex]
- Calculate Differences:
- From [tex]\(x = -1\)[/tex] to [tex]\(x = 0\)[/tex]: [tex]\(-16 - (-52) = 36\)[/tex]
- From [tex]\(x = 0\)[/tex] to [tex]\(x = 1\)[/tex]: [tex]\(-4 - (-16) = 12\)[/tex]
- From [tex]\(x = 1\)[/tex] to [tex]\(x = 2\)[/tex]: [tex]\(0 - (-4) = 4\)[/tex]
Since all these differences are positive, function [tex]\(g(x)\)[/tex] is increasing on the interval [tex]\([-1, 2]\)[/tex].
- Check if [tex]\(g(x)\)[/tex] is Negative:
- The values [tex]\(-52\)[/tex], [tex]\(-16\)[/tex], and [tex]\(-4\)[/tex] are negative, but [tex]\(g(2) = 0\)[/tex] is not negative. Therefore, function [tex]\(g(x)\)[/tex] is not entirely negative over the interval [tex]\([-1, 2]\)[/tex].
### Step 3: Compare Functions [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]
- Both Functions Increase: Both [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex] are increasing over [tex]\([-1, 2]\)[/tex].
- Negativity: Only [tex]\(f(x)\)[/tex] is negative throughout the interval.
Correct Statement: The correct comparison based on these observations is:
- B. Both functions are increasing, but function [tex]\(g(x)\)[/tex] increases at a faster average rate.
Hence, the answer is option B.
### Step 1: Analyze Function [tex]\(f(x)\)[/tex]
Function [tex]\(f(x)\)[/tex] is represented by a table:
[tex]\[
\begin{array}{|c|c|c|c|c|}
\hline
x & -1 & 0 & 1 & 2 \\
\hline
f(x) & -22 & -10 & -4 & -1 \\
\hline
\end{array}
\][/tex]
- Calculate Differences:
- From [tex]\(x = -1\)[/tex] to [tex]\(x = 0\)[/tex]: [tex]\(-10 - (-22) = 12\)[/tex]
- From [tex]\(x = 0\)[/tex] to [tex]\(x = 1\)[/tex]: [tex]\(-4 - (-10) = 6\)[/tex]
- From [tex]\(x = 1\)[/tex] to [tex]\(x = 2\)[/tex]: [tex]\(-1 - (-4) = 3\)[/tex]
Since all these differences are positive, function [tex]\(f(x)\)[/tex] is increasing on the interval [tex]\([-1, 2]\)[/tex].
- Check if [tex]\(f(x)\)[/tex] is Negative:
- All values [tex]\(-22, -10, -4, -1\)[/tex] are less than zero, so function [tex]\(f(x)\)[/tex] is negative over the interval [tex]\([-1, 2]\)[/tex].
### Step 2: Analyze 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].
- Calculate [tex]\(g(x)\)[/tex] Values:
- [tex]\( g(-1) = -18\left(\frac{1}{3}\right)^{-1} + 2 = -18 \times 3 + 2 = -54 + 2 = -52 \)[/tex]
- [tex]\( g(0) = -18\left(\frac{1}{3}\right)^{0} + 2 = -18 \times 1 + 2 = -16 \)[/tex]
- [tex]\( g(1) = -18\left(\frac{1}{3}\right)^{1} + 2 = -18 \times \frac{1}{3} + 2 = -6 + 2 = -4 \)[/tex]
- [tex]\( g(2) = -18\left(\frac{1}{3}\right)^{2} + 2 = -18 \times \frac{1}{9} + 2 = -2 + 2 = 0 \)[/tex]
- Calculate Differences:
- From [tex]\(x = -1\)[/tex] to [tex]\(x = 0\)[/tex]: [tex]\(-16 - (-52) = 36\)[/tex]
- From [tex]\(x = 0\)[/tex] to [tex]\(x = 1\)[/tex]: [tex]\(-4 - (-16) = 12\)[/tex]
- From [tex]\(x = 1\)[/tex] to [tex]\(x = 2\)[/tex]: [tex]\(0 - (-4) = 4\)[/tex]
Since all these differences are positive, function [tex]\(g(x)\)[/tex] is increasing on the interval [tex]\([-1, 2]\)[/tex].
- Check if [tex]\(g(x)\)[/tex] is Negative:
- The values [tex]\(-52\)[/tex], [tex]\(-16\)[/tex], and [tex]\(-4\)[/tex] are negative, but [tex]\(g(2) = 0\)[/tex] is not negative. Therefore, function [tex]\(g(x)\)[/tex] is not entirely negative over the interval [tex]\([-1, 2]\)[/tex].
### Step 3: Compare Functions [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]
- Both Functions Increase: Both [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex] are increasing over [tex]\([-1, 2]\)[/tex].
- Negativity: Only [tex]\(f(x)\)[/tex] is negative throughout the interval.
Correct Statement: The correct comparison based on these observations is:
- B. Both functions are increasing, but function [tex]\(g(x)\)[/tex] increases at a faster average rate.
Hence, the answer is option B.
