Answer :
To compare the two functions [tex]\( f \)[/tex] and [tex]\( g \)[/tex] over the interval [tex]\([-1, 2]\)[/tex], let's break down the problem step-by-step:
### Step 1: Check if Function [tex]\( f \)[/tex] is Increasing
The function [tex]\( f \)[/tex] is represented by specific values for given [tex]\( x \)[/tex] values in the table:
[tex]\[
\begin{array}{cccccc}
x & -2 & -1 & 0 & 1 & 2 \\
f(x) & -46 & -2 & -10 & -4 & -1
\end{array}
\][/tex]
We need to determine if [tex]\( f(x) \)[/tex] is increasing over the interval [tex]\([-1, 2]\)[/tex]:
- For [tex]\( x = -1 \)[/tex] to [tex]\( x = 0 \)[/tex]: [tex]\( f(0) = -10 \)[/tex] and [tex]\( f(-1) = -2 \)[/tex]. Since [tex]\(-10 < -2\)[/tex], [tex]\( f(x) \)[/tex] is not increasing here.
- For [tex]\( x = 0 \)[/tex] to [tex]\( x = 1 \)[/tex]: [tex]\( f(1) = -4 \)[/tex], and since [tex]\(-4 > -10\)[/tex], [tex]\( f(x) \)[/tex] is increasing.
- For [tex]\( x = 1 \)[/tex] to [tex]\( x = 2 \)[/tex]: [tex]\( f(2) = -1 \)[/tex], and since [tex]\(-1 > -4\)[/tex], [tex]\( f(x) \)[/tex] is increasing.
Since [tex]\( f(x) \)[/tex] is not increasing throughout the interval, it is not an increasing function over [tex]\([-1, 2]\)[/tex].
### Step 2: Check if Function [tex]\( g \)[/tex] is Increasing
The function [tex]\( g(x) = -18\left(\frac{1}{3}\right)^x + 2 \)[/tex].
To check if this function is increasing across [tex]\([-1, 2]\)[/tex], we calculate [tex]\( g(x) \)[/tex] at these points:
- [tex]\( g(-1) = -18 \left(\frac{1}{3}\right)^{-1} + 2 = -54 + 2 = -52 \)[/tex]
- [tex]\( g(0) = -18 \left(\frac{1}{3}\right)^{0} + 2 = -18 + 2 = -16 \)[/tex]
- [tex]\( g(1) = -18 \left(\frac{1}{3}\right)^{1} + 2 = -6 + 2 = -4 \)[/tex]
- [tex]\( g(2) = -18 \left(\frac{1}{3}\right)^{2} + 2 = -2 + 2 = 0 \)[/tex]
Observing the values, [tex]\(-52, -16, -4, 0\)[/tex], function [tex]\( g(x) \)[/tex] is increasing across the entire interval because each subsequent value is larger than the previous one.
### Step 3: Check if Functions are Negative
- For function [tex]\( f(x) \)[/tex] at [tex]\(x = -1, 0, 1, 2\)[/tex], all values [tex]\(-2, -10, -4, -1\)[/tex] are negative.
- For function [tex]\( g(x) \)[/tex] at [tex]\(x = -1, 0, 1, 2\)[/tex], the values [tex]\(-52, -16, -4, 0\)[/tex] indicate that [tex]\( g(x) \)[/tex] is negative except at [tex]\( x = 2 \)[/tex].
### Conclusion
Given the observations:
- Only function [tex]\( g \)[/tex] is strictly increasing over the interval [tex]\([-1, 2]\)[/tex].
- Only function [tex]\( f \)[/tex] is negative over the entire interval.
The correct statement is:
C. Only function [tex]\( f \)[/tex] is increasing, and only function [tex]\( f \)[/tex] is negative.
However, since [tex]\( f \)[/tex] is not actually increasing over [tex]\([-1, 2]\)[/tex], no valid statement exactly fits the observations from the given options, which were confirmed by analysis.
### Step 1: Check if Function [tex]\( f \)[/tex] is Increasing
The function [tex]\( f \)[/tex] is represented by specific values for given [tex]\( x \)[/tex] values in the table:
[tex]\[
\begin{array}{cccccc}
x & -2 & -1 & 0 & 1 & 2 \\
f(x) & -46 & -2 & -10 & -4 & -1
\end{array}
\][/tex]
We need to determine if [tex]\( f(x) \)[/tex] is increasing over the interval [tex]\([-1, 2]\)[/tex]:
- For [tex]\( x = -1 \)[/tex] to [tex]\( x = 0 \)[/tex]: [tex]\( f(0) = -10 \)[/tex] and [tex]\( f(-1) = -2 \)[/tex]. Since [tex]\(-10 < -2\)[/tex], [tex]\( f(x) \)[/tex] is not increasing here.
- For [tex]\( x = 0 \)[/tex] to [tex]\( x = 1 \)[/tex]: [tex]\( f(1) = -4 \)[/tex], and since [tex]\(-4 > -10\)[/tex], [tex]\( f(x) \)[/tex] is increasing.
- For [tex]\( x = 1 \)[/tex] to [tex]\( x = 2 \)[/tex]: [tex]\( f(2) = -1 \)[/tex], and since [tex]\(-1 > -4\)[/tex], [tex]\( f(x) \)[/tex] is increasing.
Since [tex]\( f(x) \)[/tex] is not increasing throughout the interval, it is not an increasing function over [tex]\([-1, 2]\)[/tex].
### Step 2: Check if Function [tex]\( g \)[/tex] is Increasing
The function [tex]\( g(x) = -18\left(\frac{1}{3}\right)^x + 2 \)[/tex].
To check if this function is increasing across [tex]\([-1, 2]\)[/tex], we calculate [tex]\( g(x) \)[/tex] at these points:
- [tex]\( g(-1) = -18 \left(\frac{1}{3}\right)^{-1} + 2 = -54 + 2 = -52 \)[/tex]
- [tex]\( g(0) = -18 \left(\frac{1}{3}\right)^{0} + 2 = -18 + 2 = -16 \)[/tex]
- [tex]\( g(1) = -18 \left(\frac{1}{3}\right)^{1} + 2 = -6 + 2 = -4 \)[/tex]
- [tex]\( g(2) = -18 \left(\frac{1}{3}\right)^{2} + 2 = -2 + 2 = 0 \)[/tex]
Observing the values, [tex]\(-52, -16, -4, 0\)[/tex], function [tex]\( g(x) \)[/tex] is increasing across the entire interval because each subsequent value is larger than the previous one.
### Step 3: Check if Functions are Negative
- For function [tex]\( f(x) \)[/tex] at [tex]\(x = -1, 0, 1, 2\)[/tex], all values [tex]\(-2, -10, -4, -1\)[/tex] are negative.
- For function [tex]\( g(x) \)[/tex] at [tex]\(x = -1, 0, 1, 2\)[/tex], the values [tex]\(-52, -16, -4, 0\)[/tex] indicate that [tex]\( g(x) \)[/tex] is negative except at [tex]\( x = 2 \)[/tex].
### Conclusion
Given the observations:
- Only function [tex]\( g \)[/tex] is strictly increasing over the interval [tex]\([-1, 2]\)[/tex].
- Only function [tex]\( f \)[/tex] is negative over the entire interval.
The correct statement is:
C. Only function [tex]\( f \)[/tex] is increasing, and only function [tex]\( f \)[/tex] is negative.
However, since [tex]\( f \)[/tex] is not actually increasing over [tex]\([-1, 2]\)[/tex], no valid statement exactly fits the observations from the given options, which were confirmed by analysis.