Answer :
To solve this problem, we need to compare two functions, [tex]\( f \)[/tex] and [tex]\( g \)[/tex], over the interval [tex]\([-1, 2]\)[/tex].
Step 1: Analyzing Function [tex]\( f \)[/tex]
- The function [tex]\( f \)[/tex] is provided using a table of values:
[tex]\[
\begin{array}{c|c}
x & f(x) \\
\hline
-1 & -22 \\
0 & -10 \\
1 & -4 \\
2 & -1 \\
\end{array}
\][/tex]
- To determine if [tex]\( f \)[/tex] is increasing, we check if for each consecutive pair of [tex]\( x \)[/tex], the value of [tex]\( f(x) \)[/tex] increases:
- Compare [tex]\( f(-1) = -22 \)[/tex] to [tex]\( f(0) = -10 \)[/tex] : [tex]\(-22 < -10\)[/tex]
- Compare [tex]\( f(0) = -10 \)[/tex] to [tex]\( f(1) = -4 \)[/tex] : [tex]\(-10 < -4\)[/tex]
- Compare [tex]\( f(1) = -4 \)[/tex] to [tex]\( f(2) = -1 \)[/tex] : [tex]\(-4 < -1\)[/tex]
- Since [tex]\( f(x) \)[/tex] increases as [tex]\( x \)[/tex] increases within the interval, function [tex]\( f \)[/tex] is increasing.
- Also, observe that for all [tex]\( x \)[/tex] from [tex]\([-1, 2]\)[/tex], [tex]\( f(x) \)[/tex] values are negative:
- [tex]\( f(-1) = -22 \)[/tex], [tex]\( f(0) = -10 \)[/tex], [tex]\( f(1) = -4 \)[/tex], [tex]\( f(2) = -1 \)[/tex]
Step 2: Analyzing 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]
- Let's check whether [tex]\( g(x) \)[/tex] is increasing for [tex]\( x = -1, 0, 1, 2 \)[/tex]:
- Because [tex]\( g(x) = -18 \left(\frac{1}{3}\right)^x + 2 \)[/tex], and as [tex]\( x \)[/tex] increases, [tex]\(\left(\frac{1}{3}\right)^x\)[/tex] becomes smaller, thus causing [tex]\( -18\left(\frac{1}{3}\right)^x \)[/tex] to decrease and resulting in [tex]\( g(x) \)[/tex] decreasing.
- Looking at [tex]\( g(x) \)[/tex] values, they are not always negative as values around [tex]\( x = -1 \)[/tex] and [tex]\( x = 0 \)[/tex] are positive due to the [tex]\( +2 \)[/tex] shift.
Conclusion:
- From the analysis above, we find that only function [tex]\( f \)[/tex] is increasing in the interval [tex]\([-1, 2]\)[/tex], and all the values of [tex]\( f(x) \)[/tex] are negative.
- Therefore, the correct statement is:
D. Only function [tex]\( f \)[/tex] is increasing, and only function [tex]\( f \)[/tex] is negative.
Step 1: Analyzing Function [tex]\( f \)[/tex]
- The function [tex]\( f \)[/tex] is provided using a table of values:
[tex]\[
\begin{array}{c|c}
x & f(x) \\
\hline
-1 & -22 \\
0 & -10 \\
1 & -4 \\
2 & -1 \\
\end{array}
\][/tex]
- To determine if [tex]\( f \)[/tex] is increasing, we check if for each consecutive pair of [tex]\( x \)[/tex], the value of [tex]\( f(x) \)[/tex] increases:
- Compare [tex]\( f(-1) = -22 \)[/tex] to [tex]\( f(0) = -10 \)[/tex] : [tex]\(-22 < -10\)[/tex]
- Compare [tex]\( f(0) = -10 \)[/tex] to [tex]\( f(1) = -4 \)[/tex] : [tex]\(-10 < -4\)[/tex]
- Compare [tex]\( f(1) = -4 \)[/tex] to [tex]\( f(2) = -1 \)[/tex] : [tex]\(-4 < -1\)[/tex]
- Since [tex]\( f(x) \)[/tex] increases as [tex]\( x \)[/tex] increases within the interval, function [tex]\( f \)[/tex] is increasing.
- Also, observe that for all [tex]\( x \)[/tex] from [tex]\([-1, 2]\)[/tex], [tex]\( f(x) \)[/tex] values are negative:
- [tex]\( f(-1) = -22 \)[/tex], [tex]\( f(0) = -10 \)[/tex], [tex]\( f(1) = -4 \)[/tex], [tex]\( f(2) = -1 \)[/tex]
Step 2: Analyzing 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]
- Let's check whether [tex]\( g(x) \)[/tex] is increasing for [tex]\( x = -1, 0, 1, 2 \)[/tex]:
- Because [tex]\( g(x) = -18 \left(\frac{1}{3}\right)^x + 2 \)[/tex], and as [tex]\( x \)[/tex] increases, [tex]\(\left(\frac{1}{3}\right)^x\)[/tex] becomes smaller, thus causing [tex]\( -18\left(\frac{1}{3}\right)^x \)[/tex] to decrease and resulting in [tex]\( g(x) \)[/tex] decreasing.
- Looking at [tex]\( g(x) \)[/tex] values, they are not always negative as values around [tex]\( x = -1 \)[/tex] and [tex]\( x = 0 \)[/tex] are positive due to the [tex]\( +2 \)[/tex] shift.
Conclusion:
- From the analysis above, we find that only function [tex]\( f \)[/tex] is increasing in the interval [tex]\([-1, 2]\)[/tex], and all the values of [tex]\( f(x) \)[/tex] are negative.
- Therefore, the correct statement is:
D. Only function [tex]\( f \)[/tex] is increasing, and only function [tex]\( f \)[/tex] is negative.