Answer :
To solve this problem, let's carefully analyze the given information about the functions [tex]\( f \)[/tex] and [tex]\( g \)[/tex] within the specified interval [tex]\((0, 3)\)[/tex].
### Analysis of Function [tex]\( f \)[/tex]
We have the values of function [tex]\( f \)[/tex] as given in the table:
[tex]\[ \begin{array}{|c|c|c|c|c|c|} \hline x & -1 & 0 & 1 & 2 & 3 & 4 \\ \hline f(x) & 80 & 26 & 8 & 2 & 0 & -\frac{2}{3} \\ \hline \end{array} \][/tex]
Focusing on the interval [tex]\( (0, 3) \)[/tex]:
- At [tex]\( x = 0 \)[/tex], [tex]\( f(0) = 26 \)[/tex]
- At [tex]\( x = 1 \)[/tex], [tex]\( f(1) = 8 \)[/tex]
- At [tex]\( x = 2 \)[/tex], [tex]\( f(2) = 2 \)[/tex]
- At [tex]\( x = 3 \)[/tex], [tex]\( f(3) = 0 \)[/tex]
We observe the following:
- [tex]\( f(0) = 26 \geq 0 \)[/tex]
- [tex]\( f(1) = 8 \geq 0 \)[/tex]
- [tex]\( f(2) = 2 \geq 0 \)[/tex]
- [tex]\( f(3) = 0 \geq 0 \)[/tex]
Therefore, [tex]\( f(x) \)[/tex] is non-negative for [tex]\( 0 \leq x \leq 3 \)[/tex].
We can see that the function values are decreasing from 26 to 0 in this interval. Thus, [tex]\( f(x) \)[/tex] is positive and decreasing on the interval [tex]\( (0, 3) \)[/tex].
### Analysis of Function [tex]\( g \)[/tex]
We know that [tex]\( g \)[/tex] is an exponential function passing through the points [tex]\( (0, 7) \)[/tex] and [tex]\( (3, 0) \)[/tex]. We need to check the behavior of [tex]\( g \)[/tex] on the interval [tex]\( (0, 3) \)[/tex].
Given [tex]\( g(0) = 7 \)[/tex] and [tex]\( g(3) = 0 \)[/tex]:
- Since [tex]\( g \)[/tex] is an exponential function and it is decreasing from [tex]\( 7 \)[/tex] to [tex]\( 0 \)[/tex], we can calculate the behavior by understanding the nature of exponential decay.
In general, an exponential function that decreases passes through these points would look something like [tex]\( g(x) = a e^{-bx} \)[/tex]:
- At [tex]\( x = 0 \)[/tex], [tex]\( g(0) = a = 7 \)[/tex].
- At [tex]\( x = 3 \)[/tex], [tex]\( 7 e^{-3b} = 0 \)[/tex], solving for [tex]\( b \)[/tex] is not necessary as we know it is decreasing.
Hence, [tex]\( g(x) \)[/tex] is positive and decreasing on the interval [tex]\( (0, 3) \)[/tex] because an exponential decay function always approaches zero but remains positive right up to the point where it becomes zero.
### Conclusion
Both functions [tex]\( f \)[/tex] and [tex]\( g \)[/tex] are positive and decreasing on the interval [tex]\( (0, 3) \)[/tex].
Therefore, the correct statement comparing the two functions on the interval [tex]\( (0, 3) \)[/tex] is:
[tex]\[ \boxed{\text{B. Both functions are positive and decreasing on the interval.}} \][/tex]
### Analysis of Function [tex]\( f \)[/tex]
We have the values of function [tex]\( f \)[/tex] as given in the table:
[tex]\[ \begin{array}{|c|c|c|c|c|c|} \hline x & -1 & 0 & 1 & 2 & 3 & 4 \\ \hline f(x) & 80 & 26 & 8 & 2 & 0 & -\frac{2}{3} \\ \hline \end{array} \][/tex]
Focusing on the interval [tex]\( (0, 3) \)[/tex]:
- At [tex]\( x = 0 \)[/tex], [tex]\( f(0) = 26 \)[/tex]
- At [tex]\( x = 1 \)[/tex], [tex]\( f(1) = 8 \)[/tex]
- At [tex]\( x = 2 \)[/tex], [tex]\( f(2) = 2 \)[/tex]
- At [tex]\( x = 3 \)[/tex], [tex]\( f(3) = 0 \)[/tex]
We observe the following:
- [tex]\( f(0) = 26 \geq 0 \)[/tex]
- [tex]\( f(1) = 8 \geq 0 \)[/tex]
- [tex]\( f(2) = 2 \geq 0 \)[/tex]
- [tex]\( f(3) = 0 \geq 0 \)[/tex]
Therefore, [tex]\( f(x) \)[/tex] is non-negative for [tex]\( 0 \leq x \leq 3 \)[/tex].
We can see that the function values are decreasing from 26 to 0 in this interval. Thus, [tex]\( f(x) \)[/tex] is positive and decreasing on the interval [tex]\( (0, 3) \)[/tex].
### Analysis of Function [tex]\( g \)[/tex]
We know that [tex]\( g \)[/tex] is an exponential function passing through the points [tex]\( (0, 7) \)[/tex] and [tex]\( (3, 0) \)[/tex]. We need to check the behavior of [tex]\( g \)[/tex] on the interval [tex]\( (0, 3) \)[/tex].
Given [tex]\( g(0) = 7 \)[/tex] and [tex]\( g(3) = 0 \)[/tex]:
- Since [tex]\( g \)[/tex] is an exponential function and it is decreasing from [tex]\( 7 \)[/tex] to [tex]\( 0 \)[/tex], we can calculate the behavior by understanding the nature of exponential decay.
In general, an exponential function that decreases passes through these points would look something like [tex]\( g(x) = a e^{-bx} \)[/tex]:
- At [tex]\( x = 0 \)[/tex], [tex]\( g(0) = a = 7 \)[/tex].
- At [tex]\( x = 3 \)[/tex], [tex]\( 7 e^{-3b} = 0 \)[/tex], solving for [tex]\( b \)[/tex] is not necessary as we know it is decreasing.
Hence, [tex]\( g(x) \)[/tex] is positive and decreasing on the interval [tex]\( (0, 3) \)[/tex] because an exponential decay function always approaches zero but remains positive right up to the point where it becomes zero.
### Conclusion
Both functions [tex]\( f \)[/tex] and [tex]\( g \)[/tex] are positive and decreasing on the interval [tex]\( (0, 3) \)[/tex].
Therefore, the correct statement comparing the two functions on the interval [tex]\( (0, 3) \)[/tex] is:
[tex]\[ \boxed{\text{B. Both functions are positive and decreasing on the interval.}} \][/tex]
