Select the correct answer.

Exponential function [tex]f[/tex] is represented by the table.
\begin{tabular}{|l|c|c|c|c|c|c|}
\hline
[tex]$x$[/tex] & -1 & 0 & 1 & 2 & 3 & 4 \\
\hline
[tex]$f(x)$[/tex] & 80 & 26 & 8 & 2 & 0 & -\frac{2}{3} \\
\hline
\end{tabular}

Function [tex]g[/tex] is an exponential function passing through the points [tex](0,7)[/tex] and [tex](3,0)[/tex].

Which statement correctly compares the two functions on the interval [tex](0,3)[/tex]?

A. Both functions are positive and increasing on the interval.
B. Both functions are positive and decreasing on the interval.
C. One function is positive on the interval, while the other is negative.
D. Both functions are positive on the interval, but one function is increasing while the other is decreasing.



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]