Select the correct answer.

Exponential function [tex]$f$[/tex] is represented by the table.
\begin{tabular}{|l|c|c|c|c|l|}
\hline [tex]$x$[/tex] & 0 & 1 & 2 & 3 & 4 \\
\hline [tex]$f(x)$[/tex] & 15 & 7 & 3 & 1 & 0 \\
\hline
\end{tabular}

Function [tex]$g$[/tex] is an exponential function passing through the points [tex]$(0,9)$[/tex] and [tex]$(3,0)$[/tex].

Which statement correctly compares the two functions?

A. Only function [tex]$f$[/tex] is decreasing on [tex]$[0,3]$[/tex], and both functions are positive on that interval.
B. Both functions are decreasing on [tex]$[0,3]$[/tex], and function [tex]$f$[/tex] is decreasing at a faster rate.
C. Only function [tex]$g$[/tex] is decreasing on [tex]$[0,3]$[/tex], and only function [tex]$f$[/tex] is positive on that interval.
D. Both functions are decreasing on [tex]$[0,3]$[/tex], and function [tex]$g$[/tex] is decreasing at a faster rate.



Answer :

To solve this problem, we need to analyze the properties of both functions [tex]\( f \)[/tex] and [tex]\( g \)[/tex] as described.

1. Interpret Function [tex]\( f \)[/tex]:
- The values of [tex]\( f(x) \)[/tex] are given in the table:
[tex]\[ \begin{array}{|c|c|c|c|c|c|} \hline x & 0 & 1 & 2 & 3 \\ \hline f(x) & 15 & 7 & 3 & 1 \\ \hline \end{array} \][/tex]
- We observe that [tex]\( f(x) \)[/tex] decreases from 15 to 7, then from 7 to 3, and finally from 3 to 1. Thus, [tex]\( f \)[/tex] is strictly decreasing on the interval [tex]\([0, 3]\)[/tex].
- Also, [tex]\( f(x) \)[/tex] is positive on [tex]\([0, 3]\)[/tex] because all values are greater than 0.

2. Analyze Function [tex]\( g \)[/tex]:
- Function [tex]\( g \)[/tex] is exponential and it passes through points [tex]\((0, 9)\)[/tex] and [tex]\((3, 0)\)[/tex]. This suggest that:
[tex]\[ g(x) = a \cdot b^x \][/tex]
- For [tex]\( g(0) = 9 \)[/tex], we find [tex]\( a = 9 \)[/tex] because any number to the power of 0 is 1.
[tex]\[ g(x) = 9 \cdot b^x \][/tex]
- For [tex]\( g(3) = 0 \)[/tex]:
[tex]\[ 9 \cdot b^3 = 0 \][/tex]
- Since this equation implies [tex]\(\text{ }b\)[/tex] could not be zero or positive, we conclude [tex]\(b\)[/tex] must be a negative or undefined in simple [tex]\( a\cdot b^x \ form\)[/tex]. Rather we take points and interpolate/Exponential decreasing rate.

y=a[tex]\(e^{bx}) => to simplify. 3. Check Decreasing Nature and Positivity of \( g \)[/tex]:
- We can approximate the correct function and interpolate.

4. Compare the Two Functions:
- As we already determined, function [tex]\( f(x)\)[/tex] is strictly decreasing and positive on the interval [tex]\([0, 3]\)[/tex].

For [tex]\( g(x)\)[/tex], theoretically, from [tex]\( (0,9) \to decreasing expon realizing strictly \ ). 5. Find the Rate of Decreasing: - Compare the function rates in *practical checks: A. Calculate Values of f : - f decrease ( from tab 15 to 1 ! in steps) B. Calculate potential curve decrease - \( k \emphasise \)[/tex] terms fits valid exp fitting.


Therefore, Our answer translations correct:
different checks Full Steps.

Thus Option will lie as

B: Both functions are decreasing
\(f's\ exact verification steps more practical plotting comparation.


Hence, the better realistic match point definitions and value comparations :

Select correct option * B সব়.