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.
