Answered

IDNLearn.com makes it easy to get reliable answers from experts and enthusiasts alike. Our platform offers detailed and accurate responses from experts, helping you navigate any topic with confidence.

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.

Sagot :

GinnyAnsweridn
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.
Thank you for using this platform to share and learn. Don't hesitate to keep asking and answering. We value every contribution you make. Find clear answers at IDNLearn.com. Thanks for stopping by, and come back for more reliable solutions.