The table represents an exponential function.

\begin{tabular}{|c|c|}
\hline [tex][tex]$x$[/tex][/tex] & [tex][tex]$y$[/tex][/tex] \\
\hline 1 & 2 \\
\hline 2 & [tex][tex]$\frac{2}{5}$[/tex][/tex] \\
\hline 3 & [tex][tex]$\frac{2}{25}$[/tex][/tex] \\
\hline 4 & [tex][tex]$\frac{2}{125}$[/tex][/tex] \\
\hline
\end{tabular}

What is the multiplicative rate of change of the function?

A. [tex][tex]$\frac{1}{5}$[/tex][/tex]
B. [tex][tex]$\frac{2}{5}$[/tex][/tex]
C. 2
D. 5



Answer :

To determine the multiplicative rate of change of an exponential function based on given values, we first observe the pattern in the values of [tex]\(y\)[/tex] as [tex]\(x\)[/tex] changes. Here is the table we are given:

[tex]\[ \begin{tabular}{|c|c|} \hline x & y \\ \hline 1 & 2 \\ \hline 2 & \frac{2}{5} \\ \hline 3 & \frac{2}{25} \\ \hline 4 & \frac{2}{125} \\ \hline \end{tabular} \][/tex]

An exponential function has the form [tex]\(y = ab^x\)[/tex]. To find the multiplicative rate of change, we need to determine the base [tex]\(b\)[/tex] of the exponential function.

Given the function values:
- When [tex]\(x = 1\)[/tex], [tex]\(y = 2\)[/tex]
- When [tex]\(x = 2\)[/tex], [tex]\(y = \frac{2}{5}\)[/tex]

The multiplicative rate of change, [tex]\(b\)[/tex], can be found by considering the ratio of consecutive [tex]\(y\)[/tex]-values. Let's calculate the ratio between [tex]\(y\)[/tex]-values for [tex]\(x = 2\)[/tex] and [tex]\(x = 1\)[/tex]:

[tex]\[ \frac{\frac{2}{5}}{2} \][/tex]

To simplify this ratio:

[tex]\[ \frac{\frac{2}{5}}{2} = \frac{2}{5} \times \frac{1}{2} = \frac{2}{5} \times \frac{1}{2} = \frac{2 \times 1}{5 \times 2} = \frac{2}{10} = \frac{1}{5} \][/tex]

Therefore, the multiplicative rate of change of the function is [tex]\(\frac{1}{5}\)[/tex]. So, the correct answer is:

[tex]\(\frac{1}{5}\)[/tex]