UpStudy Free Solution:
To find the multiplicative rate of change of the exponential function, we need to determine the ratio between consecutive \(y\)-values.
Given the table:
\[\begin{array} { | c| c| } \ hlinex & y \\ \hline 1 & 2 \\ 2 & \frac { 2} { 5} \\ 3& \frac { 2} { 25} \\ 4 & \frac { 2} { 125} \\ \hline \end{array} \]
Let's calculate the ratio between consecutive \(y\)-values:
\[\frac { \frac { 2} { 5} } { 2} = \frac { 2} { 5} \times \frac { 1} { 2} = \frac { 1} { 5} \]
\[\frac { \frac { 2} { 25} } { \frac { 2} { 5} } = \frac { 2} { 25} \times \frac { 5} { 2} = \frac { 1} { 5} \]
\[\frac { \frac { 2} { 125} } { \frac { 2} { 25} } = \frac { 2} { 125} \times \frac { 25} { 2} = \frac { 1} { 5} \]
The multiplicative rate of change is consistently \(\frac { 1} { 5} \).
Therefore, the correct answer is:
\(\frac { 1} { 5} \)

Supplemental Knowledge:

Exponential functions are mathematical expressions in which a constant base is raised to a variable exponent. The general form of an exponential function is \(f( x) = a \cdot b^ x\), where \(a\) is the initial value and \(b\) is the base, representing the multiplicative rate of change.
The multiplicative rate of change, or growth factor, in an exponential function indicates how much the function's value changes for each unit increase in \(x\). If the base \(b > 1\), the function represents exponential growth; if \(0 < b < 1\), it represents exponential decay.
To determine the multiplicative rate of change from a table representing an exponential function, you can calculate the ratio between consecutive outputs (function values).

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