Question

the table represents an exponential function.

x	y
2	\\(\frac{9}{8}\\)
3	\\(\frac{27}{32}\\)
4	\\(\frac{81}{128}\\)

what is the multiplicative rate of change of the function?
options: \\(\frac{3}{4}\\), \\(\frac{4}{3}\\), \\(\frac{3}{2}\\), \\(\frac{2}{3}\\)

Explanation:

Step1: Recall multiplicative rate formula

For an exponential function, the multiplicative rate of change between consecutive \(x\)-values (where \(x\) increases by 1) is \(\frac{y_{n + 1}}{y_{n}}\).

Step2: Calculate using \(x = 1\) and \(x = 2\)

When \(x = 1\), \(y=\frac{3}{2}\); when \(x = 2\), \(y=\frac{9}{8}\).
Compute \(\frac{y_{2}}{y_{1}}=\frac{\frac{9}{8}}{\frac{3}{2}}\).
Using division of fractions: \(\frac{9}{8}\times\frac{2}{3}=\frac{18}{24}=\frac{3}{4}\)? Wait, no, wait—wait, let's check again. Wait, \(\frac{9}{8}\div\frac{3}{2}=\frac{9}{8}\times\frac{2}{3}=\frac{18}{24}=\frac{3}{4}\)? Wait, no, that can't be. Wait, maybe I mixed up. Wait, no, let's check with \(x = 2\) and \(x = 3\).
When \(x = 2\), \(y=\frac{9}{8}\); \(x = 3\), \(y=\frac{27}{32}\).
\(\frac{\frac{27}{32}}{\frac{9}{8}}=\frac{27}{32}\times\frac{8}{9}=\frac{216}{288}=\frac{3}{4}\). Wait, no, that's the same. Wait, but the options have \(\frac{3}{4}\) as the first option. Wait, but let's check again. Wait, the first calculation: \(\frac{9}{8}\div\frac{3}{2}=\frac{9}{8}\times\frac{2}{3}=\frac{18}{24}=\frac{3}{4}\). Then \(\frac{27}{32}\div\frac{9}{8}=\frac{27}{32}\times\frac{8}{9}=\frac{216}{288}=\frac{3}{4}\). And \(\frac{81}{128}\div\frac{27}{32}=\frac{81}{128}\times\frac{32}{27}=\frac{2592}{3456}=\frac{3}{4}\). So the multiplicative rate is \(\frac{3}{4}\)? Wait, no, wait the options: the first option is \(\frac{3}{4}\). Wait, but let me confirm. The multiplicative rate of change for an exponential function \(y = ab^x\) is \(b\), which is the ratio of consecutive \(y\)-values when \(x\) increases by 1. So taking \(x = 1\) to \(x = 2\): \(y\) goes from \(\frac{3}{2}\) to \(\frac{9}{8}\). So \(\frac{9}{8}\div\frac{3}{2}=\frac{9}{8}\times\frac{2}{3}=\frac{18}{24}=\frac{3}{4}\). Then \(x = 2\) to \(x = 3\): \(\frac{27}{32}\div\frac{9}{8}=\frac{27}{32}\times\frac{8}{9}=\frac{216}{288}=\frac{3}{4}\). \(x = 3\) to \(x = 4\): \(\frac{81}{128}\div\frac{27}{32}=\frac{81}{128}\times\frac{32}{27}=\frac{2592}{3456}=\frac{3}{4}\). So the multiplicative rate is \(\frac{3}{4}\).

Answer:

\(\boldsymbol{\frac{3}{4}}\) (corresponding to the first option, e.g., if options are labeled A: \(\frac{3}{4}\), then A. \(\frac{3}{4}\))