To find the multiplicative rate of change for the exponential function [tex]\( f(x) = 2 \left( \frac{5}{2} \right)^{-x} \)[/tex], follow these steps:
1. Identify the function's form:
The given function is [tex]\( f(x) = 2 \left( \frac{5}{2} \right)^{-x} \)[/tex].
2. Simplify the base of the exponential term:
Notice that [tex]\( \left( \frac{5}{2} \right)^{-x} \)[/tex] can be rewritten by using the property of exponents: [tex]\( a^{-b} = \frac{1}{a^b} \)[/tex]. Hence:
[tex]\[
\left( \frac{5}{2} \right)^{-x} = \left( \frac{1}{\frac{5}{2}} \right)^x = \left( \frac{2}{5} \right)^x
\][/tex]
3. Rewrite the function with the simplified base:
The function now becomes:
[tex]\[
f(x) = 2 \left( \frac{2}{5} \right)^x
\][/tex]
4. Identify the multiplicative rate of change:
In an exponential function of the form [tex]\( f(x) = a b^x \)[/tex], the multiplicative rate of change is the base [tex]\(b\)[/tex].
In this case, the base [tex]\(b\)[/tex] of the exponential term is [tex]\(\frac{2}{5}\)[/tex].
5. Convert the fraction to a decimal:
[tex]\[
\frac{2}{5} = 0.4
\][/tex]
Thus, the multiplicative rate of change for the exponential function [tex]\( f(x) = 2 \left( \frac{5}{2} \right)^{-x} \)[/tex] is [tex]\( \boxed{0.4} \)[/tex].
