What is the multiplicative rate of change for the exponential function [tex][tex]$f(x)=2\left(\frac{5}{2}\right)^{-x}$[/tex][/tex]?

A. 0.4
B. 0.6
C. 1.5
D. 2.5



Answer :

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].