What is the multiplicative rate of change for the exponential function [tex]f(x)=2\left(\frac{5}{2}\right)^{-x}[/tex] ?
A. 0.4
B. 0.6
C. 1.5
D. 2.5
Answer (2)
Rewrite the function in the standard exponential form f ( x ) = a ⋅ b x .
Use the property a − x = a x 1 = ( a 1 ) x to rewrite the function as f ( x ) = 2 ( 5 2 ) x .
Identify the base b in the rewritten function. The base b represents the multiplicative rate of change.
State the multiplicative rate of change: 0.4 .
Explanation
Understanding the Problem We are given the exponential function f(x)=2\[\frac{5}{2}\]^{-x} and asked to find its multiplicative rate of change. The multiplicative rate of change is the base of the exponential function when it is written in the form f ( x ) = a ⋅ b x , where a is the initial value and b is the multiplicative rate of change.
Rewriting the Function To find the multiplicative rate of change, we need to rewrite the given function in the standard form f ( x ) = a ⋅ b x . We can use the property a − x = a x 1 = ( a 1 ) x to rewrite the function as follows:
f ( x ) = 2 ( 2 5 ) − x = 2 ( 5 2 ) x
Identifying the Multiplicative Rate of Change Now that the function is in the form f ( x ) = a ⋅ b x , we can identify the base b , which represents the multiplicative rate of change. In this case, b = 5 2 .
Stating the Answer The multiplicative rate of change is 5 2 , which is equal to 0.4.
Examples
Exponential functions are used to model various real-world phenomena, such as population growth, radioactive decay, and compound interest. The multiplicative rate of change represents the factor by which the quantity changes for each unit increase in the independent variable. For example, if a population grows exponentially with a multiplicative rate of change of 1.2, it means the population increases by 20% each year. Understanding the multiplicative rate of change helps us predict future values and analyze trends in these phenomena.
The multiplicative rate of change for the function f ( x ) = 2 ( 2 5 ) − x is found to be 0.4 after rewriting the function in the standard exponential form. This is equivalent to the base of the function when expressed as f ( x ) = a ⋅ b x . Hence, the chosen option is A. 0.4.
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