Which value of a in the exponential function below would cause the function to shrink?
[tex]f(x)=a\left(\frac{3}{2}\right)^x[/tex]
A. [tex]\frac{4}{5}[/tex]
B. [tex]\frac{5}{4}[/tex]
C. [tex]\frac{3}{2}[/tex]
D. [tex]\frac{7}{4}[/tex]
Answer (1)
The exponential function f ( x ) = a ( 2 3 ) x will shrink if 0 < a < 1 .
Check which of the given values of a satisfy the condition 0 < a < 1 .
The given values are 5 4 , 4 5 , 2 3 , 4 7 .
Therefore, the value of a that would cause the function to shrink is 5 4 .
Explanation
Understanding the Problem We are given an exponential function f ( x ) = a ( 2 3 ) x and asked to find which value of a would cause the function to 'shrink'. In this context, 'shrink' means that the function values are smaller than they would be if a = 1 . Since the base 2 3 is greater than 1, the exponential part ( 2 3 ) x will always increase as x increases. Therefore, for the function to 'shrink', the value of a must be between 0 and 1, i.e., 0 < a < 1 .
Analyzing the Options We are given the following options for a : 5 4 , 4 5 , 2 3 , 4 7 . We need to determine which of these values falls between 0 and 1.
Checking Each Value Let's analyze each option:
5 4 = 0.8 . This value is between 0 and 1, so 0 < 5 4 < 1 .
4 5 = 1.25 . This value is greater than 1.
2 3 = 1.5 . This value is greater than 1.
4 7 = 1.75 . This value is greater than 1.
Final Answer Only 5 4 is between 0 and 1. Therefore, the value of a that would cause the function to shrink is 5 4 .
Examples
Exponential functions are used to model various real-world phenomena, such as population growth, radioactive decay, and compound interest. In the context of population growth, the factor 'a' can represent the initial population size. If 'a' is less than 1, it indicates that the initial population is smaller than a reference value, causing the overall population size to be smaller compared to a scenario where 'a' is equal to 1. Similarly, in compound interest, 'a' can represent the initial investment. If 'a' is less than 1, it means the initial investment is a fraction of a standard investment amount, leading to a smaller accumulated value over time.
