Which function represents exponential decay?
$f(x)=0.5 e^x$
$f(x)=0.5\left(\frac{5}{4}\right)^x$
$f(x)=\frac{5}{4}(0.5)^x$
$f(x)=\frac{4}{5}(2)^x$
Answer (1)
Exponential decay occurs when the base of the exponential function is between 0 and 1.
Examine each function to determine if it represents exponential decay.
f ( x ) = 0.5 e x represents exponential growth.
f ( x ) = 4 5 ( 0.5 ) x represents exponential decay, since 0 < 0.5 < 1 .
The function that represents exponential decay is f ( x ) = 4 5 ( 0.5 ) x .
Explanation
Understanding Exponential Decay We are given four functions and asked to identify which one represents exponential decay. Exponential decay occurs when the base of the exponential function is between 0 and 1, or when the coefficient of x in the exponent is negative. The general form of an exponential function is f ( x ) = a b x or f ( x ) = a e k x , where a is the initial value. For exponential decay, we need 0 < b < 1 or k < 0 .
Analyzing Each Function Let's examine each function to determine if it represents exponential decay.
f ( x ) = 0.5 e x : Here, the base is e a pp ro x 2.718 , which is greater than 1. Thus, this function represents exponential growth, not decay.
f ( x ) = 0.5 ( 4 5 ) x : The base is 4 5 = 1.25 , which is also greater than 1. This function also represents exponential growth.
f ( x ) = 4 5 ( 0.5 ) x : The base is 0.5 = 2 1 , which is between 0 and 1. Therefore, this function represents exponential decay.
f ( x ) = 5 4 ( 2 ) x : The base is 2 , which is greater than 1. This function represents exponential growth.
Identifying the Correct Function Therefore, the function that represents exponential decay is f ( x ) = 4 5 ( 0.5 ) x .
Examples
Exponential decay is a mathematical concept that describes the decrease in a quantity over time. A common real-world example is the decay of radioactive isotopes. For instance, if you have a sample of a radioactive material, the amount of the material decreases exponentially over time. This is used in carbon dating to determine the age of ancient artifacts. Another example is the depreciation of a car's value over time, where the car loses a certain percentage of its value each year.
