Which is an exponential decay function?
[tex]f(x)=\frac{3}{4}\left(\frac{7}{4}\right)^x[/tex]
[tex]f(x)=\frac{2}{3}\left(\frac{4}{5}\right)^{-x}[/tex]
[tex]f(x)=\frac{3}{2}\left(\frac{8}{7}\right)^{-x}[/tex]
[tex]f(x)=\frac{1}{3}\left(-\frac{9}{2}\right)^x[/tex]
Answer (1)
Exponential decay functions have the form f ( x ) = a x where 0 < b < 1 .
Rewrite functions with negative exponents as f ( x ) = a ( b 1 ) x and check if b 1 < 1 .
Eliminate functions with negative bases, as they are not exponential functions.
The exponential decay function is f ( x ) = 2 3 ( 7 8 ) − x .
Explanation
Understanding Exponential Decay We are given four functions and asked to identify which one represents exponential decay. An exponential decay function has the form f ( x ) = a x where 0"> a > 0 and 0 < b < 1 . If 1"> b > 1 , then f ( x ) is an exponential growth function. If b < 0 , then f ( x ) is not an exponential function. If the exponent is − x , i.e. f ( x ) = a − x , then we can rewrite it as f ( x ) = a ( b − 1 ) x = a ( b 1 ) x , so we need to check if b 1 < 1 for decay.
Analyzing Each Function Let's analyze each function:
f ( x ) = 4 3 ( 4 7 ) x : Here, a = 4 3 and b = 4 7 . Since 1"> 4 7 = 1.75 > 1 , this is an exponential growth function, not decay.
f ( x ) = 3 2 ( 5 4 ) − x : We can rewrite this as f ( x ) = 3 2 ( 4 5 ) x . Here, a = 3 2 and b = 4 5 . Since 1"> 4 5 = 1.25 > 1 , this is an exponential growth function, not decay.
f ( x ) = 2 3 ( 7 8 ) − x : We can rewrite this as f ( x ) = 2 3 ( 8 7 ) x . Here, a = 2 3 and b = 8 7 . Since 0 < 8 7 < 1 , this is an exponential decay function.
f ( x ) = 3 1 ( − 2 9 ) x : Here, a = 3 1 and b = − 2 9 . Since b = − 2 9 < 0 , this is not an exponential function.
Conclusion Therefore, the exponential decay function is f ( x ) = 2 3 ( 7 8 ) − x .
Examples
Exponential decay is a mathematical concept that describes the decrease of a quantity over time. A practical example is the decay of radioactive isotopes, which are used in carbon dating to determine the age of ancient artifacts. The half-life of a radioactive substance is the time it takes for half of the substance to decay. Understanding exponential decay helps scientists accurately estimate the age of materials by measuring the remaining amount of the isotope. This principle is also applicable in pharmacology, where it helps determine how quickly a drug's concentration decreases in the body over time.
