Which is a shrink of an exponential growth function?
[tex]f(x)=\frac{1}{3}(3)^x[/tex]
[tex]f(x)=3(3)^x[/tex]
[tex]f(x)=\frac{1}{3}\left(\frac{1}{3}\right)^x[/tex]
[tex]f(x)=3\left(\frac{1}{3}\right)^x[/tex]
Answer (1)
An exponential growth function has the form f ( x ) = a b x where 0"> a > 0 and 1"> b > 1 .
A shrink of a function f ( x ) is given by c f ( x ) where 0 < c < 1 .
Examine each function to determine if it is a shrink of an exponential growth function.
The function that represents a shrink of an exponential growth function is f ( x ) = 3 1 ( 3 ) x .
Explanation
Understanding the Problem We are given four exponential functions and asked to identify which one represents a shrink of an exponential growth function.
Exponential Growth Function An exponential growth function has the form f ( x ) = a b x where 0"> a > 0 and 1"> b > 1 .
Shrink of a Function A shrink (or vertical compression) of a function f ( x ) is given by c f ( x ) where 0 < c < 1 .
Identifying the Correct Function We need to identify which of the given functions can be written in the form c b x where 0 < c < 1 and 1"> b > 1 or can be obtained from such a function by a horizontal or vertical scaling.
Analyzing Each Function
f ( x ) = 3 1 ( 3 ) x : This is of the form c b x with c = 3 1 and b = 3 . Since 0 < c < 1 and 1"> b > 1 , this is a shrink of an exponential growth function.
f ( x ) = 3 ( 3 ) x : This is of the form c b x with c = 3 and b = 3 . Since 1"> c > 1 and 1"> b > 1 , this is a stretch of an exponential growth function.
f ( x ) = 3 1 ( 3 1 ) x : This is of the form c b x with c = 3 1 and b = 3 1 . Since 0 < c < 1 and 0 < b < 1 , this is a shrink of an exponential decay function.
f ( x ) = 3 ( 3 1 ) x : This is of the form c b x with c = 3 and b = 3 1 . Since 1"> c > 1 and 0 < b < 1 , this is a stretch of an exponential decay function.
Conclusion Therefore, the function that represents a shrink of an exponential growth function is f ( x ) = 3 1 ( 3 ) x .
Examples
Imagine you're observing the population of a certain bacteria in a petri dish. If the bacteria's growth is naturally exponential, but you introduce a factor that slows down the growth (like a mild antibiotic), the population increase might be a 'shrink' of its usual exponential growth. This means the population still grows exponentially, but at a rate that's a fraction of what it would have been without the inhibiting factor. Understanding exponential growth and its shrinks can help predict and control various real-world phenomena, from population dynamics to financial investments.
