Which is a stretch of an exponential growth function?
[tex]f(x)=\frac{2}{3}\left(\frac{2}{3}\right)^x[/tex]
[tex]f(x)=\frac{3}{2}\left(\frac{2}{3}\right)[/tex]
[tex]f(x)=\frac{3}{2}\left(\frac{3}{2}\right) x[/tex]
[tex]f(x)=\frac{3}{3}\left(\frac{3}{2}\right)[/tex]
Answer (2)
Exponential growth functions have the form f ( x ) = a ⋅ b x , where 0"> a > 0 and 1"> b > 1 .
The first function is an exponential decay function because b = 3 2 < 1 .
The second and fourth functions are constant functions.
Assuming a typo, the third function f ( x ) = 2 3 ( 2 3 ) x is an exponential growth function because 0"> a = 2 3 > 0 and 1"> b = 2 3 > 1 .
Therefore, the answer is: f ( x ) = 2 3 ( 2 3 ) x (assuming a typo in the original question).
Explanation
Understanding Exponential Growth We are given four functions and asked to identify which one is a stretch of an exponential growth function. An exponential growth function has the form f ( x ) = a ⋅ b x , where 0"> a > 0 and 1"> b > 1 . A stretch of an exponential growth function will also have this form. We need to examine each function to see if it fits this form.
Analyzing Function 1 Let's analyze the first function: f ( x ) = 3 2 ( 3 2 ) x . Here, a = 3 2 and b = 3 2 . Since 0 < b < 1 , this is an exponential decay function, not a growth function.
Analyzing Function 2 Now, let's examine the second function: f ( x ) = 2 3 ( 3 2 ) . This simplifies to f ( x ) = 1 . This is a constant function, not an exponential function.
Analyzing Function 3 Next, let's analyze the third function: f ( x ) = 2 3 ( 2 3 ) x . This is a linear function, not an exponential function. However, if there's a typo and the function is meant to be f ( x ) = 2 3 ( 2 3 ) x , then a = 2 3 and b = 2 3 . Since 0"> a > 0 and 1"> b > 1 , this would be an exponential growth function.
Analyzing Function 4 Finally, let's examine the fourth function: f ( x ) = 3 3 ( 2 3 ) . This simplifies to f ( x ) = 1 ( 2 3 ) = 2 3 . This is a constant function, not an exponential function.
Conclusion Based on the analysis, if we assume that the third function has a typo and is actually f ( x ) = 2 3 ( 2 3 ) x , then it is a stretch of an exponential growth function.
Examples
Exponential growth functions are used to model various real-world phenomena, such as population growth, compound interest, and the spread of diseases. For example, if a population of bacteria doubles every hour, the population can be modeled by an exponential growth function. Similarly, if you invest money in an account that earns compound interest, the amount of money in the account will grow exponentially over time. Understanding exponential growth can help you make informed decisions about investments, healthcare, and other important aspects of life.
The function that represents a stretch of an exponential growth function is f ( x ) = 2 3 ( 2 3 ) x , assuming a typo in the third function. This is because it satisfies the criteria of having both 0"> a > 0 and 1"> b > 1 .
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