Which equation represents an exponential function that passes through the point $(2,80)$?
$f(x)=4(x)^5$
$f(x)=5(x)^4$
$f(x)=4(5)^x$
$f(x)=5(4)^x$
Answer (2)
The exponential function that passes through the point ( 2 , 80 ) is f ( x ) = 5 ( 4 ) x . This conclusion is reached after testing each equation to see if they meet the criteria of both being an exponential function and passing through the specified point. The other equations either do not fit the exponential form or do not pass through the point successfully.
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Identify the general form of an exponential function: f ( x ) = a x .
Substitute x = 2 into each equation and check if f ( 2 ) = 80 .
Eliminate polynomial functions and identify exponential functions.
The exponential function that passes through ( 2 , 80 ) is f ( x ) = 5 ( 4 ) x .
Explanation
Understanding the Problem We are given four equations and a point ( 2 , 80 ) . Our goal is to identify which equation represents an exponential function that passes through this point. An exponential function has the general form f ( x ) = a x , where a is the initial value and b is the base.
Checking Each Equation Let's examine each equation to determine if it's an exponential function and if it passes through the point ( 2 , 80 ) .
f ( x ) = 4 ( x ) 5 : This is a polynomial function, not an exponential function, because the variable x is raised to a constant power. Substituting x = 2 , we get f ( 2 ) = 4 ( 2 ) 5 = 4 ( 32 ) = 128 . This does not pass through ( 2 , 80 ) .
f ( x ) = 5 ( x ) 4 : This is also a polynomial function, not an exponential function. Substituting x = 2 , we get f ( 2 ) = 5 ( 2 ) 4 = 5 ( 16 ) = 80 . This passes through ( 2 , 80 ) , but it is not an exponential function.
f ( x ) = 4 ( 5 ) x : This is an exponential function because a constant (5) is raised to a variable power x . Substituting x = 2 , we get f ( 2 ) = 4 ( 5 ) 2 = 4 ( 25 ) = 100 . This does not pass through ( 2 , 80 ) .
f ( x ) = 5 ( 4 ) x : This is an exponential function. Substituting x = 2 , we get f ( 2 ) = 5 ( 4 ) 2 = 5 ( 16 ) = 80 . This passes through ( 2 , 80 ) .
Conclusion From the above analysis, we see that f ( x ) = 5 ( 4 ) x is the only exponential function that passes through the point ( 2 , 80 ) .
Examples
Exponential functions are incredibly useful in modeling real-world phenomena, such as population growth, radioactive decay, and compound interest. For instance, if you invest money in a bank account with compound interest, the amount of money you have over time can be modeled by an exponential function. Understanding exponential functions helps you predict how your investment will grow or how quickly a population will increase.
