Which set of ordered pairs could be generated by an exponential function?
(1,1),$\left(2, \frac{1}{2}\right),\left(3, \frac{1}{3}\right),\left(4, \frac{1}{4}\right)$
(1,1),$\left(2, \frac{1}{4}\right),\left(3, \frac{1}{9}\right),\left(4, \frac{1}{16}\right)$
$\left(1, \frac{1}{2}\right),\left(2, \frac{1}{4}\right),\left(3, \frac{1}{8}\right),\left(4, \frac{1}{16}\right)$
$\left(1, \frac{1}{2}\right),\left(2, \frac{1}{4}\right),\left(3, \frac{1}{6}\right),\left(4, \frac{1}{8}\right)$
Answer (2)
Set 1 does not represent an exponential function because f ( 3 ) = 4 1 = 3 1 .
Set 2 does not represent an exponential function because f ( 3 ) = 16 1 = 9 1 .
Set 3 represents an exponential function f ( x ) = ( 2 1 ) x .
Set 4 does not represent an exponential function because f ( 3 ) = 8 1 = 6 1 .
Therefore, the set of ordered pairs that could be generated by an exponential function is ( 1 , 2 1 ) , ( 2 , 4 1 ) , ( 3 , 8 1 ) , ( 4 , 16 1 ) .
Explanation
Understanding the Problem We are given four sets of ordered pairs and asked to identify which set could be generated by an exponential function. An exponential function has the form f ( x ) = a b x where a is the initial value and b is the base. We need to check if there exist constants a and b such that the given points satisfy the exponential function.
Analyzing Each Set Let's analyze each set of ordered pairs:
Set 1: ( 1 , 1 ) , ( 2 , 2 1 ) , ( 3 , 3 1 ) , ( 4 , 4 1 ) If ( 1 , 1 ) and ( 2 , 2 1 ) are on the curve, then a b 1 = 1 and a b 2 = 2 1 . Dividing the second equation by the first gives b = 2 1 . Then a ( 2 1 ) = 1 , so a = 2 . Thus f ( x ) = 2 ( 2 1 ) x . Check if the other points satisfy this equation. f ( 3 ) = 2 ( 2 1 ) 3 = 4 1 = 3 1 , so this is not an exponential function.
Set 2: ( 1 , 1 ) , ( 2 , 4 1 ) , ( 3 , 9 1 ) , ( 4 , 16 1 ) If ( 1 , 1 ) and ( 2 , 4 1 ) are on the curve, then a b 1 = 1 and a b 2 = 4 1 . Dividing the second equation by the first gives b = 4 1 . Then a ( 4 1 ) = 1 , so a = 4 . Thus f ( x ) = 4 ( 4 1 ) x . Check if the other points satisfy this equation. f ( 3 ) = 4 ( 4 1 ) 3 = 16 1 = 9 1 , so this is not an exponential function.
Set 3: ( 1 , 2 1 ) , ( 2 , 4 1 ) , ( 3 , 8 1 ) , ( 4 , 16 1 ) If ( 1 , 2 1 ) and ( 2 , 4 1 ) are on the curve, then a b 1 = 2 1 and a b 2 = 4 1 . Dividing the second equation by the first gives b = 2 1 . Then a ( 2 1 ) = 2 1 , so a = 1 . Thus f ( x ) = ( 2 1 ) x . Check if the other points satisfy this equation. f ( 3 ) = ( 2 1 ) 3 = 8 1 and f ( 4 ) = ( 2 1 ) 4 = 16 1 . This set of points could be generated by an exponential function.
Set 4: ( 1 , 2 1 ) , ( 2 , 4 1 ) , ( 3 , 6 1 ) , ( 4 , 8 1 ) If ( 1 , 2 1 ) and ( 2 , 4 1 ) are on the curve, then a b 1 = 2 1 and a b 2 = 4 1 . Dividing the second equation by the first gives b = 2 1 . Then a ( 2 1 ) = 2 1 , so a = 1 . Thus f ( x ) = ( 2 1 ) x . Check if the other points satisfy this equation. f ( 3 ) = ( 2 1 ) 3 = 8 1 = 6 1 , so this is not an exponential function.
Examples
Exponential functions are incredibly useful for modeling various real-world phenomena, such as population growth, radioactive decay, and compound interest. For instance, if you invest money in a bank account that offers compound interest, the amount of money you have will grow exponentially over time. Similarly, the decay of a radioactive substance can be modeled using an exponential function, allowing scientists to predict how much of the substance will remain after a certain period.
The set of ordered pairs that can be generated by an exponential function is ( 1 , 2 1 ) , ( 2 , 4 1 ) , ( 3 , 8 1 ) , ( 4 , 16 1 ) . This set satisfies the exponential function condition f ( x ) = ( 2 1 ) x . Therefore, the answer is Set 3.
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