Which set of ordered pairs could be generated by an exponential function?
(1,1), (2, 1/2), (3, 1/3), (4, 1/4)
(1,1), (2, 1/4), (3, 1/9), (4, 1/16)
(1, 1/2), (2, 1/4), (3, 1/8), (4, 1/16)
(1, 1/2), (2, 1/4), (3, 1/6), (4, 1/8)
Answer (1)
Test each set of ordered pairs to see if they fit the form f ( x ) = a b x .
For Set 1, derive f ( x ) = 2 ( 2 1 ) x and find that it does not fit all points.
For Set 2, derive f ( x ) = 4 ( 4 1 ) x and find that it does not fit all points.
For Set 3, derive f ( x ) = ( 2 1 ) x and confirm that it fits all points.
For Set 4, derive f ( x ) = ( 2 1 ) x and find that it does not fit all points.
Conclude that only Set 3 can be generated by an exponential function: ( 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 determine which set could be generated by an exponential function. An exponential function has the form f ( x ) = a b x where a and b are constants. We need to test each set of ordered pairs to see if there exist constants a and b that satisfy the equation for all pairs in the set.
Plan of Action Let's analyze each set of ordered pairs to determine if they can be generated by an exponential function. We'll use the form f ( x ) = a b x and check if consistent values of a and b can be found for each set.
Analyzing Set 1 Set 1: ( 1 , 1 ) , ( 2 , f r a c 1 2 ) , ( 3 , f r a c 1 3 ) , ( 4 , f r a c 1 4 ) .
If this set follows an exponential function f ( x ) = a b x , then: f ( 1 ) = a b 1 = 1 f ( 2 ) = a b 2 = f r a c 1 2 Dividing the second equation by the first, we get b = f r a c 1 2 . Substituting b into the first equation, we get a ( f r a c 1 2 ) = 1 , so a = 2 . Thus, f ( x ) = 2 ( f r a c 1 2 ) x .
Now, let's check if this function holds for the other points: f ( 3 ) = 2 ( f r a c 1 2 ) 3 = 2 ( f r a c 1 8 ) = f r a c 1 4 n e f r a c 1 3 .
Therefore, this set cannot be generated by an exponential function.
Analyzing Set 2 Set 2: ( 1 , 1 ) , ( 2 , f r a c 1 4 ) , ( 3 , f r a c 1 9 ) , ( 4 , f r a c 1 16 ) .
If this set follows an exponential function f ( x ) = a b x , then: f ( 1 ) = a b 1 = 1 f ( 2 ) = a b 2 = f r a c 1 4 Dividing the second equation by the first, we get b = f r a c 1 4 . Substituting b into the first equation, we get a ( f r a c 1 4 ) = 1 , so a = 4 . Thus, f ( x ) = 4 ( f r a c 1 4 ) x .
Now, let's check if this function holds for the other points: f ( 3 ) = 4 ( f r a c 1 4 ) 3 = 4 ( f r a c 1 64 ) = f r a c 1 16 n e f r a c 1 9 .
Therefore, this set cannot be generated by an exponential function.
Analyzing Set 3 Set 3: ( 1 , f r a c 1 2 ) , ( 2 , f r a c 1 4 ) , ( 3 , f r a c 1 8 ) , ( 4 , f r a c 1 16 ) .
If this set follows an exponential function f ( x ) = a b x , then: f ( 1 ) = a b 1 = f r a c 1 2 f ( 2 ) = a b 2 = f r a c 1 4 Dividing the second equation by the first, we get b = f r a c 1 2 . Substituting b into the first equation, we get a ( f r a c 1 2 ) = f r a c 1 2 , so a = 1 . Thus, f ( x ) = ( 1 ) ( f r a c 1 2 ) x = ( f r a c 1 2 ) x .
Now, let's check if this function holds for the other points: f ( 3 ) = ( f r a c 1 2 ) 3 = f r a c 1 8 f ( 4 ) = ( f r a c 1 2 ) 4 = f r a c 1 16 This set can be generated by an exponential function.
Analyzing Set 4 Set 4: ( 1 , f r a c 1 2 ) , ( 2 , f r a c 1 4 ) , ( 3 , f r a c 1 6 ) , ( 4 , f r a c 1 8 ) .
If this set follows an exponential function f ( x ) = a b x , then: f ( 1 ) = a b 1 = f r a c 1 2 f ( 2 ) = a b 2 = f r a c 1 4 Dividing the second equation by the first, we get b = f r a c 1 2 . Substituting b into the first equation, we get a ( f r a c 1 2 ) = f r a c 1 2 , so a = 1 . Thus, f ( x ) = ( 1 ) ( f r a c 1 2 ) x = ( f r a c 1 2 ) x .
Now, let's check if this function holds for the other points: f ( 3 ) = ( f r a c 1 2 ) 3 = f r a c 1 8 n e f r a c 1 6 .
Therefore, this set cannot be generated by an exponential function.
Conclusion After analyzing all four sets, we found that only the third set, ( 1 , f r a c 1 2 ) , ( 2 , f r a c 1 4 ) , ( 3 , f r a c 1 8 ) , ( 4 , f r a c 1 16 ) , can be generated by an exponential function, specifically f ( x ) = ( f r a c 1 2 ) x .
Examples
Exponential functions are incredibly useful for modeling various real-world phenomena, such as population growth, radioactive decay, and compound interest. For example, imagine you invest $1000 in a savings account that offers an annual interest rate of 5%, compounded annually. The amount of money you have in the account each year can be modeled by an exponential function. Understanding exponential functions helps you predict how your investment will grow over time, making it a powerful tool for financial planning.
