Which set of ordered pairs could be generated by an exponential function?
$\left(-1,-\frac{1}{2}\right),(0,0),\left(1, \frac{1}{2}\right),(2,1)$
$(-1,-1),(0,0),(1,1),(2,8)$
$\left(-1, \frac{1}{2}\right),(0,1),(1,2),(2,4)$
$(-1,1),(0,0),(1,1),(2,4)$
Answer (1)
Exponential functions have the form f ( x ) = a ⋅ b x , where a is the initial value and b is the base.
Sets containing the point (0,0) cannot be exponential functions.
Test the remaining sets to see if they fit the form f ( x ) = a ⋅ b x .
The set ( − 1 , 2 1 ) , ( 0 , 1 ) , ( 1 , 2 ) , ( 2 , 4 ) can be generated by the exponential function f ( x ) = 2 x .
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. The function must pass the vertical line test, meaning that each x-value has only one y-value. We can test each set of ordered pairs to see if there is an exponential function that passes through all the points.
Solution plan Check each set of ordered pairs to see if they could be generated by an exponential function of the form f ( x ) = a ⋅ b x .
Analyzing Set 1 Set 1: ( − 1 , − 2 1 ) , ( 0 , 0 ) , ( 1 , 2 1 ) , ( 2 , 1 ) . Since the point (0,0) is in the set, this cannot be an exponential function because exponential functions are never 0.
Analyzing Set 2 Set 2: ( − 1 , − 1 ) , ( 0 , 0 ) , ( 1 , 1 ) , ( 2 , 8 ) . Since the point (0,0) is in the set, this cannot be an exponential function because exponential functions are never 0.
Analyzing Set 3 Set 3: ( − 1 , 2 1 ) , ( 0 , 1 ) , ( 1 , 2 ) , ( 2 , 4 ) . If this is an exponential function, then f ( x ) = a ⋅ b x . From the point (0,1), we have f ( 0 ) = a ⋅ b 0 = a = 1 . So f ( x ) = b x . From the point (1,2), we have f ( 1 ) = b 1 = 2 , so b = 2 . Thus, f ( x ) = 2 x . Check the other points: f ( − 1 ) = 2 − 1 = 2 1 and f ( 2 ) = 2 2 = 4 . This set of ordered pairs could be generated by the exponential function f ( x ) = 2 x .
Analyzing Set 4 Set 4: ( − 1 , 1 ) , ( 0 , 0 ) , ( 1 , 1 ) , ( 2 , 4 ) . Since the point (0,0) is in the set, this cannot be an exponential function because exponential functions are never 0.
Conclusion Therefore, the set of ordered pairs that could be generated by an exponential function is ( − 1 , 2 1 ) , ( 0 , 1 ) , ( 1 , 2 ) , ( 2 , 4 ) .
Examples
Exponential functions are used to model population growth, radioactive decay, and compound interest. For example, if a population doubles every year, the population can be modeled by the exponential function P ( t ) = P 0 ⋅ 2 t , where P 0 is the initial population and t is the number of years. Understanding exponential functions helps us predict future values and make informed decisions.
