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)
Check each set of ordered pairs to see if they fit the form of an exponential function f ( x ) = a × b x .
For the first set, the point (0,0) implies a = 0 , which means the function would be f ( x ) = 0 for all x , contradicting the other points.
Similarly, the second and fourth sets also have (0,0), so they cannot be exponential functions.
The third set ( − 1 , 2 1 ) , ( 0 , 1 ) , ( 1 , 2 ) , ( 2 , 4 ) fits the exponential function f ( x ) = 2 x .
Therefore, the correct set is ( − 1 , 2 1 ) , ( 0 , 1 ) , ( 1 , 2 ) , ( 2 , 4 ) .
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.
Exponential Function Form 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 values for a and b that satisfy the given ordered pairs for each set.
Analyzing Set 1 Set 1: ( − 1 , − 2 1 ) , ( 0 , 0 ) , ( 1 , 2 1 ) , ( 2 , 1 ) . If x = 0 , then f ( 0 ) = a × b 0 = a = 0 . This implies f ( x ) = 0 for all x , which contradicts the other points. So, this set cannot be generated by an exponential function.
Analyzing Set 2 Set 2: ( − 1 , − 1 ) , ( 0 , 0 ) , ( 1 , 1 ) , ( 2 , 8 ) . Similar to set 1, if x = 0 , then f ( 0 ) = a × b 0 = a = 0 . This implies f ( x ) = 0 for all x , which contradicts the other points. So, this set cannot be generated by an exponential function.
Analyzing Set 3 Set 3: ( − 1 , 2 1 ) , ( 0 , 1 ) , ( 1 , 2 ) , ( 2 , 4 ) . If x = 0 , then f ( 0 ) = a × b 0 = a = 1 . So f ( x ) = b x . If x = 1 , then 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 . All points satisfy the function f ( x ) = 2 x .
Analyzing Set 4 Set 4: ( − 1 , 1 ) , ( 0 , 0 ) , ( 1 , 1 ) , ( 2 , 4 ) . If x = 0 , then f ( 0 ) = a × b 0 = a = 0 . This implies f ( x ) = 0 for all x , which contradicts the other points. So, this set cannot be generated by an exponential function.
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 incredibly useful for modeling phenomena that grow or decay at a rate proportional to their current value. For instance, consider the growth of a bacteria colony. If the colony doubles every hour, we can model its size using an exponential function. Starting with an initial population, the function helps predict the colony's size at any given time, which is crucial in medical and environmental studies.
