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 (2)
Check each set of ordered pairs to see if they fit the form of an exponential function f ( x ) = a x .
Set 1, 2, and 4 have (0,0), which means a = 0 , making f ( x ) = 0 for all x , contradicting other points.
Set 3 has ( − 1 , 2 1 ) , ( 0 , 1 ) , ( 1 , 2 ) , ( 2 , 4 ) . If x = 0 , a = 1 . If x = 1 , b = 2 . Thus f ( x ) = 2 x , which satisfies all points.
Therefore, the set that could be generated by an exponential function 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. An exponential function has the form f ( x ) = a x where a is a constant and b is a positive constant not equal to 1. We need to check each set of ordered pairs to see if there exist constants a and b that satisfy the equation for all pairs in the set.
Analyzing the Sets Let's analyze each set of ordered pairs to determine if they can be generated by an exponential function.
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. Thus, this set cannot be generated by an exponential function.
Analyzing Set 2 Set 2: ( − 1 , − 1 ) , ( 0 , 0 ) , ( 1 , 1 ) , ( 2 , 8 ) . If x = 0 , then f ( 0 ) = a b 0 = a = 0 . This implies f ( x ) = 0 for all x , which contradicts the other points. Thus, 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 = b = 2 . So f ( x ) = 2 x . Check if the other points satisfy this: f ( − 1 ) = 2 − 1 = 2 1 , f ( 2 ) = 2 2 = 4 . All points satisfy f ( x ) = 2 x . Thus, this set could be generated by an exponential function.
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. Thus, 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 various real-world phenomena, such as population growth, radioactive decay, and compound interest. For example, if you invest money in a bank account that offers compound interest, the amount of money you have will grow exponentially over time. Understanding exponential functions helps you predict how your investment will grow, plan for the future, and make informed financial decisions.
The set of ordered pairs that could be generated by an exponential function is Set 3: ( − 1 , 2 1 ) , ( 0 , 1 ) , ( 1 , 2 ) , ( 2 , 4 ) . This set satisfies the conditions of an exponential function. Other sets either contradict the form of an exponential function or equate to zero for all x.
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