Which set of ordered pairs could be generated by an exponential function?
(0,0),(1,1),(2,8),(3,27)
(0,1),(1,2),(2,5),(3,10)
(0,0),(1,3),(2,6),(3,9)
(0,1),(1,3),(2,9),(3,27)
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: ( 0 , 0 ) , ( 1 , 1 ) , ( 2 , 8 ) , ( 3 , 27 ) does not fit because f ( 0 ) = 0 implies f ( x ) = 0 for all x , contradicting other points.
Set 2: ( 0 , 1 ) , ( 1 , 2 ) , ( 2 , 5 ) , ( 3 , 10 ) does not fit because f ( 0 ) = 1 and f ( 1 ) = 2 imply f ( x ) = 2 x , but f ( 2 ) = 4 e q 5 .
Set 3: ( 0 , 0 ) , ( 1 , 3 ) , ( 2 , 6 ) , ( 3 , 9 ) does not fit because f ( 0 ) = 0 implies f ( x ) = 0 for all x , contradicting other points.
Set 4: ( 0 , 1 ) , ( 1 , 3 ) , ( 2 , 9 ) , ( 3 , 27 ) fits the exponential function f ( x ) = 3 x . Thus, the answer is ( 0 , 1 ) , ( 1 , 3 ) , ( 2 , 9 ) , ( 3 , 27 ) .
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 the initial value and b is the base. We need to check each set of ordered pairs to see if they fit this form.
Analyzing Set 1 Let's analyze the first set of ordered pairs: ( 0 , 0 ) , ( 1 , 1 ) , ( 2 , 8 ) , ( 3 , 27 ) . If x = 0 , f ( 0 ) = a 0 = a = 0 . This implies f ( x ) = 0 for all x , but this contradicts the other points (e.g., ( 1 , 1 ) ). So, this set cannot be generated by an exponential function.
Analyzing Set 2 Now, let's analyze the second set of ordered pairs: ( 0 , 1 ) , ( 1 , 2 ) , ( 2 , 5 ) , ( 3 , 10 ) . If x = 0 , f ( 0 ) = a 0 = a = 1 . So f ( x ) = x . Then f ( 1 ) = 1 = 2 , so = 2 . Thus f ( x ) = 2 x . Check f ( 2 ) = 2 2 = 4 , but the set has the point ( 2 , 5 ) . So, this set cannot be generated by an exponential function.
Analyzing Set 3 Let's analyze the third set of ordered pairs: ( 0 , 0 ) , ( 1 , 3 ) , ( 2 , 6 ) , ( 3 , 9 ) . If x = 0 , f ( 0 ) = a 0 = a = 0 . This implies f ( x ) = 0 for all x , but this contradicts the other points (e.g., ( 1 , 3 ) ). So, this set cannot be generated by an exponential function.
Analyzing Set 4 Finally, let's analyze the fourth set of ordered pairs: ( 0 , 1 ) , ( 1 , 3 ) , ( 2 , 9 ) , ( 3 , 27 ) . If x = 0 , f ( 0 ) = a 0 = a = 1 . So f ( x ) = x . Then f ( 1 ) = 1 = 3 , so = 3 . Thus f ( x ) = 3 x . Check f ( 2 ) = 3 2 = 9 and f ( 3 ) = 3 3 = 27 . This set of ordered pairs can be generated by the exponential function f ( x ) = 3 x .
Conclusion Therefore, the set of ordered pairs that could be generated by an exponential function is ( 0 , 1 ) , ( 1 , 3 ) , ( 2 , 9 ) , ( 3 , 27 ) .
Examples
Exponential functions are incredibly useful for modeling various real-world phenomena, such as population growth, radioactive decay, and compound interest. For example, if a bacteria population doubles every hour, we can model its growth using an exponential function. Similarly, the decay of a radioactive substance can be modeled using a decreasing exponential function. Understanding exponential functions helps us make predictions and analyze these phenomena effectively.
The set of ordered pairs that can be generated by an exponential function is ( 0 , 1 ) , ( 1 , 3 ) , ( 2 , 9 ) , ( 3 , 27 ) because it follows the form of an exponential function with base 3. Other sets do not maintain consistent multiplicative growth. Thus, the correct option is the fourth set.
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