Which set of ordered pairs could be generated by an exponential function?
A. (1,1), (2, 1/2), (3, 1/3), (4, 1/4)
B. (1,1), (2, 1/4), (3, 1/9), (4, 1/16)
C. (1, 1/2), (2, 1/4), (3, 1/8), (4, 1/16)
D. (1, 1/2), (2, 1/4), (3, 1/6), (4, 1/8)
Answer (2)
Test each set of ordered pairs to see if they fit the form of an exponential function f ( x ) = a b x .
For the set ( 1 , 2 1 ) , ( 2 , 4 1 ) , ( 3 , 8 1 ) , ( 4 , 16 1 ) , find a and b using the first two points: a = 1 and b = 2 1 , so f ( x ) = ( 2 1 ) x .
Verify that the remaining points satisfy the function: f ( 3 ) = ( 2 1 ) 3 = 8 1 and f ( 4 ) = ( 2 1 ) 4 = 16 1 .
Conclude that the set ( 1 , 2 1 ) , ( 2 , 4 1 ) , ( 3 , 8 1 ) , ( 4 , 16 1 ) can be generated by an exponential function: ( 1 , 2 1 ) , ( 2 , 4 1 ) , ( 3 , 8 1 ) , ( 4 , 16 1 ) .
Explanation
Understanding Exponential Functions 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 general form f ( x ) = a b x , where a and b are constants. We need to test each set to see if we can find values for a and b that satisfy all the points in the set.
Analyzing Set 1 Let's analyze the first set of ordered pairs: ( 1 , 1 ) , ( 2 , 2 1 ) , ( 3 , 3 1 ) , ( 4 , 4 1 ) . If these points come from an exponential function f ( x ) = a b x , then we must have f ( 1 ) = a b 1 = 1 and f ( 2 ) = a b 2 = 2 1 . Dividing the second equation by the first, we get ab a b 2 = 1 1/2 , which simplifies to b = 2 1 . Since ab = 1 , we have a ( 2 1 ) = 1 , so a = 2 . Thus, f ( x ) = 2 ( 2 1 ) x . Now let's check if the remaining points satisfy this function. f ( 3 ) = 2 ( 2 1 ) 3 = 2 ( 8 1 ) = 4 1 . But the third point is ( 3 , 3 1 ) , and 4 1 = 3 1 . Therefore, this set of ordered pairs cannot be generated by an exponential function.
Analyzing Set 2 Now let's analyze the second set of ordered pairs: ( 1 , 1 ) , ( 2 , 4 1 ) , ( 3 , 9 1 ) , ( 4 , 16 1 ) . If these points come from an exponential function f ( x ) = a b x , then we must have f ( 1 ) = a b 1 = 1 and f ( 2 ) = a b 2 = 4 1 . Dividing the second equation by the first, we get ab a b 2 = 1 1/4 , which simplifies to b = 4 1 . Since ab = 1 , we have a ( 4 1 ) = 1 , so a = 4 . Thus, f ( x ) = 4 ( 4 1 ) x . Now let's check if the remaining points satisfy this function. f ( 3 ) = 4 ( 4 1 ) 3 = 4 ( 64 1 ) = 16 1 . But the third point is ( 3 , 9 1 ) , and 16 1 = 9 1 . Therefore, this set of ordered pairs cannot be generated by an exponential function.
Analyzing Set 3 Now let's analyze the third set of ordered pairs: ( 1 , 2 1 ) , ( 2 , 4 1 ) , ( 3 , 8 1 ) , ( 4 , 16 1 ) . If these points come from an exponential function f ( x ) = a b x , then we must have f ( 1 ) = a b 1 = 2 1 and f ( 2 ) = a b 2 = 4 1 . Dividing the second equation by the first, we get ab a b 2 = 1/2 1/4 , which simplifies to b = 2 1 . Since ab = 2 1 , we have a ( 2 1 ) = 2 1 , so a = 1 . Thus, f ( x ) = ( 2 1 ) x . Now let's check if the remaining points satisfy this function. f ( 3 ) = ( 2 1 ) 3 = 8 1 , which matches the third point. f ( 4 ) = ( 2 1 ) 4 = 16 1 , which matches the fourth point. Therefore, this set of ordered pairs can be generated by an exponential function.
Analyzing Set 4 Now let's analyze the fourth set of ordered pairs: ( 1 , 2 1 ) , ( 2 , 4 1 ) , ( 3 , 6 1 ) , ( 4 , 8 1 ) . If these points come from an exponential function f ( x ) = a b x , then we must have f ( 1 ) = a b 1 = 2 1 and f ( 2 ) = a b 2 = 4 1 . Dividing the second equation by the first, we get ab a b 2 = 1/2 1/4 , which simplifies to b = 2 1 . Since ab = 2 1 , we have a ( 2 1 ) = 2 1 , so a = 1 . Thus, f ( x ) = ( 2 1 ) x . Now let's check if the remaining points satisfy this function. f ( 3 ) = ( 2 1 ) 3 = 8 1 . But the third point is ( 3 , 6 1 ) , and 8 1 = 6 1 . Therefore, this set of ordered pairs cannot be generated by an exponential function.
Conclusion Based on our analysis, only the third set of ordered pairs, ( 1 , 2 1 ) , ( 2 , 4 1 ) , ( 3 , 8 1 ) , ( 4 , 16 1 ) , can be generated by an exponential function.
Examples
Exponential functions are incredibly useful for modeling various real-world phenomena. For instance, they can describe population growth, where the number of individuals increases at an accelerating rate. Similarly, exponential decay is used to model the decrease in the amount of a radioactive substance over time. In finance, compound interest follows an exponential pattern, allowing investments to grow significantly over the long term. Understanding exponential functions helps us make predictions and informed decisions in diverse fields.
Set C: ( 1 , 2 1 ) , ( 2 , 4 1 ) , ( 3 , 8 1 ) , ( 4 , 16 1 ) is the only set that can be generated by an exponential function, specifically f ( x ) = ( 2 1 ) x . The other sets do not satisfy the properties of exponential functions. Therefore, the chosen option is C.
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