Which set of ordered pairs could be generated by an exponential function?
(1,1), (2, 1/2), (3, 1/3), (4, 1/4)
(1,1), (2, 1/4), (3, 1/9), (4, 1/16)
(1, 1/2), (2, 1/4), (3, 1/8), (4, 1/16)
(1, 1/2), (2, 1/4), (3, 1/6), (4, 1/8)
Answer (1)
An exponential function has the form f ( x ) = a x .
Test each set of ordered pairs to see if there exist constants a and b that satisfy the equation for all pairs in the set.
Set 3: ( 1 , 1 2 ) , ( 2 , 1 4 ) , ( 3 , 1 8 ) , ( 4 , 1 16 ) can be generated by the exponential function f ( x ) = ( 1 2 ) x .
The set of ordered pairs that could be generated by an exponential function is ( 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 form f ( x ) = a x where a is a constant and b is the base. 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 Set 1 Let's analyze the first set of ordered pairs: ( 1 , 1 ) , ( 2 , 1 2 ) , ( 3 , 1 3 ) , ( 4 , 1 4 ) . If this set is generated by an exponential function f ( x ) = a x , then we must have f ( 1 ) = 1 , f ( 2 ) = 1 2 , f ( 3 ) = 1 3 , and f ( 4 ) = 1 4 . From f ( 1 ) = 1 , we have a 1 = 1 . From f ( 2 ) = 1 2 , we have a 2 = 1 2 . Dividing the second equation by the first, we get b = 1 2 . Substituting this into the first equation, we get a ( 1 2 ) = 1 , so a = 2 . Thus, f ( x ) = 2 ( 1 2 ) x . Now let's check if this function generates the other ordered pairs. f ( 3 ) = 2 ( 1 2 ) 3 = 2 ( 1 8 ) = 1 4 . However, the third ordered pair is ( 3 , 1 3 ) , and 1 4 e q 1 3 . Therefore, this set cannot be generated by an exponential function.
Analyzing Set 2 Now let's analyze the second set of ordered pairs: ( 1 , 1 ) , ( 2 , 1 4 ) , ( 3 , 1 9 ) , ( 4 , 1 16 ) . If this set is generated by an exponential function f ( x ) = a x , then we must have f ( 1 ) = 1 , f ( 2 ) = 1 4 , f ( 3 ) = 1 9 , and f ( 4 ) = 1 16 . From f ( 1 ) = 1 , we have a 1 = 1 . From f ( 2 ) = 1 4 , we have a 2 = 1 4 . Dividing the second equation by the first, we get b = 1 4 . Substituting this into the first equation, we get a ( 1 4 ) = 1 , so a = 4 . Thus, f ( x ) = 4 ( 1 4 ) x . Now let's check if this function generates the other ordered pairs. f ( 3 ) = 4 ( 1 4 ) 3 = 4 ( 1 64 ) = 1 16 . However, the third ordered pair is ( 3 , 1 9 ) , and 1 16 e q 1 9 . Therefore, this set cannot be generated by an exponential function.
Analyzing Set 3 Now let's analyze the third set of ordered pairs: ( 1 , 1 2 ) , ( 2 , 1 4 ) , ( 3 , 1 8 ) , ( 4 , 1 16 ) . If this set is generated by an exponential function f ( x ) = a x , then we must have f ( 1 ) = 1 2 , f ( 2 ) = 1 4 , f ( 3 ) = 1 8 , and f ( 4 ) = 1 16 . From f ( 1 ) = 1 2 , we have a 1 = 1 2 . From f ( 2 ) = 1 4 , we have a 2 = 1 4 . Dividing the second equation by the first, we get b = 1 2 . Substituting this into the first equation, we get a ( 1 2 ) = 1 2 , so a = 1 . Thus, f ( x ) = ( 1 2 ) x . Now let's check if this function generates the other ordered pairs. f ( 3 ) = ( 1 2 ) 3 = 1 8 , which matches the third ordered pair. f ( 4 ) = ( 1 2 ) 4 = 1 16 , which matches the fourth ordered pair. Therefore, this set can be generated by an exponential function.
Analyzing Set 4 Now let's analyze the fourth set of ordered pairs: ( 1 , 1 2 ) , ( 2 , 1 4 ) , ( 3 , 1 6 ) , ( 4 , 1 8 ) . If this set is generated by an exponential function f ( x ) = a x , then we must have f ( 1 ) = 1 2 , f ( 2 ) = 1 4 , f ( 3 ) = 1 6 , and f ( 4 ) = 1 8 . From f ( 1 ) = 1 2 , we have a 1 = 1 2 . From f ( 2 ) = 1 4 , we have a 2 = 1 4 . Dividing the second equation by the first, we get b = 1 2 . Substituting this into the first equation, we get a ( 1 2 ) = 1 2 , so a = 1 . Thus, f ( x ) = ( 1 2 ) x . Now let's check if this function generates the other ordered pairs. f ( 3 ) = ( 1 2 ) 3 = 1 8 . However, the third ordered pair is ( 3 , 1 6 ) , and 1 8 e q 1 6 . Therefore, this set cannot be generated by an exponential function.
Conclusion Only the third set of ordered pairs can be generated by an exponential function.
Examples
Exponential functions are incredibly useful for modeling various real-world phenomena, such as population growth, radioactive decay, and compound interest. For instance, imagine you invest 1 , 000 ina s a v in g s a cco u n tt ha t o ff ers anann u a l in t eres t r a t eo f 5 A = P(1 + r)^t$, where A is the final amount, P is the principal, r is the interest rate, and t is the time in years, is a direct application of exponential functions.
