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)
Exponential functions have the form f ( x ) = a × b x .
Sets containing the point (0,0) cannot be generated by an exponential function.
Check each set to see if there exists an exponential function that passes through all the points.
The set ( − 1 , 2 1 ) , ( 0 , 1 ) , ( 1 , 2 ) , ( 2 , 4 ) can be generated by the exponential function f ( x ) = 2 x .
( − 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 × b x , where a is the initial value and b is the base. We need to check each set of ordered pairs to see if there exists an exponential function that passes through all the points in the set.
Analyzing Each Set Let's analyze each set of ordered pairs:
Set 1: ( − 1 , − 2 1 ) , ( 0 , 0 ) , ( 1 , 2 1 ) , ( 2 , 1 ) .
Since ( 0 , 0 ) is in the set, f ( 0 ) = a × b 0 = a = 0 . This means f ( x ) = 0 for all x , which contradicts the other points. So, this set cannot be generated by an exponential function.
Set 2: ( − 1 , − 1 ) , ( 0 , 0 ) , ( 1 , 1 ) , ( 2 , 8 ) .
Similar to set 1, since ( 0 , 0 ) is in the set, this set cannot be generated by an exponential function.
Set 3: ( − 1 , 2 1 ) , ( 0 , 1 ) , ( 1 , 2 ) , ( 2 , 4 ) .
If f ( x ) = a × b x , then f ( 0 ) = a × b 0 = a = 1 . So f ( x ) = b x . Then f ( − 1 ) = b − 1 = b 1 = 2 1 , so b = 2 . Thus, f ( x ) = 2 x . Check the other points: f ( 1 ) = 2 1 = 2 and f ( 2 ) = 2 2 = 4 . This set can be generated by the exponential function f ( x ) = 2 x .
Set 4: ( − 1 , 1 ) , ( 0 , 0 ) , ( 1 , 1 ) , ( 2 , 4 ) .
Since ( 0 , 0 ) is in the set, this set cannot be generated by an exponential function.
Final Answer 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 instance, 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 future values based on current trends, making it a powerful tool in finance, science, and many other fields.
The set of ordered pairs that can be generated by an exponential function is ( − 1 , 2 1 ) , ( 0 , 1 ) , ( 1 , 2 ) , ( 2 , 4 ) , which corresponds to the function f ( x ) = 2 x .
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