An electric device delivers a current of [tex]$15.0 A$[/tex] for 30 seconds. How many electrons flow through it?
Answer (1)
Test each given function with the points in the table.
f ( x ) = 2 1 ( 4 ) x does not match the table.
f ( x ) = 4 ( 4 ) x does not match the table.
f ( x ) = 4 ( 2 1 ) x matches the table.
f ( x ) = 2 1 ( 2 1 ) x does not match the table.
The correct function is f ( x ) = 4 ( 2 1 ) x .
Explanation
Understanding the Problem We are given a table of x and f ( x ) values and asked to identify the exponential function that represents these values. The general form of an exponential function is f ( x ) = a x , where a is the initial value and b is the base. We will test each of the given options to see which one fits the points in the table.
Testing Option 1 Let's test the first option: f ( x ) = 2 1 ( 4 ) x . If we plug in x = − 2 , we get f ( − 2 ) = 2 1 ( 4 ) − 2 = 2 1 ( 16 1 ) = 32 1 . This does not match the table, which gives f ( − 2 ) = 16 . So, this option is incorrect.
Testing Option 2 Now let's test the second option: f ( x ) = 4 ( 4 ) x . If we plug in x = 0 , we get f ( 0 ) = 4 ( 4 ) 0 = 4 ( 1 ) = 4 . This matches the table. However, if we plug in x = 1 , we get f ( 1 ) = 4 ( 4 ) 1 = 4 ( 4 ) = 16 , which does not match the table, which gives f ( 1 ) = 2 . So, this option is incorrect.
Testing Option 3 Next, let's test the third option: f ( x ) = 4 ( 2 1 ) x . If we plug in x = 0 , we get f ( 0 ) = 4 ( 2 1 ) 0 = 4 ( 1 ) = 4 . This matches the table. If we plug in x = 1 , we get f ( 1 ) = 4 ( 2 1 ) 1 = 4 ( 2 1 ) = 2 . This matches the table. If we plug in x = 2 , we get f ( 2 ) = 4 ( 2 1 ) 2 = 4 ( 4 1 ) = 1 . This matches the table. If we plug in x = − 1 , we get f ( − 1 ) = 4 ( 2 1 ) − 1 = 4 ( 2 ) = 8 . This matches the table. If we plug in x = − 2 , we get f ( − 2 ) = 4 ( 2 1 ) − 2 = 4 ( 4 ) = 16 . This matches the table. Therefore, f ( x ) = 4 ( 2 1 ) x is the correct function.
Testing Option 4 Finally, let's test the fourth option: f ( x ) = 2 1 ( 2 1 ) x . If we plug in x = 0 , we get f ( 0 ) = 2 1 ( 2 1 ) 0 = 2 1 ( 1 ) = 2 1 , which does not match the table, which gives f ( 0 ) = 4 . So, this option is incorrect.
Conclusion Therefore, the exponential function represented by the values in the table is f ( x ) = 4 ( 2 1 ) x .
Examples
Exponential functions are used to model various real-world phenomena, such as population growth, radioactive decay, and compound interest. For example, if you invest 4000 inana cco u n tt ha tp a ys 50 f(x) = 4000(1.5)^x$. This function tells you how your investment grows exponentially over time.
