Identify Exponential Function from Ordered Pairs

An exponential function has the general form , where is the initial value (the -intercept when ) and is the base (the common ratio).

We will test each set of ordered pairs to see if they can fit this form for some constants and .

A crucial feature of most standard exponential functions is that if is a point, then . The -intercept is . If is a point, then , which implies . If , the function is , which is the constant function . If the set contains points other than , it cannot be the function .

Let’s examine the options:

Option 1:
This set contains , which implies . If , then for all . However, the set also contains , where . Therefore, this set cannot be generated by an exponential function.

Option 2:
This set also contains , implying and thus for all . Since and have , this set cannot be generated by an exponential function.

Option 3:
This set contains .
Using :
For :

So, the function must be .
Now we use the other points to find :
For :

The potential function is . Let’s check the remaining points with :
For :

This matches the point .
For :

This matches the point .
Since all points fit the form , this set could be generated by an exponential function.

Option 4:
This set contains , implying and thus for all . Since and have , this set cannot be generated by an exponential function.

The only set that fits the form is Option 3.

The correct set of ordered pairs is .

The final answer is the third option.