Question

Does the data set display exponential behavior? * {(0, 1), (1, 3), (2, 9), (3, 27)}

Answer 1

Yes, it does.

Step-by-step explanation

We want to check if the data set displays an exponential behavior.

An exponential function is one in which the values of the range (y values) increase by a certain factor.

The general form of an exponential function is:

`y=a\cdot b^x`

where a is the starting value

b = factor.

Now, we have to compare the data set with this kind of function.

To do that, we have to find a mock function of the data set using the first two data points to test each x value (domain) for each y value.

Basically, we will replace x in the function with a value and see if we get the correct y.

Therefore, when x = 0:

`\begin{gathered} y=a\cdot b^0 \\ y=a\cdot1 \end{gathered}`

From the data set, we see that, when x = 0, y = 1:

`\begin{gathered} \Rightarrow1=a\cdot1 \\ a=1 \end{gathered}`

That is the value of a.

Now, let us try when x = 1:

`\begin{gathered} \Rightarrow y=1\cdot b^1 \\ y=b \end{gathered}`

From the data set, we see that, when x = 1, y = 3:

`\begin{gathered} \Rightarrow3=b \\ b=3 \end{gathered}`

Now, we can say that we have an exponential function to test with:

`y=3^x`

So, let us test for the remaining values of x and y and see if they match the function.

`\begin{gathered} \text{when x = 2:} \\ y=3^2 \\ y=9 \\ \text{when x = 3:} \\ y=3^3 \\ y=27 \end{gathered}`

As we can see, each x value that goes into the function yields the exact y value as the data set. This means that the exponential function works for it.

Hence, the data set displays an exponential behavior.