Domain and Range with Functions

This week we did a very quick unit that was all about graphing and functions. We learned many new things from how to interpret graphs to how to write functions. The part that I thought was very important was showing the domain and the range of a graph while determining if a graph/relation is a function or not. This is a fundamental skill to learn when dealing with relations as it opens a lot of doors. Within this topic we also learned about what it means when there is a closed dot and what it means when there is an open dot on a graph, I will go further into detail later on in the examples.

Here is the first example question we are going to be working with:

Note: each box is equal to 1.

Step 1 – finding the domain: first off we are going to look at the domain of the graph. The domain of the graph is essentially what part of the x axis is being used. So we are going to look at how far the the line goes to the left, in this case it goes to -2 on the x axis. Then we look how far the line goes to the right, and in this case it goes up to 2 on the x axis. Now that we know our values we need to look at what the points look like. If the dots are closed then the expression will involve greater/less than or equal to signs. However, in this case we have two open dots so the expression will involve only greater than or less than signs.

Step 2 – writing the expression for the domain: to write an expression for the domain of a graph you need to first write the lowest number (-2) is less than x. Than another less than sign is needed followed by the largest number (2). This means that the points on the x axis are going to be somewhere in between -2 and 2. These signs may vary depending on the graph, but this is what is needed for this particular situation.

Step 3 – finding the range: when finding the range we are going to look at the y axis, so the opposite of the domain. To find the range we are going to look at how far the line goes down on the y axis, in this case it goes down to -2. Then we need to see how high the line goes up on the y axis, as you can see it goes up to 2. Now that we know our values we need to look at what the points look like. If the dots are closed then the expression will involve greater/less than or equal to signs. However, in this case, just like in the domain we have two open dots so the expression will involve only greater than or less than signs.

Step 4 – writing the expression for the range: to write an expression for the range of a graph you need to first write the lowest number (-2) is less than y. Than another less than sign is needed followed by the largest number (2). This means that the points on the y axis are going to be somewhere in between -2 and 2. Once again, these signs may vary depending on the graph, but this is what is needed for this particular situation.

Step 5 – determining if the graph/relation is a function: to determine if we have a function or just a relation it is actually quite simple and noticeable. In order to figure this out we can use something called the vertical line test. In the vertical line test you can imagine a line is being run through the graph (or you can use a ruler) and if the line runs into the x axis twice at the same point on the x axis, then it does not pass the vertical line test. This means the graph we are looking at is a relation since it did not pass the test. Another way you can think about it is if the same x value has two different y values it is not a function, it is only a relation. In some cases you may have to determine if the graph is continuous, discrete, or neither but this is not necessary here as the graph is not a function. 

 

Example 2:

Now we are going to look at another graph with some slightly different features.

Here is the example question we are going to be working with:

Note: each box is equal to 1.

Step 1 – finding the domain: first off we are going to look at the domain of the graph. The domain of the graph is essentially what part of the x axis is being used. So we are going to look at how far the the line goes to the left, in this case it goes to. In this graph it’s a little different because there are arrows at the ends of the lines on the graph. This means that graph keeps on going, and in this case it keeps going sideways, along the x axis. So in this case the domain would just be “all real numbers” since the x values could really be anything. No greater than, less than, greater than or equal to, less than or equal to signs are needed. Unlike in the previous example where we had a step 2 of writing the expression we are just going to skip that part and find the range since the domain can be written in words.

Step 2 – finding the range: next we are going to take a look at the range of the graph, what part of the y axis is being used. Once again we are going to look at how far the line goes up and how far it goes down. Like I said previously there are arrows on the end of the lines of the graph which means the graph keeps on going. However the graph does begin somewhere and it doesn’t go higher than that, it only goes below, that number is 1.

Step 3 – writing the expression for the range: to write an expression for the range of a graph you need to first write the lowest number which we don’t actually know since the graph keeps on going forever (to infinity). Because of this only one sign will be needed and it will be a less than or equal to sign because there are no open dots in the graph. The equation will look like this:

Step 4 – determining if the graph/relation is a function: to determine if we have a function or just a relation it is actually quite simple and noticeable. In order to figure this out we can use something called the vertical line test. In the vertical line test you can imagine a line is being run through the graph (or you can use a ruler) and if the line runs into the x axis twice at the same point on the x axis, then it does not pass the vertical line test. Another way you can think about it is if the same x value has two different y values it is not a function, it is only a relation. This graph does pass the vertical line test since all of the x values have their own “partners” or y values. Since this graph turns out to be function we will need to determine if it is continuous, discrete, or neither of these. This graph is going to be continuous since all the dots are connected in a line. Whereas if there were just random unconnected points the graph would be discrete.

 

Summary:

That is how you find the domain and range of different graphs, both relations and functions. I believe that this is an essential skill when working with graphs. I used it many times throughout the unit and it was very useful to me. Once you learn this skill it is fairly simple, you just have to make sure to pay close attention when making your observations.