Hi everyone, and welcome back. Up to this point, we spent a lot of time talking about functions, and in this video, we're going to be taking a look at how we can find the domain and range of a function based on its graph. Now, when encountering finding the domain and range of functions, it can often seem a bit tricky, but in this video, we're going to be going over some analogies and examples that will hopefully make this topic a lot more clear. So let's get into this.
When finding the domain of a graph, you're looking for the allowed x values that you can have. And when finding the range, you're looking for the allowed y values. Now, there's a little trick that you can use, and that is known as the squish strategy. What you can do if you have the domain of a graph is you can take your graph and squish it down to the x-axis. So if you squish things down to the x-axis, that will tell you the domain. So, let's say we have this graph down here. And notice how we took this curve, and we literally squished it down to the x-axis so only some sort of line or set of lines remains. This tells you the domain. Now, when finding the range, you can take your graph and squish it down to the y-axis. So notice for our graph, we've squished it down to the y and this tells you the range.
Now, there are a couple of different ways we can write the domain and range of a graph. One of the common things is to use interval notation. Interval notation represents the domain and range using these brackets and parentheses. And you can actually see this in the example we have above. So notice that it looks like our domain goes from a value of negative 4 to positive 5 on the x-axis. So this is what we said the domain was. And for our range, it looks like we go from about negative 1 to positive 2 on the y-axis. So that's using interval notation. Now, there's another notation you can use called set builder notation, and set builder notation uses these inequality symbols as opposed to the brackets and parentheses.
Now, something you need to keep in mind whenever using either one of these notations is that whenever you see these brackets or inequality symbols with the bar underneath, that means that you want to include whatever value you're looking at. So, let's say we have this graph, for example. The values that you want to include are either closed dots or solid lines or curves. These are values that you always want to include in your domain and range. Now whenever you see these parentheses or inequality symbols without the bar underneath, this means that you do not want to include those values. And values that you do not want to include are situations where you see a hole in your graph. So anytime you see this hole or this open circle, that means you do not want to include it whereas anytime you see a solid dot or a solid line or curve, you do want to include it.
So given all this information, let's see if we can solve an example. So here we're asked to determine the domain and range of the graph below and to express our answer using interval notation. Now, what I'm first going to do is see if I can find the domain of this graph. And by the way, this is the same graph as you can see for both these diagrams. But for finding the domain, recall that what we want to do is take our graph and squish it down to the x-axis. So, going down here, if we take this graph and we imagine squishing every single point down to the x-axis, we're going to get a graph that looks something like this. We'll have a line that goes from negative 4 all the way to an x value of 0 and then we'll have another line that goes from positive 1 all the way to 4. Now notice how there's an open circle here, so we have to include the open circle there as well. And then everywhere else, we have a closed circle because these circles were all closed. So looking at this squished graph, our domain is going to go from negative 4 to 0 on our x-axis and we need to include both these values since we have 2 closed circles. Now, I can see that another one of these lines is going to go from positive one because this is an x value of 1 right here, to 4. So we're going to have 1 to 4,