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 different ways we can write the domain and range of a graph. One of the common things is to use interval notation. And 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 one 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 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 one 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 gonna have 1 to 4. We're going to include this positive one since we have a closed circle there, but our open circle is going to cause us to draw a parenthesis here because we do not want to include that value. Now notice how we have multiple intervals here. Whenever you have multiple intervals or a jump in your graph, you need to use something called a union symbol, which looks like this. So since we have multiple intervals, we need to put a union symbol between these to join the 2 intervals together. So this right here is the domain of our graph. Now let's see how we can find the range. And recall when finding the range, we need to take our graph and squish it down to the y axis. So if I take this other graph we have drawn over here and I squish all these values down to the y, what we're first going to end up with is a line that goes from 123 on the y axis and then we're going to end up with another line that goes from negative 3 all the way to negative 1 and we'll have an open circle at negative 1 since we have the open circle right there. So our range is going to go from negative 3 to negative one and keep in mind we'll include the negative 3 but we do not include the negative one since we have an open circle and our range will also go from positive one to 3. And we need to include both of these values since we have closed dots or I should say we have a solid line or curve, but still you want to include that value. And we'll put the union symbol, and then this is the range of our graph. So that is how you can find the domain and range of a function based on its graph. Let me know if you have any questions, and thanks for watching.

Struggling with Business Calculus?