Domain and range A1.3.1.2

Added: 2026-06-02

[00:00:01]next we're gonna be looking at relations and domain and range a relation is any pairing between two sets of numbers one set is called the domain one set is called the range when you look at a relation there's lots of ways to show it you could show it in a table in a list and a mapping diagram in a graph lots of different ways to show the same thing but here's an example with the table the domain values are the axes or the inputs or the independent values the range is the Y values or the outputs or the dependent values now another way I could show this same relation rather than a table I could make a scatter plot by plotting each of these points so plotting the point 2 3 & 4 5 & 5 6 6 6 and seven nine so that would be the same relation shown as a scatterplot or I could also show you this same relation as a set when you write it as a set you would write each ordered pair in parentheses so two three four five five six six six and seven nine and then to make it a set you take that list of ordered pairs and you put it in a set of curly brackets or one last way that you could represent this is you could show it as a mapping diagram in a mapping diagram you'll have one circle or curved type shape because it's more elliptical than circular and you would list your input values or your X's there so our X values are 2 4 5 6 & 7 and then you have another one for your Y values which are 3 5 6 & 9 notice how I didn't write the six twice I just wrote it once for the Y values and then you draw arrows showing which ones go together so the ordered pair 2 3 you'd connect with an arrow 4 5 5 6 6 6 you just have two arrows going to the 6 rather than writing it twice and then 7 9 so those are all different ways to show that same data but it doesn't change the domain and range so in all of these questions the domain is this list of X's so if you were asked to find the domain you would write your answer as a set and you'd say that the domain is 2 4 5 seven and the range the range is just all of your y-values no matter how it would be represented you could either get it from the graph the table the set notation or the mapping diagram just listing out all over the Y's and those are three five six nine again notice we didn't write the six twice just one time here's another graph this graph is rather than separate points it's a lot of points connected it's an infinite number of points and if you're asked to find the domain and the range here what we're looking for is an interval so to find the domain remember that the domain is the X values so you're going to look along the x-axis and you're going to see the smallest value along the x axis that would be right here at negative 11 and then the largest value of the graph along the x axis which would be right here at 9 and everything in between there is included so there's a few ways you could write this in interval notation you would write this as all of the numbers from negative 11 to 9 or in set notation you'd write it as all numbers X such that X is between negative 11 and 9 for the range you're looking for the Y values so we look for the lowest point on our graph and that's right here which is at negative 1 along the y-axis and the highest point on our graph which is right here which is at 3 on the y axis so the range as an interval would be negative 1 2 3 or all the values Y such that Y is between 1 & 3 this is a very similar example but you'll notice this time that the endpoints are open circles rather than colored in circles that's just gonna change one very small piece of our notation so if we go to look at our X's smallest value along the x-axis is here at negative 6 largest value is here at 7 but because those are open circles they're not actually included you're just getting really close to those points so then the domain rather than saying that it goes from negative 7 or negative 6 to 7 and including those points we want to exclude them in the last problem we did a bracket in this problem we're gonna do parenthesis so the bracket means that the endpoint was colored in and the parentheses means that it was not if you wanted to use interval notation you'll remember that from the last one as well where we said X was between the negative 6 and the 7 you just wouldn't put an equal sign as part of that inequality to find the range you're looking for your highest point in your lowest point so here's your lowest point on the y-axis and your highest point on the y-axis notice that these points that I'm at these are not open circles so that means those values would be included so the range would include all of those numbers from negative 5 to 5 and you would use the bracket instead of the parenthesis or if you were going to use the set notation you would have the equal signs in your set now if you had liked one of these circles colored in at the end that would just change something to a bracket or a parenthesis so like if I colored this one in what that would do is it would change this negative six would get changed to a bracket because now it's included and this negative six here would have an equal sign because it's now included