 Given some relationship between variables, we define the following. The domain of f, our relationship, is the set of allowable input values, while the range of f is the set of possible output values. Now typically we can view our relationship f as an algebraic formula, but another way to represent f is as a set of ordered pairs. So f, our relationship, might be the collection of ordered pairs of 1, x2, y2, and so on. In this case, we treat the x values as the inputs and the y values as the outputs. And this leads to a useful idea. If our functional relationship is given as ordered pairs, the inputs are going to be the x values, and the outputs are going to be the y values. For example, let's find the domain and range of the relationship. Remember, definitions are the whole of mathematics. All else is commentary, so we'll pull in our definition of domain and range. And since we're given this relationship as a set of ordered pairs, it's useful to remember that if our functional relationship is given as ordered pairs, the inputs are the x values, and the outputs are the y values. So the domain is the set of input values. Those are the set of x values, the first coordinates. That would be these. So our domain is the set 3, 2, 1, 8. Meanwhile, the range is the set of output values. Those are going to be the y values, these. And so our range is 7, 5, 4, 11. Or we might take another example, and here it's useful to keep in mind that in a set, we should list each element only once. In other words, we don't have to and shouldn't list an element more than once. So again, the domain is the set of input values, which will be 3, 1, and 8, where even though we have 2, 3s as inputs, we don't need to list them twice. The range is the set of output values, and so the range will be 7, 5, 4, 9. We can take another viewpoint as well. If the function is a set of ordered pairs, we can also graph these ordered pairs. And in that case, our domain is the set of all possible x values, and the range is the set of all possible y values. And so this leads to another idea. If our function or relationship is given as a graph, the inputs are the x values, and the outputs are the y values. So for example, find the domain and range of the relationship graph. So definitions are the whole of mathematics. All else is commentary. We'll pull in our definition of domain and range. And since we're given a graph of our relationship, it's useful to remember that our domain, our inputs are going to be the x values, and our outputs are going to be the y values. So let's start with the domain. The domain is the set of x values, and if we look at this graph, we see the following. The least x value of any point in the graph is x equal to minus 2, and that occurs here on the left. The graph extends to the right, and the x values go up to, but don't include, x equal 3, which is at this point. And every x value between minus 2 and 3 is included has some point on the graph, and so our domain is either minus 2 less than or equal to x strictly less than 3, if we want to write this in inequality notation. Or in interval notation, we could write this as x is in the interval minus 2 included, up to 3 not included. Now the range is going to be the set of y values, and so if we look at our graph, we look for the least and highest y value, and we see the least y value is y equal to minus 3, which occurs at this point, and the highest y value is going to be here at this point, which is where y is equal to 1, and there is a point there, and every y value in between minus 3 and 1 is included in some point, and so y is greater than or equal to minus 3, and less than or equal to 1, or in interval notation, y is in the interval minus 3 included, up to 1 also included. Or take a look at this graph. So we take a look at our graph, and we see the least x value occurs here or here at x equal to minus 2, and we move to the right, and the greatest x value occurs here at the point x equal to 2, and there are points for all x values in between minus 2 and 2, so our domain is x greater than or equal to minus 2 and less than or equal to 2, or in interval notation, x is in minus 2 included, up to 2 included. How about our y values? The least y value is y equals minus 3, and there is a point there. The greatest y value is y equals 3, and there's a point there as well, and we see that there are points with y values between minus 3 and 3. Well, actually there aren't. There's this gap in the graph, and what this gap means is there are some y values for which there are no points. So we have to look a little bit more carefully, and so if we do that, we see there are points with y values between y equals minus 3 and y equals negative 1, and there are points with y values between y equals 1 and y equals 3, and so our range actually consists of two parts, y greater than or equal to minus 3, less than or equal to minus 1, or y greater than or equal to 1, less than or equal to 3, which we can write in interval notation.