 Given a function or relation, it's useful to define the following. We define the domain of a function or relation as the set of all possible input values. On the other hand, the range of a function or relation is the set of all possible output values. For example, let x be the length of one side of a square, and a the area of a square. Using x as input and a as output is a function of x, and what is the domain and range of the relation? To determine whether a is a function of x, we need to decide whether any given input x produces a unique output a. So let's think about that. Now it will be convenient to express this relationship as an algebraic equation if we can do that. So the relationship between the area of a square and its side length is going to be a equals x squared. And the advantage to algebra is the following. If we have a specific value of x, we're going to find the area by finding x squared. But since this is just an arithmetic formula, there's only one possible value for x squared. And so given any value of x, there is one and only one possible value for x squared. So a is a function of x. How about the domain and range? Since the length of one side of a square could be any positive real number, the domain is all positive real numbers. Since these are inputs, we write this as x greater than zero. Now at this point we do something that seems a little strange. This is based on the following idea. Strict inequalities are sometimes difficult to work with. So we often allow equality provided the algebraic expression allows it. In this particular case, since we can find x squared even if x is equal to zero, will allow our domain to be x greater than or equal to zero. How about the range? If x is a non-negative real number, then x squared is also a non-negative real number. Well since x squared is the area and the area is the output, that means the range is all non-negative real numbers. And so the range is a greater than or equal to zero. Now we could represent a function or a relation in terms of an equation, but sometimes that's not always possible. And so there are two other ways that we represent functions and relations. One is through set notation. We could represent functions and relations by giving a list of ordered pairs. For example, if we do this, we use the following convention. Suppose x, y is one of the ordered pairs in a function or relation. Then x, our first value, is read as the input value, and y, our second value, is read as the output value. For example, suppose I have a function or a relation given by this. Let's find the output for an input value of two, the input required to obtain an output value of four, and the output for an input value of six. So if we want the output for an input of two, then we want to find an ordered pair where the first term is two. So we'll look at our ordered pairs, and we'll try to find one with first term two. And that's this one. So our input value of two has output value of five. What if we want to get an output value of four? If we want an output value four, we look for an ordered pair where the second term is four. So we'll look and find this one, negative three, four. And so our input value of negative three will give an output value four. But wait, there's more. We have a second ordered pair with an output value of four, this one, five, four. And so we might add that the input value five will also give an output value four. And finally, if we want the output for an input value of six, we look for an ordered pair where the first term is six. And it's this, no, not that, there isn't one. And it's important to realize we can't just make up facts. If there isn't an ordered pair with input value six, we have to say that the relation is undefined for the input value six. If we have a function or relation given as a list of ordered pairs, we can find the domain and range, and then decide whether or not we have a function. So remember the domain is the set of all possible input values. And as ordered pairs, the inputs are the first entries of all the ordered pairs. So the domain will be the set of all the first entries. So our first entries are one, two, negative three, and one. But we've already listed one, and so we don't list any element twice. So our domain is just going to be one, two, negative three. Remember the range is the set of all possible output values. And so if we're given a list of ordered pairs, the outputs are the second entries of all the ordered pairs. And so the range is going to be three, five, four, and seven. Now to determine if f is a function, we check to see if any input has more than one output. So remember, the inputs are the first terms in all of the ordered pairs. And the thing to notice here is the input one appears in two ordered pairs. One three and one seven. But they have different outputs. And so we might say the following. Since the input one has two possible outputs, three and seven, this is not a function. One other thing we could do is we might also graph a relation. And since the points on a graph correspond to ordered pairs x, y, then our input will be all possible x values and the output will be all possible y values. So for example, suppose we have a graph. So let's see if we can determine the domain and range. So remember the domain is the set of all possible input values. That's the set of all possible x values. And our range is the set of all possible output values, which is to say the set of all possible y values. And so it will help to identify the least and greatest values of x and y. So the leftmost point has x-coordinate minus three. That's because we could read the coordinates of this point as minus three, one. Meanwhile, the rightmost point has x-coordinate two. Because the coordinates of this point are two, two. And all points in between have an x-coordinate between minus three and two, without exception. And so that means the domain, our possible x values, is going to be negative three, less than or equal to x, less than or equal to two. Meanwhile, the lowest point on the graph has y-coordinate minus one. And the highest point has y-coordinate five. And so that means our range is minus one to five.