 One of the more important concepts in later mathematics is the function concept. So here's some useful terms. A relation is an equation that includes two or more variables. So for example, 3x plus 7 equals 12. Well, there's only one variable in this, so this is not a relation. This is actually just an equation. But if I throw in a second variable, x squared plus y squared equals 8xy, well not only is this an equation, you can tell because there's this equal sign here, but it's also a relation because it has two variables in it. And so here's one, which is also an equation and a relation in ABC3 variables and so on. Now, if we substitute in values for all but one of the variables, we can at least try to solve for the remaining variable. So for example, if I take this equation, this relation, x squared plus y squared equals 8xy, and if I let x equals 5, then this relation becomes this equation in one variable, and I can attempt to solve for a value of the remaining variable. Likewise, if I take my equation and relation, one half AB equals C, if I let A equals 5, C equals 20, then this relation becomes this equation in one variable, and I can attempt to solve for that last variable. Now, this allows us to introduce the notion of a function. A relation in two variables is a function of x, if when I substitute in a specific value of x, then I get an equation that has at most one solution, y. So for example, this relation, x squared plus y squared equals 25, well it's an equation, it has the equals, it has two variables, so this is a relation in x and y. But if I substitute in a specific value of x, if x equals 7 or something like that, what I end up with is an equation that is quadratic in y, and generally speaking, this quadratic equation will have two solutions. And because I have two solutions, I do not have a function of x. On the other hand, let's take a look at something like this. Again, it's an equation because it has the equals, there's two variables, so it's a relation in my two variables x and y, and if I substitute in a specific value of x, what I'm going to get is I'm going to get the principal square root of some number. And the principal square root has at most one value. If I'm taking the square root of a negative number, it has no real values, but if I'm taking the square root of a positive number, I'll have one actual value, so what I have is a function of y. And the notation that we'll introduce here is that we'll write y equals f of x, and this letter f can change, f, g, h, j, q, m, whatever you want to put here, but the variable x is here. And we use this notation to indicate that we have a function, and a given value of x will produce at most one value of y. And again, it's possible that a given value of x might produce no values of y, and that still allows us to have a function. So for example, let's take a look at the following, determine which of the following are functions of x. And again, what that means is if I supply a specific value of x, I want to make sure that I have at most one value of y. So how can we do this? Well, so let's take a look at our relation, and again, the idea is that if it's a function of x, then what I want is a specific value of x to give me at most one value of y. So suppose x is a specific number, and so let's go through this one step at a time. x plus two, a little bit of analysis goes a long way, x plus two is a sum, and sums have at most one value. There's not a choice. If I add two numbers, I get a sum, and I don't get a bunch of possibilities, and you get to choose which one. Now the next part of our relation, one over x plus two, well that's a quotient, and again I have at most one value for a quotient. The other part of this relation, absolute value of x minus seven, x minus seven itself is a difference. Again, there's at most one value. Absolute value of x minus seven, well again there's at most one value to that expression. And finally this is a sum of two things, and again there's a sum, there's at most one value. So there's at most one value for this thing, there's at most one value for this thing, and when I add two things together, there's at most one value, and so that says for any given value of x, for any specific value of x, this expression has at most one value, and so that tells me that y, which is equal to this expression, has at most one value. So there's at most one value for y, so that tells me y is a function of x. Well let's take a look at the other one, for y cubed plus three, y plus five equals x. Again if I let x have a specific numerical value, then what I get is I get y cubed plus three, y plus five equals some number. I have a third degree equation in y, and our previous problem solving experience says that in general this type of equation will usually have more than one solution, and so that tells me that in general, for any given value of x, there's going to be more than one value of y, and so that tells me that y is not a function of x, because I have the possibility of more than one value of y for any value of x, y is not going to be a function of x. We can also take a look at functions graphically, and again if I have a relation in two variables, I can show this as a graph where every point on the graph satisfies the relation. For example, if I have this relation in one two variables, it's an equation, it has two variables, so it's also a relation between the two variables, and if I were to try and graph this, what I'm going to see is I'm going to get a circle with a certain center, five negative three, and a radius of five, and likewise, if I try to graph something else, I have the graph y equals three x minus four squared plus eight, again this turns out to be the graph of a, this is an equation, there's two variables, so it's a relation in two variables, and this will have the graph of a parabola with a particular vertex, and one of the things we can do is we can ask the question, suppose I have the graph of a relation, do I also have the graph of a function? So for example, let's take a look at this graph, so I have a graph that shows a relation between x and y, and because there's two variables, I can ask whether I have a function of x or whether I have a function of y, and it's possible I might have a function of both x and of y, I might have a function of neither variable. So to determine whether the graph shows a function of x, what I want to do is I want to determine if any specific x value could give me more than one y value. So let's think about this, I'll what I'll do is I'll take a point on the graph, and if it's possible, so I have a point on the graph x, y, and again what I want, if I have a function of x, I want to make sure that this value of x gives me at most one value of y, and it's often easy to determine if this is false, if this value of x gives me more than one y value, then there's going to be a second point with the same x value, but a different y value on the graph. Now let's, to find those points, let's consider a vertical line through our point. So there's a vertical line through the point, and for most points on the graph, I can see that this vertical line is going to intersect at a second point on the graph, x, y, prime, and what that says is I now have two points on the graph, the same value of x, but more than one y value, and so for most values of x, there's going to be more than one value of y, and so I do not have a function of x. What about the question of whether this is a function of y? So again, same reasoning, different problem. What I want to do is given a specific value of y, can I find more than one value of x? And what that means is given a specific point, x, y on the graph, maybe there's a second point with the same y value, the same height, but a different x value, the different horizontal extension. And so to find that, I'll look at a horizontal line through that point, x, y, and it doesn't appear that there's another point with the same y coordinate, but a different x coordinate, but let's make sure we'll draw a bunch more horizontal lines, and it appears that in general, there is not going to be a second point with the same y coordinate, and so no value of y will give me more than one value of x, and so it appears that I do in fact have a function of y.