 Suppose f of x is a function of x. So remember that a function is a relationship between variables that has one output for any given input. So for any input value x, there's a unique output value f of x. Now if we look at the graph of y equals f of x, then any input value x gives a unique output value y. And what this means is that we can decide from the graph whether we have the graph of a function. So for example, we have a graph. Does the graph shown represent that of a function of x? And let's explain why or why not. So consider any point on the graph. How about this one? Now this particular point has a particular x value. And the thing to notice is that for the particular x value of this point, there appears to be one and only one y value, since there are no points directly above it or directly below it. So whatever determines the y values of the points appears to give a unique value of y for any x. So this is the graph of a function of x. How about a graph like this? Well again, let's consider any point on the graph. And here we see something different. For some points, there are no other points with the same x value. So for this point and this point, there are no points above it and no points below it. But for others, there are additional points with the same x value. These are directly above it or directly below it. And the important thing to recognize here is that for these points, the same x value gives more than one y value. So whatever determines y can't be a function. And this leads to an important result, sometimes called the vertical line test. The graph of a function of x can't cross over itself. In other words, we can't have a portion of the graph that's above another portion of the graph. And this leads to something called the vertical line test. And we won't tell you what that is because it's important to remember, don't memorize procedures, understand concepts. The important idea in this case is that graphs that do cross over themselves have multiple values of y for the same value of x.