 So let's put it all together and try to find the graphs of some functions. Before we do that, let's consider things a little bit further. 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. So let's consider this graph. We want to graph y equals f of x. Equals means replaceable, so we'll replace f of x with x plus 4 over x minus 3. And this is the graph of A. We don't know what this is a graph of. This isn't a familiar type of object. And so we should run and find out what we can. So we do know how to find the x and y intercepts, so let's find them. The y intercepts occur when x equals 0. Equals means replaceable, so we'll replace x with 0 and find y. And remember, the y intercept is a point, so we need to specify both coordinates. And if it's not written down, it didn't happen. The x intercepts occur when y is equal to 0. So again, equals means replaceable, and we'll replace y with 0. And now we have to do a little bit more work to solve this equation. Since we have this denominator x minus 3, we'll multiply both sides by the denominator, and then solve the resulting equation. Again, the x intercept is a point, so we have to specify both the x and y coordinates. The x intercept is minus 4, 0. Now, since this is a rational expression, there could also be forbidden values. Anything that makes the denominator 0. So let's run and find out what makes the denominator 0. And so we conclude the value x equals 3 is forbidden. Now, except for the forbidden values, we can use any other value for x, so the domain is all real numbers except for x equals 3. So let's start to sketch our graph. We'll plot the x and y intercepts. The forbidden value means that no point on the graph can have x equals 3. And we'll indicate this by drawing the line x equals 3, and, well, this were a real line, we'd run an electric current through it, but we'll just have to remember that the graph can't touch this line. Now, the x and y intercepts and the forbidden values are important points, but if we want to get a picture of what the graph really looks like, we have to consider what happens between those points. To find out, let's find points between the important points. It's useful to remember how we handled inequalities by picking test values. So here, to the left of this point, negative 4 is 0, we can let x equal minus 10,000. Remember, we want to either go big or go home because it's actually easier to deal with large numbers. So x is a large negative number, so y is going to be a large negative number divided by a large negative number. While we don't know the magnitude, we don't know whether this is large or small, we do know it will be positive. So again, the secret to graphing is plot first then label, so we have a point somewhere to the left that's positive. So we'll put it here. Between negative 4 is 0 and 0, negative 4 is 3, we can let x equal, how about negative 1? So we'll let x equal negative 1 and run and find out that y is 3 over negative 4, so negative 1, negative 3 fourths is a point on the graph. Between 0, negative 4 thirds and x equals 3, we can let x equal, well, how about 1? And so y will be 5 over negative 2, so 1, negative 5 halves is on the graph. Beyond x equals 3, we can let x equal 10 billion, and so y will be a large positive number divided by a large positive number, and again, while we don't know whether the result will be large or small, we do know it will be positive. So we'll graph a point somewhere off to the right that's above the x-axis. So now let's put everything together. Since the function is defined for all other x-values, there have to be points on the graph between the graph points, not crossing this line x equals 3. Since this is a function, we can't double back. And remember, we already found all the x and y intercepts, so since there are no other intercepts, we can't cross the axis anywhere we haven't already done so. So while there are many ways we could go from one side to the other, we'll sketch the simplest. Wait, that can't be right, we're crossing the x-axis again, so we've got to adjust this line, and this gives us a preliminary sketch of what our graph looks like. Now later on, we'll learn more about what the graphs of functions look like so our preliminary sketch will be refined into a better sketch later.