 A picture is worth a thousand words, and in mathematics the equivalent of a picture is a graph, and the equivalent of a word is a function, so can we graph functions? Well, I hope so, otherwise this will be a very short video. So let's lay down some introductions. If f of x is a function of x, we can speak of the graph of y equals f of x. And just like when we graphed equations, what we mean by this is we mean the graph that consists of all points x, y, where y equals f of x. Well, with the exception of the use of function notation, this is really the same thing we did when we looked at graphs the first time. So let f of x be some function of x. We can determine which points are on the graph of y equals f of x. So the underlying concept remains the same. If the point 5, 3 is on the graph of y equals f of x, then x equals 5, y equals 3 will satisfy the equation y equals f of x. So we'll check. So we know we're dealing with y equals f of x equals, means, replaceable. So instead of f of x, I can replace it with x cubed minus 7x squared plus 15x plus 10. So equals means replaceable. I claim x equals 5, y equals 3, so we can replace x with 5 and y with 3. Now, we should be at least careful. We don't actually know that the point 5, 3 is on the graph, so we don't really know whether 3 is equal to this mess. So we'll put that question mark there and we'll do some calculations. And this statement, unless you're a politician, is false. And since this is false, 5, 3 is not on the graph of y equals f of x. Again, if it's not written down, it didn't happen. And so what we might write down here is that since f of 5 is not equal to 3, then 5, 3 is not on the graph of y equals f of x. Similarly, we can check to see if 220 is on the graph of y equals f of x. So equals means replaceable. So again, making all of our replacements, doing a little arithmetic. And this is a true statement, so 220 is on the graph of y equals f of x. And again, we can summarize it since f of 2 is equal to 20, then 220 is on the graph of y equals f of x. How about some intercepts? Let's find the x and y intercepts of the graph of y equals f of x, where f of x is x squared minus 6x minus 8. So the y intercepts will occur where the graph of y equals f of x crosses the y axis. So remember that for any point on the y axis, x is equal to 0. So we want to find the points 0, y that satisfy the equation y equals f of x. Equals means replaceable, so let's replace and solve. So when x equals 0, y equals negative 8, and so the point 0, negative 8 is on the graph of y equals f of x, and it's going to be the y intercept. The x intercept will occur where the graph of y equals f of x crosses the x axis. And remember, for any point on the x axis, y is equal to 0. So we want to find the points where y equals 0 on the graph of y equals f of x. Equals means replaceable, so we'll replace and solve, which gives us our x coordinates. And remember, that's just half a point. We need to include the y coordinates to have the full x intercepts. And so our x intercepts will be, what about the x and y intercepts of the graph of this function? So the y intercept will be the point or points where the graph crosses the y axis, so equals means replaceable, and so the y intercept is 0, 7. How about the x intercepts? The x intercept will be the point or points on the graph where y is equal to 0, so replacing and solving. Now, while we do have a solution, the thing to remember is that any point x, y in the plane must have real coordinates. Since the solutions to this equation do not give us real coordinates x, then there is no point where the graph crosses the x axis. There are no x intercepts. A similar situation happens for the graph of y equals f of x, where f of x is 1 over x. The x intercept will be where y is equal to 0, so replacing and solving we find. We can't solve the equation, so there is no x intercept. We can try to find the y intercept, which is where x is equal to 0, replacing. But we can't, since we can't let x be 0, and so there is no y intercept. Let's see if our picture, our geometry, can give us some insight into our words, the algebra. So here we have a graph, and let's see if we can solve based on the graph of y equals f of x, f of x equals 0. Also, let's see if we can find f of 0. So the thing to remember is that since the graph shows y equals f of x, then any point on the graph x, y satisfies the equation y equals f of x. We want to solve f of x equals 0, equals means replaceable, we want to find points on the graph where y equals 0. Those points will be on the x-axis, 5, 0, minus 3, 0, and 2, 0. Now this time, because we're actually trying to solve an equation in x, we only need to give the x values. So the solutions will be negative 3, 2, and 5. Now we'd like to find f of 0, well we know that f of x equals y. So let's try to solve f of 0 equals y, so we want to find the point on the graph where our x value is 0, and this point will be the point on the y-axis, 0, 30. So again, equals means replaceable, we want to find f of 0, f of 0 is y, and our y-coordinate is 30. So f of 0 is 30.