 So in this video, I want to talk about the vertical line test So imagine we have a set of points in the xy plane and it gives us the graph of some equation Or it's just it's a collection of points, right? Well, that graph will be the graph of a function if and only if every vertical line intersects the graph at Most one point it could be that you have an inner vertical line outside the domain of the graph So therefore doesn't intersect it at all. That's fine, but we're looking for graphs whose Will intersect vertical lines only at one point. So you look at this first example this This this yellow line right here This is the graph of the equation y equals 2x minus 1 although we don't need to worry about that equation right now You'll notice that as you start drawing vertical lines here along this graph It intersects the graph only at a unique point for each line right different lines will have different Intersections but each line intersects the graph at only one place and therefore this graph would Get a pass a pass with the so-called vertical line test which was to say that this is a function This is the graph of a function a Function graph will always pass this vertical line test and the reason behind that is if there were two different Two different points that shared the same vertical line So we have like this right here two comma say four and two comma say two Right here these two points because this is a graphical representation of the function This would tell us that f of two equals four, but it would also equal f of two would equal to it can't be both, right? The function evaluation has to make a decision It's got to be one or the other f of two either equals four or it equals two or something else make a choice already If our graph failed the vertical line test that would be the predicament. We will be trapped in right there We'd be trying to sign two different y coordinates to the same x-coordinate The second picture right here. This is a parabola. In fact, it's the parabola y equals x squared You can see that every vertical line intersects the graph at exactly one point So this one also is going to be a pass to the vertical line test So this is also the graph of a function Finally though examples see here. This is the graph of the equation y squared equals x q plus 1 We can actually see here that there are instances where this graph Does not pass the vertical line test This has a situation where the y-coordinate is one and the y-coordinate is negative one. It can't be both, right? x equal f of zero would have to be one or negative one can't be both, right? You have to make a decision so with regard to the vertical line test This is actually a fail. It fails the vertical line test. And so that tells us it's not a function Determine whether a graph is a function or not is as simple as running this vertical line test Now I do want to make one one slight disclaimer right now that when you look at the two the two graphs that were functions The first one passes the horizontal line test, but the second one fails The horizontal line test that has no bearing on whether it's a function or not I mean heck the the elliptic curve here it passes the horizontal line test to be a function You must pass the vertical line test and you don't have to worry about anything else It does turn out that the horizontal line test will have some bearing on Properties of the function, but that's a topic for another day And so we can very easily test whether a graph gives us a function or not from the vertical line test algebraically This is a little bit more challenging, but that's a topic for another video