all right so we're going to do just two things in this video the first is the uh vertical line test explaining what that is and drawing a piecewise function with three parts okay so let's start with the one so we've seen before algebraically that something like y squared equals x plus one is not a function that is y is not a function of x so the way we saw that was we plugged in say something like x equals uh zero and then we got y squared is equal to 1 which meant that y would be plus or minus 1. so which meant that for a single value of x we had two values of Y well well we can another way to do this is to think about the graph of x equals y squared minus 1. now even though it's natural usually to um plug in a value of x and then solve for y you can do it the other way too so for example if you plug in y equals um minus one or one you get x equals zero so you have the point zero one zero minus 1 which we just solved for actually so let's put those values on the graph and if you plug in y equals zero then you get X equaling minus 1. and this actually is this is going to be a sideways graph it's going to be a parabola opening up sideways this is actually the vertex point of this graph so we actually had something like this so the fact that there is two points right this point Y equals one and Y equals negative one for the particular value x equals zero which we saw algebraically can be visually thought of as realizing that this vertical line or actually a lot of these vertical lines have crossed a graph at multiple points and that exactly corresponds to the idea of that a particular x value corresponding to two different y values or multiple y values so if a graph or just a curve even if we don't know the formula if a graph curve has the property that a vertical line particular vertical line just just has to be one you don't need multiple ones if a vertical line passes through it at multiple points then it is not a function I.E Y is not a function of x so we say then that it fails then it fails the vertical line test then it fails the vertical line test so if you take all of the graphs that we normally look at you know things like Y equals X cubed or y equals the square root of x all of these past the vertical line tests meaning if you draw a vertical line either it doesn't pass through it at all so for example this line right here doesn't pass through the graph at all or a line passes through the graph only once and that these past the vertical line test so these pass the vertical line test meaning they are functions and they should be because we can write them explicitly as y as a function of x okay so this is the vertical line test and you could do it for any you know if you had some random um graph even like this so this is just a random curve I don't know what it is but notice if you take a line say right here something like that it passes through it multiple times so this would fail the vertical line test even though say a line over here would only go through it once so all right so all you need is one line that crosses to the graph vertically multiple times and if it fails to be a function all right let's move to the second item which is a function that is made up of multiple pieces so let's say we have negative 2X cubed and a constant function one let's say this is X less than or equal to one um one less than x less than two and x squared are equal to two okay so this is my function I've defined it piecewise in three pieces and less than or equal to one so less than or equal to one my graph is a negative 2x so here's x equals one but at one if I plug in one I would plug in 1 into here I'll get negative two so it would have a point one comma negative two right here it would be a closed Circle here because it's including notice this includes one I have that equal to sign right there and then I have negative 2x so I'm going um well let's plot maybe one more point if I plug in 0 I have zero so there's another point right here zero zero I draw this line which is the line negative 2X now here what I do is I plug in 1 again but I note that one is technically not part of this graph I have one Cube which is one so I have the point one and one here but this is an open circle because I'm not including that point if I had the point say 1.5 I would have 1.5 cubed and at 2 although I want to reach 2 2 cubed is eight so essentially I'm looking at the rest of the graph that grows through an 8 is somewhere up here so we would be if we had part of the graph we would be looking at something the X cubed graph we would have something like this and it would go up through eight of course we don't have this piece because that's not part of the graph it's only from one to two so we only select the piece between one and two we have open circles at 1 1 and 2 8 because we're not technically not including where we have um uh the 0.1 and the 0.2 in X cubed so notice these open circles at 1 1 and 2 8 and those correspond exactly to the fact that we have a strict less than sign one less than x less than two then at 2 if you plug in 2 actually you get one so we get a solid Circle here and then it's just a constant function so it's y equals one all the way for the rest of time so this is the graph of f of x notice it's still if you ever fail the vertical line test if you ever you know if if this graph if this green part went down a little bit further then it would fail the vertical line test I would be able to draw a red line I would be able to draw a line here and it would it would make it not a function so if you have you know if you ever make a graph too long it'll fail the vertical line test so we have to delete this piece and now it is indeed a function and this is how you draw the graph okay so that's it for piecewise functions we're going to move into applications of lines and linear functions in the next video