hey there welcome to another video in this video we're gonna discuss the vertical line test it could be very very short you've probably heard of the vertical line test before if you haven't that's not a big deal I'll explain it very well I'm gonna focus more on the why it works than that how're you doing I'm explaining both of them but mostly the why it works that's how we retain things by understanding them so what is the vertical line test why does it work and how do you do it the vertical line test is a way that we check whether the graph of some sort of mathematical expression or equation is a function or not so when we when we look at a graph we should be able to determine immediately whether it's a function or whether it's not so how we do that is by understanding the interplay that functions have between inputs and outputs remember four functions one input gives you one an output that's it so what we're gonna focus on is why the vertical line test actually works in order to start we've gotta know that the input axis I'm not going to really talk about X's and Y's then put axis is the horizontal axis unless we switch that around then ordinal accesses is for inputs so with that in mind if this is the input axis and so the output axis when we pick a number to plug in on the input axis it's kind of a horizontal idea is if this was X you'd say all right exit equals 5 is here in x equals negative 4 is over here and we move horizontally to locate different inputs however for any given input the output will be somewhere above or below it that's why our output axis we call vertical because it has this idea that our outputs will be strung along somewhere along this direction above or below each individual input that's how the rectangular coordinate system works you go over to input and you up or down for your output we all know this because we can graph points at this point so if our input axis is horizontal and at every given input the output will be somewhere above or below this a vertical idea then all we have to do is go over to you give an input you go right here's my here's an input and we checked vertically how many outputs are given to us so we look above it is their input or sorry is their output above or below it is their an output below our idea is that if one input must give us one output and our inputs are somewhere along the horizontal line we're gonna go somewhere along the horizontal line we're going to pick a point and then we go right look above below it because if there's an output for that particular input it's going to appear vertically the output axis is vertical there for all of our outputs must be vertically given above or below that input and if we find one input great at that point that's a functional relationship if we find two outputs that's a problem that's given it's a non function for that input and that ruins the whole thing for today and this whole thing's not a function because right here see this at that point well we have two outputs for one input and we do this for the entire graph so that's why the vertical line test works it's based on the idea that for functions one inputs got to give you one output since inputs are located horizontally we pick a point on this horizontal axis anywhere that you want and once you pick a point basically we do for the whole axis once you pick a point on there then you look above and below because their outputs must be strung along vertically if we touch our graph at more than one point well then at that input you are getting two or more outputs that is a non function it's giving you more than one output for that single input we're gonna practice a few of them but that's basically the rundown here so take a look if there are our diagonal line our input axis horizontal has purposely not labeled these x and y because it doesn't really matter our input axis is horizontal or output axis is vertical we look at every point along our horizontal axis and think oh right well what's going on here what's where would our outputs be located at this point outputs would be vertical well if my outputs are vertical and I started crossing my graph right here that says hey there's an output for that given input isn't there that says if this input I would get that number out and this input I would get that number out and this I would get that number out because for every given input on the input axis our vertical axis where it touches the graph that would be our output are we touching the graph more than one time for any given one input I know that we touch the graph three times but that was for three different inputs that's fine and if we continue across the whole axis no matter what even here that's they the output is on the axis no problem and we look vertically and everywhere we touch the graph that is the output for the specific input that we've chosen every single time that we have checked an input and we do this across the whole axis we are getting one output right there that says that one inputs give us 1 output it's not touching the graph twice the vertical line test here says that that is a function what about the next one imagine some vertical lines imagine choosing and input here in here in here in here all the way down here and imagine vertical lines at every single one of those inputs those vertical lines are the outputs or the representation of where the output would be for that given input and where we hit the graph here and here here and here those are the outputs for those given inputs that we've chosen are we getting more than one point on a vertical line the vertical line represents outlets for that input so if we're getting more than one touching upper graph more than one point on a vertical line we would have a non function well I got invest that way in the up here well no we're not we're getting one point for this vertical line one point for that one one point for this one one point to that one there's never a case where we're getting two outputs for any given input this is a function that one graph the back what do you think that if you look at the alright taking input here great inputs are on the horizontal axis no problem and output would be somewhere along a vertical line the vertical axis is the output axis our outfits would be a vertical idea having chosen and input there you go all right well here's my output axis like here's where my outputs would be let's look at the graph there's output there there's an output there those are the two outputs on that vertical oh wait those are the two outputs on the vertical line for that input so I chose my input I looked at all the potential all the outputs that we have and we cross the graph twice same for that graph there are two given outputs for that one give an input right there you're done you stop and you say that's not a function because I put in one input I chose one point on the input axis I looked at vertically above and below it and it gave me two points out that's not in the function this one for sure as a function we can imagine some vertical lines of all these different inputs we think through and go you know what every vertical line you don't even have to draw them you can imagine them every vertical line that we think about is only going to have one time crossing that graph that means that every of though every one of those inputs that I chose for those vertical lines or every possible input on that horizontal axis is probably a better way to put it for every possible input on the horizontal axis and I imagine an infinite number of vertical lines there's never a case where a vertical line is going to have two outputs it's going to touch our graph twice that does not happen here this is a function because for every possible input I'm getting one output I don't let I need to backtrack per second is it okay to get no outputs for an input yeah and it can still be a function actually if we think about square roots and a square root looks like this even though these inputs over here are not defined for that square root they create imaginary numbers so when we plug in negatives instead into a square root we learn this in there's nothing there for real numbers so it doesn't show up graphically on a real coordinate plane this does not say that you don't have a function and this over here for every possible input we have one output that is also a function so don't I should have probably mentioned that when we get to there but don't let it be the casing man I'm not getting any inputs therefore it's not a function that's not the case the cases is this where you have inputs on your horizontal axis and you have something that's defined for your function so where your inputs give you an outputs there's only one that's the idea so where the inputs are giving you outputs there's only one output and not two so function yes function yes function now there's two function yes function note for this input or lots of them you're getting two outputs for that input you're done that's not a function lastly how about the vertical line test can a vertical line pass the vertical classic vertical line test can the vertical line pass a vertical line test well let's think about it what if I pick it in four over here are there any other inputs russer outputs for that input no that's fine if you are not worried about inputs not giving us outputs worried about where the inputs are giving us outputs they're giving us only one so we check all these hey there's there's no problem with any of these vertical lines but as soon as I get took that input right there and say hey what's the one output for that one input you go there's lots of outputs for that one input in fact every point of vertical is an output but that's a non function and I hope that makes sense to you I hope that that you're seeing the interplay between inputs and outputs on the vertical line test that we're checking inputs that's a horizontal idea so you locate one and you look above and below it for a now and if it has outputs there can only be one output from it so yes yes no that's a problem yes that's fine no that's a that's not fine I know that's not fine fruit of line does not pass through blind test here this is certainly a function so a square root says I'm not getting any outputs therefore the verb like this doesn't really apply but where I get outputs for my inputs there's only one output for every given input that's how the vertical line test works that's why it works and so we're going to be using that from time to time just to check to see whether we have functions or not or when we're graphing like especially piecewise functions to make sure that we're not actually accidentally overlapping or refs because that wouldn't be a function so I hope it makes sense I'll see you for the next video you