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in this episode you’ll learn a graphical

method for determining whether or not a graph represents a function it’s

amazingly simple and so you got to try this let’s do some math [Music]

okay we’re going to begin with the definition of a function

uh which we covered in previous videos but i want to just quickly review it so

function from a set a to a set b is a relation that assigns to each element x

in the set a a unique value y in the set b

and the set a is called the domain and the range is the set of output values so

we’ve talked about this in the previous videos

but in this video here this episode here we’re going to uh talk about uh a

graphical way of determining if this function is if this definition is

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satisfied so for example what if we have something like this what if we have a

2 here and we have a 4 here and let’s say we have a minus 7 here

and let’s say the function or the graph has a point here and a point here on the

graph let’s make that a little bit more there so it’s going through

so this is the point here 2 4 and here’s the point 2 minus 7. and you may wonder

can such a thing happen on a on a function with a function and the

answer is no because the input two would

have two different output values so i’ll call these the x’s right here and this

the y’s right here and we see for this input of two there are two outputs

if we can make the situation even worse by adding say a minus 11 as another

point on the equation or the graph sorry on the graph

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and so you know this is not going to represent a function these three points

right here are not going to represent a function as we did in previous episodes

we graph or we represented this as a set of ordered pairs 2 4 and 2 minus 7

and 2 and minus 11. so this is a set of ordered pairs it’s a relation

but um if you if if you name this relation say s and then you try to talk

to s and you give s input of two it will s will reply back with three outputs

four minus seven and minus 11. so this is an example of something that’s not a

function now if i took away these two points here

and just had this point here that ordered pair all by itself would be a function

because if i input 2 into t it gives me only one

output it doesn’t really matter what the output is in terms of whether or not

it’s a function it is having one output t only has one

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output for that given input forgive for every given input in fact okay so we’re

going to look at a graphical way of determining all of this rather than

looking at a set of ordered pairs because a set of ordered pairs here is

finite and what if you’re dealing with infinite number of points you would like

a more intuitive way of determining if this technical definition right here is

is working is true or not satisfied by given relation

now before we do that though i just want to mention though that in the previous

episodes we covered uh what functions are in detail in this video here

and in this video here we talked about piecewise defined functions so we worked

out lots of graphs and we even took a look of how to graph them using computer

in this in this episode here and then in this episode right here we

uh in the last episode um we talked about finding the domain and range of a

function so these are three really good uh

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episodes right here to watch before we continue on

so i hope you check them out the series is called functions and their graphs

step-by-step tutorials for beginners so i’m assuming that you’ve already checked

out these videos and now we’re going to give a graphical way of trying to

determine if something is a function or not so let’s get started so the first

is what is the vertical line test so we’re going to determine if the graph

passes the vertical line test so i’m going to give you six graphs right here

and we’re going to determine if they pass the vertical line test so i’ll call

this one here um a or let’s go with one and then let’s say here

let’s say we just have a line going through here like that

and then number two we’re going to have here a

now if you’ve seen the vertical line test or you heard about it before but

you’re not quite sure then pause the video when you get all six here and

you know think and try to think on on your own

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which one of these past the vertical line test i’m obviously going to explain

to you in a moment this is a sideways parabola it’s upside

down parabola this is just a line and for 4 here we’ll have like a function that

just like this and then for 5 we’ll have a graph that looks like this

i kind of gave it away i called that a function there so forget i said that

and then this one will look something like this

or maybe i should draw that one first yeah try to draw a good shape first uh

it’s still kind of oval all right that’s good enough

something that looks like that and then this one right here

i’ll put a restriction on this one right here so i’ll say

looks like an upside down parabola but i’m going to put restriction here and a

restriction here and let’s just say this is 2 and -2 just

to give it some numbers there all right so

for these six right here pause the video make a conjecture say which one passes

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which one does it pass all right all right so now let’s go on and let’s see

which one passes the vertical line test what exactly is the vertical line test

okay so vertical line test means we’re drawing vertical lines and you can draw

a vertical line wherever you want and you try to draw a vertical line it

as many of them as you want as many as you can so i’m just doing this to

represent a vertical line a vertical line and we’re going to ask the question

how many times does the vertical line cross the graph how many times does it

cross for example if i draw a vertical line here it crosses one time if i draw

a vertical line here it crosses one time if i draw a vertical line here it

crosses one time in fact if you keep thinking about it if you keep drawing

vertical lines how many times does a vertical line cross

this graph right this graph in black here so no matter how many times you draw a

vertical line it only crosses the graph once this keeps going down this keeps

going up or the whole graph keeps going down forever but

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no matter how many times you draw a vertical line it’s only going to cross

once now if it crosses once or less it’s going to pass

but as soon as you find a vertical line that crosses twice or more

it’s going to fail it’s going to fail the vertical line

test so that’s what the vertical line test is so let’s go on to number two so

this one passes all right so let’s look at number two

here so i draw a vertical line over here it doesn’t even cross so zero is okay

i’ll draw a vertical line over here it doesn’t even cross i’ll draw a vertical

line over here it crosses once i draw a vertical line over here crosses once i

just keep drawing vertical lines over and over again until you convince

yourself how many times will it cross at most if it crosses at most zero or one

time then it is a function if it crosses twice or more

then it automatically fails so to fail all you need to do is to find one

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vertical line where it crosses twice or more then it will fail this one right

here also passes how about number three here

so i draw a vertical line over here zero doesn’t cross at all

here doesn’t cross at all here zero so i keep going until i

convince myself that it will not cross twice or more i draw one here here it

crosses only one time one times okay now as soon as i draw one here though it

fails the vertical line test it cross twice or more it’s going to fail so this

one fails this one fails the vertical line test

what about this one right here no matter where i draw a vertical line it only

crosses once only crosses once this passes

this passes the vertical line test now what does it mean to say you pass the

vertical line test it means you have a function it means that every input will

have a unique output so what about this right here

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we found a vertical line it crosses twice

it fails this graph right here does not represent a function

what about this one right here so if i draw a vertical line over here zero

that just means we’re not in the domain crossover over here it’s not in the

domain so right here it crosses the graph one time

here one time even right here on the boundary one time this one here passes

the vertical line test also passes the vertical line test so that’s what that

means is this graph right here can represent a function and we can

um or it does represent a function the graph of this right here does represent

a function right here all right so there’s uh six examples

there hope you got them all right um and let’s do some more

so let’s get you six more here now so now that you’ve seen what it is let’s

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test to see if you’re getting it down or not by

looking at six more so this first one here is going to be something that looks

like this so i’ll call that number one for example three here

and then number two um let’s look look something like this and then

something looks like that and then three here let’s go down here

so let’s look at something that looks like this this is going up here like this

and it’s getting really close and it goes like that and then number four

um let’s look at something like this this one’s uh shifted down here like that

and then number five let’s look at something looks like that

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well messed that up let me try that again all right that’s good enough

and then something that looks like this right here

so i’m going to come in here like this all right

so suppose this looks symmetrical although that’s not required for this

example here all right so there’s six examples there

so pause the video now and see if you uh which ones you um

like and uh which ones you could figure out here

all right so let’s go through these right here so number one does it pass or

fail so let’s look at um let’s use a red marker this time

so or how about orange all right so this one passes no matter where i draw a

vertical line it will only cross one or less times so this one passes so i’ll

put a check mark for passes number two uh well actually let’s just

say passes some people like to skim videos right so number two uh right here

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so no matter where i draw a vertical line it will only cross once or more or

sorry once or less so this one also passes this one right here number four draw

vertical lines right here no matter where i draw a vertical line it will

cross once or less this passes this one right here um

no matter where i draw a vertical line it crosses now it doesn’t go actual

vertical here it looks like it does uh potentially because just of my sloppy

drawing but it’s not meant to it’s meant to get really really close and then it

passes through right there so if it does if it did go vertical i

need to be more clear about that so i’m going to say this one passes here

because it didn’t get vertical there this one fails right here i draw a

vertical line right there cross twice twice or more so this fails

and then this one right here um well actually maybe i didn’t draw

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this one here good enough i kind of ran out of space let me try again

to make it extra clear that it’s not a function

and so i’ll just put the arrows on here this one kind of reminds you of a

hyperbola if you know what that is but you don’t need to know what that is if

you draw a vertical line i found one here that crossed twice or more

and then so it fails so if you can find a vertical line that

crosses twice or more it fails so that means that this graph here cannot

represent a function this graph can represent can represent can these two

right here fail all right so there we go there’s six more examples

and last thing is let’s try six more all right so this time i’ll start

numbering at seven so let’s get this erased real quick

all right let’s try seven more or six more all right so seven here will look

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like this it’s gonna come in here like that and then it’s gonna bounce

and then eight will look like this come in here like this

will come in here look like this and number nine will

let’s draw something fun let’s draw something that looks like this about

like that you see these when you start working in polar coordinates later on

something looks like that looks like a flower almost

and then how about something that looks like that

and it’s symmetrical down there if we could get that symmetrical there

and then how about something that looks like

this this is a step function so it’s going to be i’m going to draw the step

function in red and so we can see the actual step so i’m

going to say open and then closed closed right here and then it’s going to

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step up and then go across and close it’s going to step up and go across and

close it’s going to keep repeating that pattern right there

so let’s just do it four times right there so we can see that

so it’s like a step function it’s we’re not including the origin and then we’re

including all these and then we’re including the end point and then we step

up and go across again um so and then the 12th example here

let’s look at something that levels off and see

about that so we got levels off right there and then and then it comes down

and does like a parabola thing make that a little curvy

all right so something that looks like that all right so which one of these

let’s get a guess for you write it down your paper

guess which one it is pause the video try to figure it out don’t go fast

and then we’ll do it okay all right so here we go i’m gonna do it right here

number seven right here now when i’m looking at number seven right

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here it doesn’t go vertical right here it’s just going towards the origin and

it doesn’t go vertical right here so if it had done something like this

and it went straight down and then something like that

and then this was the axis right there through here

the y-axis and the x-axis so if it went straight vertical down like that then

that would not be a function right there

but that’s not what that’s doing so this one right here passes so let’s put

passes in red here number eight here now um when i mar when i mark it right here

i’m actually missing the isotopes here so i’m going to dash this in with an

isotope so we’ll learn what isotopes are better later

but what what that means is that this graph is going to keep going

down but it’s not going to cross it’s not going to cross the x-axis

and this is going to keep getting closer and closer to the y-axis but it’s not

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going to cross the y-axis so we start drawing vertical lines

we cannot draw one that crosses twice or more so this one passes all right

so what about this one right here so i’m going to draw a vertical line and

it fails it fails because it crosses once twice

three times four times twice or more is bad so this fails right here

as soon as i find one that crosses twice or more it’s not a function this one

right here fails all right so what about this one right here the step function

now because these are open and this is closed if i draw a vertical line right

through here it will only cross one time if i draw a vertical line through this

part right here only one time if i draw a vertical line

over here zero times right so this passes

this is called a vertical step function and there’s lots of examples of them

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depending upon how much you’re stepping and so on um

if you step out of uniform you know like one two three four that’s

a very nice function right there and it goes off one and two and three and four

so you can you can scale it just like you can scale any kind of staircase you

might find you can make these longer you can make the steps bigger there’s all

kinds of step functions but here’s example of one it passes the vertical

line test now if i fill this in right here if i fill that in then i could have

found a vertical line that crossed twice and then that wouldn’t have been a

function but that’s not what we were given right so this step function right

here it has to be closed right here where the other one meets right there

all right so there we go that passes the vertical line test

this one right here this last example right here

if i draw a vertical line over here it crosses only once or more

even where it’s connected it only crosses once

no matter where i draw a vertical line so this one right here

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passes the vertical line test all right and so there we go 18 examples of

of of graphs and we determine each one if it passed or failed the vertical line

test so i hope you enjoyed this video and i look forward to seeing you in the

next episode thanks for watching if you enjoyed this video please like

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