The Vertical Line Test for Functions (And Why Its So Useful)

Video Series: Functions and Their Graphs (Step-by-Step Tutorials for Precalculus)

(D4M) — Here is the video transcript for this video.

00:00
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

00:01
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

00:02
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

00:03
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

00:04
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

00:05
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

00:06
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

00:07
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

00:08
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

00:09
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

00:10
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

00:11
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

00:12
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

00:13
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

00:14
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

00:15
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

00:16
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

00:17
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

00:18
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

00:19
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
and subscribe to my channel and click the bell icon to get new video updates

About The Author
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David A. Smith (Dave)

Mathematics Educator

David A. Smith is the CEO and founder of Dave4Math. His background is in mathematics (B.S. & M.S. in Mathematics), computer science, and undergraduate teaching (15+ years). With extensive experience in higher education and a passion for learning, his professional and academic careers revolve around advancing knowledge for himself and others. His work helps others learn about subjects that can help them in their personal and professional lives.

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