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in this episode you’ll learn what a one-to-one function is and we’ll

practice using the horizontal line test yes that’s right there is a horizontal

line test and it will help us determine if an inverse relation is a function

let’s do some math [Music] hi everyone welcome back i’m dave

so we’re going to begin by talking about

what a one-to-one function is and i also

want to remind you about what a function is

so if you’re really really clear on what a function is

this will be somewhat of a review but i’m going to spend most of our time

talking about what a one-to-one function is so let’s start off with

the definition here a function is one-to-one if every element in the range

corresponds to only one element of the domain

so this is what a one-to-one function is

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now i’m hoping that this kind of rings a bell because it’s very similar to the

definition of what a function is so a function definition can be said to be

a function a relation is a function if every element in the domain

corresponds to only one element in the range and so you can kind of see how

these how this has been switched so let me give you a graphical uh way of

looking at all this before we go on with more words

so if i start off here with a collection of objects let’s call this here set a

and we have over here another collection of objects let’s call this set here b

and so we have a bunch of things in here and so these are called the inputs

and f is going to have some way of associating a

relationship between all of these elements right here and some of the

elements over here and so we’ll collect all these elements

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over here that are the outputs right here and we’ll collect them all together

and this will be called the range so that’s called the range of f the range of f

that’s these set these elements right here hey i’ll just shade it in red

this is the range these are the outputs so if you collect together a bunch of

things right here maybe the function f doesn’t care about

all of them so these are the outputs right here so i’ll just say outputs

all right and so what does a function do what does a function make sure because

we have there’s lots of ways to to to relate things so i’ll say right

here that each input everything over here has a unique output so there’s only

one thing over here that it can go to so

if i start looking at this element right here and i ask hey f

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where does this guy go to right here so if this guy right here let’s call him

let’s call this one right here x1 and this one here x2

and let’s say f1 x1 goes to this output right here there it is right there

and then i’ll ask the question f hey f where does x2 go right because f is the

one making the assignment so x2 where where where does it go well

f can say it can go to anything but in order for f to be a function x2 cannot

go here can it yes in fact it can but what f cannot say

is that x1 can go to another one over here if x1 goes to this one right here

then it cannot go to any of the other ones in other words i cannot draw an arrow

let’s call this here one x1 i cannot call make another arrow and go to a

different one over here that would not be a function if f went x1 to here and f

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took x1 to over here also and those are different then that would violate being

a function so that’s what a function is right

x1 can only go to one thing over here now what about

what does it mean to say one to one function so that’s more than just being a

function a function says every input has only one output

but what now we’re going to do is reverse it and say that each output

every element in the range right each output can only come from one element in

the domain so can we now say that x2 can go here

so if we say x2 goes to this guy the same one that x1 did then that would

violate being a one-to-one function so one-to-one function says this cannot

happen that every element in the range even this one right here every element

in the range can only have come from one element in the domain so this is not

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allowed this would be example of two different things went to one thing so

that would be a two to one assignment right there but a one-to-one assignment

means everything over here in the range comes from only one to one

element over here and because it’s a function

every element over here goes to only one over here so it’s a one to one

back and forth but one two one all right so um yeah the words here

really matter and i want you to see not only by diagram

um you know what the definition means but also the words um because sometimes

you get tripped up in the words and then you have to go make a diagram to help

you remember what’s going on which is certainly fine especially when you’re

learning it i would definitely rely more on upon diagrams to help you

remember it but you also need to be able to look at the words so what’s another

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way of saying these words right here perhaps using symbols so if we look at

something like this right here a function is one to one means if

and now i’m going to try to erase this so we can see the picture and the words

so if a function is one to one so this is the set a i’ll just put it over here

so this is a set a here a function is one to one means that

if x1 and x2 are different then their outputs have to be different

in other words i cannot have this this happening right here

we cannot take both of these they’re different

to the same one over here if they’re different over here then they have to be

different over here and that’s what this right here says in symbols

so this would be one to one here if everything over here no matter what

it is not just these two but any two that i pick over here if they’re

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different over here then their outputs over here have to be different so that’s

what it means to be one to one so here i worded it in terms of range and domain

and here i worded it in terms of inputs and outputs

using you know the function notation here

so we can say one more way because this is an implication statement here

it’s an if p then q and um and there’s another way of saying

saying this right here we can use the contrapositive so not q implies not p

so if i take the negation of this that will imply the negation of this right

here and so that’s the next way that you might see the definition of one to one

the function is one to one and now i have to get rid of this means

the negation of this if the negation of that right there then

the negation of this one right here so in other words if these are equal over

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here then these have to be equal over here so any one of these three statements

right here can be taken as the definition of one to one

all these three statements right here mean the same thing here and so now i

also want to contrast and compare them to the original definition that you’re

given in terms of what is a function so a relation

is a function if every element in the domain

corresponds to only one element in the range in other words every input has a

unique output and so then the to change this from right here we would

say a relation is a function means if they’re not equal here

then they’re if they’re different here then that has to imply that they’re

different here and the last one is a relation is a

function and then we can just use contrapositive again if x1 is equal to

x2 then f of x1 has to be equal to x two so these three right here have the same

meaning as of uh as what it means to be a

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function and these three right here have the same meaning in terms of what it

means to be a one-to-one function so really um it takes a little bit of time

um especially but if you want to go beyond you know pre-calculus and

calculus that you really want to have these three statements all in your mind

right here um very fluid uh meaning uh understanding of these and it takes a

little bit of time to get used to rearranging all the words and knowing

the difference between the between them so let’s practice that um

in terms of getting some intuition and by working out some exercises alright so

first thing we’re going to do is play the game

is this relation one to one function or not right so i’m going to give you a

relation and we have to ask the question

is it one is it a one-to-one function so for our first example right here let’s

call this example a and let’s look at the set right here a minus seven four

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and then we’re going to look at minus one nine and then zero five

and then minus two one and then 5 minus minus 5.

all right so this is a relation in other words it’s a set i use set notation and

it’s a set of ordered pairs all right so now we can ask is it a one-to-one

function but notice that that’s two questions first is is it a function

and then the second question is is it a one-to-one function

all right so let’s first ask the question is it a function so function

one to one function so those are the two questions so is it a function

so to check if it’s a function every input has to have a unique output so i’m

really worried about the number of outputs not exactly what the output is

so minus seven has only one output minus five only has one output

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zero only has one output minus two has only one output and five

has only one output if we had another point in here something like zero seven

then this would not be a function because i would have two different outputs

for for zero right if i input zero i’m going to get two different outputs

but this is what we have so this right here is a function so i’ll put check

so it is a function so now the question is is they function

a one-to-one function in other words and now i’m looking

backwards now i’m looking at the four if i input four or if the output is four

how many inputs will i have going to it well if i look at the last condition

right here or the last second entry and all the ordered pairs

uh this is the only one that has a four here so i check them all

they only have one right here so now i check the nine

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and this is the only one where the second entry is in a nine

and then minus five that’s the only one with what sorry five and then one

and then minus five so this one here is one to one function right here

now if we had something like three minus five

so this would be a function right here because every input would have a unique

output but this would not be a one to one function because for the output of

minus five there isn’t a unique input i have two different inputs so this one

would not be a function but this one right here is a function is a one-to-one

function all right so now let’s look at our next example here let’s look at 9 1

and then -2 7 and then 7-4 and then 3-9 and then 2 7.

okay so pause the video and see if you can figure it out is this a function

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and is this a one-to-one function which one is correct

right so in order to have this one you have to first check this one

if this one fails right here then you don’t have a one to one function because

you don’t even have a function so let’s check function first nine has only one

output minus two has only one seven only has one three only has one and two only

has one so this is a function so now i’m just going to check the

outputs one only has one uh and one input seven only had oh

nope if i look right here and right here the output 7 does not have a unique

input so this is not a one-to-one function right there

all right let’s look at one more example how about let’s look at -6 1

and then four minus nine and then zero eleven and then minus two seven

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and then four five and then eight one now i want to show you how to do this

one right here two different ways so the first way is the same way we did the

other two so pause the video and see if you can get it right

all right so let’s check for function first six minus six has only one output

four has only one output zero has only one output minus two has only one output

minus four has only and eight has only one output so this is definitely a

function now is it a one-to-one function so now i’m going to check the outputs

here i have a one and nope nope here’s another one so the output of one

does not have a unique input so this one’s not a one-to-one function so

here’s an example of a function that is not one-to-one function now

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i said i would solve this two different ways so once you check that it is a

function i can then go find the inverse relation

so this is something that you can do if you want but you don’t have to it’s just

a slightly different way so i’m finding the inverse relation

right here in other words i’m just switching all the x’s and y’s

i’m switching all the outputs to inputs and i ran out of room um so 7 minus 2

and then comma and then five minus four so i’m

switching all the inputs to outputs and all the outputs to inputs

and i didn’t do that last one right this should be one eight

all right so let me just check in real quick all right so good so

this one we check that it is a function so then i’m going to find the inverse

relation and i’m going to check if this is a function

so i check one and i have to say nope it’s not a function because it’s got two

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different outputs for this one input so that’s not a function

not a function so either way you’re going to get the same conclusion

the inverse is not a function which means it’s not one to one

so i’d like to like just kind of clarify that

um in a in a lit in a second but first i’m going to go on to what is the

horizontal line test let’s practice using the horizontal line test because

not always are you going to be given a finite number of

ordered pairs what if you’re given an infinite number of ordered pairs

so what is the horizontal line test so also you know what is the vertical

line test we covered that in a previous episode

so let’s just draw a line through here let’s call it y equals i don’t know say

something like uh 2x or something so does this graph right here pass the

vertical line test and the answer is yes let’s remember

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what the vertical line test is the vertical line test means if you draw

a vertical line no matter where you draw it if it only crosses one or less

then it passes the vertical line test if

you could in other words if you can find at least one vertical line where it

crosses twice or more then it fails and so this graph right here passes the

vertical line test no matter where i draw a vertical line it will only cross

once so it passes the vertical line test and

remember what that means is that this is the graph that that can represent a

function and so i’ll go ahead and name that function f

so this is a function right here now this graph represents a function so

now we can ask the question is it a one-to-one function

and we have infinitely many ordered pairs on here so we can’t sit there and

look at inputs and outputs and go through a list

now we need a graphical way of saying is is this a one-to-one function and the

graphical way is the horizontal line test it’s very similar to the vertical

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line test except now i’m drawing horizontal lines so if you can find a

horizontal line that crosses the graph twice or more then it fails so look at

this graph right here no matter where we draw a horizontal line it’s only going

to cross once so this passes the horizontal line test

and what that means is this function is one to one

and what we’re also going to see is that this function that means that this

function right here has an inverse function

so let’s look at another one how about something that looks like this

y equals x squared we can ask does it pass the vertical line test

does this graph right here pass the vertical line test no matter where i

draw a vertical line it only crosses once or less

so yes this passes the vertical line test that means we have a function

that means this graph can represent a function

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now does it pass the horizontal line test so if we can find a horizontal line

that crosses twice or more then it will fail the horizontal line test and look

that’s easy there it failed i cross twice or more it just takes twice to fail

right so that fails so this is not a one-to-one function and the reason why

we can see that it’s not one-to-one is look at this height right here let’s say

the height here is four and the inputs are two and minus two

two squared is four and minus two squared is also four

so here is an output that doesn’t have a unique input

so this right here is not a one-to-one function

not a one-to-one yes it is a function but it is not a one-to-one function here

all right and so in fact let’s just write down here um you know

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how long does it take to be not a one-to-one function right here so i’m

drawing a function i’m drawing a function and as soon as i start to cut back on

itself right there it fails it’s not a one-to-one function anymore all right so

yeah let me draw a couple more examples here let me get rid of this real quick

so let me draw a couple of graphs for you and you tell me is it um

one to one or not all right so i’ll draw this right here it’s gonna go up here

like this is it one to one or not it just keeps going up is it one to one

so no matter where i draw a horizontal line it only crosses once or if i draw a

horizontal line down here it doesn’t even cross at all so this passes the

horizontal line test what about if i draw something that looks like this

just keeps going up so this looks like it’s going to bottom

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out or or become horizontal but it doesn’t it gets infinitely close but

it’s still always it’s just always increasing right there so it’s just always

increasing so i dash it with a horizontal isotope to to show that it’s

always increasing and so this one also passes what about if i do something like

an upside down absolute value function so does this pass the horizontal line

test and the answer is no we found a horizontal line crosses twice

so this one fails what if we have something that looks like a cubic

well some cubics might go like that so that would pass the horizontal line test

what if our cubic looks like something like this though

so then that would fail the horizontal line test that crosses twice or more so

it fails fails the horizontal line test all right so now

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let’s make the connection between one to one and inverse functions right now

so if a function is one to one in other words if it passes the

horizontal line test that’s another way of saying that then the

inverse function exists and this composition right here holds

now in a previous episodes we talked about what the composition is and we

talked about verifying uh whether something was an inverse function so

those are previous episodes that i highly recommend you check out if you

haven’t seen them yet so this is making the connection though

between one to one and inverse functions if it is one to one

then the inverse has to exist and the inverse satisfies these right here

so what about if we consider this function right here is this one to one

function does it pass the horizontal line test

if it does we’ll know the inverse exists and this is important for the next

episode because in the next episode i’m going to show you that if you know the

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inverse exists then i’ll show you a way to actually find it

so let’s look at this function right here now also from previous episodes we

talked about parent functions and how to apply transformations so we know the

parent function here is to the third power so we can graph something like

this right here we have it memorized so in terms of what it looks like so it’s

just gonna go through here like this and now we also know what the

transformations are here we’re going to shift left to and we’re going to shift

down one so i recommend checking out those videos

also so i’m going to shift left to so we’re taking this point right here and

shifting it left and so now it’s going to look like this

but it’s also being shifted down one so let’s move it down one so minus two

minus one and then it’s gonna have the same

shape right here where it’s increasing and then it’s still increasing so i’m

gonna keep the same shape right there and then now where’s across the

x-axis right here let’s be a little bit precise where is it cross-x axis so

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[Music] where is this whole thing zero right here

or where does it cross the y axis also so when this is 0 that’s going to be a 7

right so that would be when this is 0 that would be 8 minus 1 so 7.

so it’s going to cross somewhere like that and just go up

so there’s about a 7 right there all right so yeah it’s going to curve

like that and then go up like that all right in any case this passes the

horizontal line test no matter where you draw a horizontal line no matter where

it only crosses once so right here it does not look like this just want to

make this clear it does not look like that where there’s some arc to it right

there that’s going to make it fail so it just keeps increasing and then it

keeps increasing like that right there so this passes a horizontal line test

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passes the horizontal line test so one to one it is so one to one

and so what this is saying is that the f inverse exists

now from what we’ve seen before i even showed you a step-by-step method on how

to find the inverse but we’ve talked about it informally what would we think

the inverse is informally just you know trying to undo everything

that’s being done and remember the way that you want to think about that is

what happens when you input what are the steps that you do

and you try to undo those steps so first when i input something i’m going

to add 2. that’s the first thing i’m going to do is add 2.

so the inverse the last thing it will do will be to subtract 2. so i’m going to

put a minus 2 all the way out here because that’s the last thing it’s going

to do and that’s because f the first thing you do is add 2. so now the last

thing so i’m going to do a bunch of stuff but then the last thing i’m going

to do is -2 all right and then what do we do after

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that here then we’re going to cube it and then i’m going to subtract 1. so i’m

subtracting one does the last thing over

here so over here it should be the first thing is to undo subtracting 1. so i’m

going to say x plus 1 and then i’m going to say cube root of that

and then the last thing will be minus 2. so this would be my guess for the

inverse function now how do we know that that’s right how

do we know that this is the function and that this is the inverse function it’s

because both of these compositions will hold

if i check this one right here and this one right here and i combine them

together using composition it turns out to be the identity function in other

words whatever you put in is what you get out so again we did a whole episode on

verifying when this function right here and this inverse function right here if

they actually satisfy these two things i highly recommend checking that out

all right so that’s what this right here is saying if you know it passes the

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horizontal line test then you know it’s one to one and then you know the f

inverse exists all right so now let’s look at

kind of putting all this together all this um language here and all these

ideas put them putting them together so here we go so the graph of a function

passes horizontal line test then the function is one to one if a function is

one to one then the f inverse exists if the domain of f is uh okay so this is

just making the connection between these two what does it mean to be inverse

right so the domain of this one is the range because that’s what inverse means

it means switch x’s and y’s so if the domain of f

and then you switch x’s and y’s so that’s the range of the inverse and the

range of f is the domain of f inverse and then the next one is for a function f

and it’s inverse you can always verify that these are two are inverses of each

other by checking these two composition equations making sure that this function

00:28

and this function there both are the identity function

and then now for the last one here the graphs of f and f inverse are

symmetric with respect to the line y equals x and that was the previous

episode we went over how to show that by hand and also by using python so let’s

talk about this function again just one more time we saw what the graph of this

looks like it’s been shifted to and then down and so it looks something

like this right here where this point right here was minus 2 minus 1

and it passes the horizontal line test so that means this function is 1 to 1

which means this function has an inverse which means

that the domain of this function will be the range of the inverse

and the range of this function the outputs will be equal to the domain of

the inverse and we can verify that this is the

inverse by checking these two equations and then last but not least is if you

00:29

graph the inverse function and the original function you’ll get symmetry

with respect to the line y equals x all right so there’s all five of those

put together right there all right so now one last thing

um in this episode which is hugely important is because sometimes

people don’t want to give up they don’t want to say hey that doesn’t pass the

horizontal line test it doesn’t have an inverse well we’re going to make an

inverse anyways so let’s look at something like this

we’re going to say here is x minus 4 and then we’re going to square it so x

minus 4 squared now what does this look like well

we have a minus four here so it’s been shifted right four units and it just

looks like our squared a regular square function right here so it’s just going

to come through here like that and then go back up here like that and there’s

our graph right there and as you can see this does not pass the horizontal line

test if i throw in a horizontal line there it fails

00:30

so this does not pass the horizontal line test which means it’s not one to

one f is not one to one f fails the horizontal line test

f does not have an inverse function f does not have and inverse function

let’s just put little little bullet points here right if it’s not one to one

f fails horizontal line test f does not have an inverse function so

those three things mean the same thing so we’re going to be hard-headed though

we’re going to look at this function which i’ll call capital f

just to distinguish it from this one so i’m going to keep the same thing

but now i’m going to put a restriction on the domain

00:31

so this is chopped right here at 4 so let’s just look at the right branch

or perhaps you’d like to look at the left branch

so i could actually take either the left or the right or in fact we could

restrict it any other way if we want we could restrict it five or greater but i

think it’s pretty natural to just look at the right half at four so

i’m going to say here’s my restriction right here so this

is going to be a new function which i’m calling capital f it looks just like

this function right here except we’re only taking the right branch so i’m

saying x is greater than or equal to 4. so capital f is just this part right

here i’ll go and put this in blue well no blues for those dots so there so

i’ll say this one right here is red here so this part right here is just the

right branch that’s the graph of capital left right there

yeah so there’s the graph of capital f right there now if i just look at

capital f the one in part in red that does pass the horizontal line test

00:32

it it will always pass the line test so this function right here f is one to one

f passes the horizontal line test f does have an inverse oops sorry

those should all should be capital fs f capital f is one to one capital f does

pass horizontal line test and then capital f inverse exists

and so this is a great time to mention that in the very next episode i’m going

to show you how to find the inverse and we can stop guessing informally about

how what the inverse is i think that’s valuable though because if you can

informally figure out what the inverse is that

gives you a lot of intuition in terms about what the inverse is how it undoes

it and and not but we really want a formal method step-by-step instructions

that you can follow to get to the inverse and that’s we’re going to cover

in the next episode so i look forward to seeing you then

let me know how you think about this video give me some comments below i’d

00:33

love to read them and have a great day see you next time

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