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in this episode we’ll practice sketching

the graph of a function and its inverse on the same set of axes the symmetry is

amazing you got to see this let’s do some math [Music]

hi everyone welcome back i’m dave we’re going to start off by uh asking

the question what does the inverse look like

now by inverse i mean inverse function and before we before we get started i

want to say that this episode is a part of the series

uh functions and their graphs step-by-step tutorials for beginners

so find the link below to the full playlist

below in the description and so in the last episode we talked about what

inverse functions are and so we want to continue

on um with that discussion so so far we talked about you know what inverses are

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and how to verify if two functions are inverses or not

and so this episode we’re going to talk about uh graphing functions

um and their inverses on the same set of axes and we’re going to do that by hand

and also by looking at python and seeing how to do that

so yeah let’s get started um so i’m going to start us off by looking at

a function here and i’m going to use f of x equals 2x plus 3 and the inverse is

the function x minus 3 over 2. now in the previous episode we talked about

an informal way of finding the inverse just by

sort of guessing and then verifying um and so we did lots of examples of

that and so this is what i would guess would be the inverse

and the reason why is because if we input an x here the first thing we’re

going to do is multiply by two and then we’re going to add a three so how do you

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undo all that if you’ve multiplied by two first and then you added that to a

three so i’m going to first undo the three and and do that first right so the

reverse order and then divide by two so i’m reversing the order and i’m

reversing the operations all right and so we could check that

these are inverses and we did lots of practice of that last video so we would

check that both of these compositions are the identity this right here

and this right here and so we practiced that

uh in with lots of examples in the last video so we we could check that

but i’m of course in this video i’m going to concentrate on more of what

they look like so let’s graph both of these on the same

set of axes and so we can see what’s happening here so let’s graph

2x plus 3 um and here we go so let’s look at that

so here we’re going to have 2x plus 3 and i’m going to plot some points like

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um you know what happens when x is 0 so then we’re going to get a 3 right here

and then what happens when x is one then we’re going to get out here a five

and so let’s put that up here so this looks this looks like a three

and then at one we’re at a five so we got this these two points there and then

what happens when we’re at minus one at minus one we’re going to be um

at a height of one so that should go through um 2x plus 3 and you know

let’s just put here the 0 down here it’s about what minus 1.5

right there and then how about one more over here how about

when we’re at minus four that will give us um a minus eight plus

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three so that gives us like a a minus four right here right um

add a minus five so let’s say minus five is about right there

so that will give us a um or let’s go to minus four right here

that will give us a minus eight and then

plus three so that would be like a minus five right there yeah minus five so

let’s leave a minus four on here and say minus five right here

and so basically what we have here is this line going through here 2x plus 3

and let’s put this in blue all right and so then we got this line

going right through here at this point and this point and this point

and this point right there and so this is the graph of

um f of x equals 2x plus 3. let’s put that in blue there

all right and so now let’s graph this inverse right here let’s plot a couple

points and see how it’s different so first off

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we’re going to have this zero right here which is what -1.5

zero so now it’s one point minus one point five in and zero so so this point

right here becomes on the inverse becomes about right there let’s say

and so then we have the point here uh 0 3 and now that becomes the point here

3 0 so let’s put it about right there so we’ve got two points on our inverse so

far in fact let’s go ahead and see if we can make those red so there’s

the in two points there and then what about this point right here so so the um

on the y-axis we got five we got one five right so now we’re going to go up

here and get it 5 1. so this was a 3 and then this was about a 5 right here so

say there’s about a 5 right there and then about a 1 right there

and so looks like we’re coming through there like that

um and so let’s see if we can just put a line through there

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and then it’s going to come down and and uh let’s see here about -5 right here

um let’s see minus 5 right here we’re going to have the minus 4 right there so

let’s put it in about right there okay so yeah so that would be minus 5

minus 8 over 2 so minus 4 right there all right so this would be the inverse

function right here inverse function which should be x minus 3 over 2

okay so yeah so if we try to understand this graph and are these two graphs and

how they’re related what happens is is these graphs are symmetric about the line

y equals x which just passes right through here

so i’ll just say y equals x right there that’s the line here in orange and

yeah so it’s just going to pass right through there and there’s where it

intersects right there y equals x there’s that line right there and so

it’s just symmetric about this line right here these two graphs

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the graph of a function and its inverse is always symmetric about the line y

equals x and so yeah just here’s a couple of um you know we have x and f of x

and we got the points here 1 5 zero three minus three over two zero

and minus four minus five just to give us a couple points and then we have the

table for the f inverse which is of course just the you know five one

three zero zero minus three over two minus five minus four right so you get

this table immediately once you know the function and once you know

that it has an inverse function um yeah and so that’s actually

something we’re going to do on the next episode is we’re going to

look to see how to determine if a function in fact has an inverse and

we’re going to do that graphically we talked about that a little bit in the

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previous video but yeah so now let’s look at one more example here so let’s

do this let’s do one more by hand so there’s our first one let me get rid

of this for us all right and so in the next one here

we’re going to be looking at the function here uh f of x is equal to

uh let’s go with um square root of x plus 2

and its inverse function is going to be x squared minus 2. with the caveat that

x is [Music] you know got some restriction right here because

this right here is not going to be the exact inverse right here

so what’s going to be the restriction right here

upon this right here so this right here is going to be like something like um

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[Music] you know what’s the range of this function right here will be

0 and greater and so that’s the domain over here so this is

x squared minus two but then x has to be greater than or equal to zero

so these are functions these are inverse functions

um but don’t don’t forget that uh domain right there if you if you leave off the

domain and just say x squared minus 2 that these are not inverse functions in

so this this domain right here that is is the natural domain which is just x is

greater than or equal to minus 2. you can substitute in minus 2 and greater

that’s the natural domain you don’t need to mention that but if you leave this

off the natural domain of this function is all real numbers so for them to be

inverses we would have to make sure and specify the domain for this right here

all right so let’s see how these uh functions right here look so

this right here we know from previous episodes it just looks like the square

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root function but it’s been shifted uh to the left by

two units so it’s going to look something like this it’s just going to

be a square root looking function square

root there we go it’s just going to keep increasing

but it increases slower and slower maybe i’ll just uh you know put it something

like that right there right so it’s just always increasing and right here is a

minus 2 right here so now how does this function right here look like

so this looks like a parabola right it’s just y equals x squared right but it’s

been shifted down 2 units so i’m gonna go one two and so then now it’s just been

it’s just going to look like a parabola right here like this just

like that but it’s only x is greater than or equal to two

uh sorry x greater than equal to zero so we’re only going to take the right

branch of it so i’m gonna have this just coming up here like that right there

so this is the graph of f of x x plus two and this is the graph of f inverse

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which is x squared minus two x greater than or equal to zero so

yeah we can put this one here in red we can shade it red if we want and

it just keeps wanting to roll in the way and we can shade this one here in blue

if we want and notice that what the range and domain are for these right here

so this point um minus 2 0 now becomes 0 minus 2

those and you pick any point on here x y and then you get the other point right

here if this one’s x y then this one will be y x so if we look

through the y equals x line i’ll pass right

through there y equals x line so these two graphs are symmetric

about the line y equals x any two functions if they’re inverses of

each other they’re symmetric about the line y equals x

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and we can plot a lot more points and you could see for example what would

this point right here be where would this one right here cross

the y axis so that’s when x is 0 so that’s the square root of 2 right there

so the height is square root of 2. and so what would this point right here

be so that’s this is the point here 0 square root of 2 so now this would be

square root of 2 0 right there so that tick mark right there is square root of

2 right there all right and so yeah so graphing them by hand it can be fun

especially when you start looking at the trig functions the trig and the inverse

trig functions then it gets to be a lot of fun or if you’re sketching

exponentials and then the inverses will turn out to be logarithm functions

so graphing them uh on the same set of axes is also fun

so yeah there we go so let’s take a look at how to

also do this using python so let’s see how to do that now so let’s go to python

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here and uh let’s get rid of this stuff right here real quick so using python

now if you’ve never used python before we’re using a python notebook and

there’s a link below in the description if you’ve never used a python notebook

before and how you can open up and start using your first python notebook it’s

it’s free and you can do that and you can just follow along in this

video or if you’ve been following along on the whole series maybe by now you

have already got your python notebook set up in any case here’s our setup that

we’re going to use for this video right here we’re going to import the matte

plot library splt and we’re going to import the numpy as np

and so let’s go ahead and execute that cell right there shift enter to execute

so i just got to type that up exactly as you see it

now if you’ve watched the uh series so far you know this function right here by

now it’s how i customize my axes i call this the precalculus axes definition

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right here and it’s because the python axes by default i don’t like them

and so i just want to customize my axes right here all right and so now i’m

going to make up a definition that’s going to help us plot some

functions and their inverses on the same set of axes right here so i’m going to

this function right here was is going to be defined right here and it’s going to

be called plot functions with an s so in the previous episode i used one

that was called plot function and so i added an s right here and then we’re

going to input two functions right here function one function two

and when we plot them we’re gonna have for our x values we’re gonna have a min

and a max and so i’m gonna call this x minimum x maximum and then i’m gonna

have an increment value right here and you can make this finer if you want like

0.00 if you’d like it to be less choppy all right and so yeah let’s just type up

this definition right here which is basically going to plot two

functions right here and then it’s going to show them right here so we’re going

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to plot with the x’s the same x values y one and same values for x y two

all right there we go we executed that uh input four

and so now let’s look at some examples so let’s go over here and

so now we’ve done our definition we’ve done our setup now let’s go look at some

examples real quick so in these examples right here we’re going to show that f

and g are inverse functions and here’s the first example right here

and so this right here makes sense from what we said before let’s see if we can

get bigger right here this makes sense from what we said

before right um in a previous episode we said you know what’s the inverse of 2x

how do you undo multiplying something by 2 well you divide it by two right so it

seems like these would be inverse functions right so um let’s uh execute

this definition right here so i’m going to define f uh function f1 which will be

my 2x 2 times x and then i’ll um execute my definition for g1 so this is g

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example one so i call it function g1 and basically it’s just take an x in and

then divide it by two and and send it out in any case there’s my definitions

of my functions and i’m going to go ahead and plot it and i’m doing this all

in one cell and so there’s the plot functions

there’s there’s the f1 there’s the g1 right the f1 has slope 2

and it’s the steeper one so that one going through right through there is is uh

y equals two x and then this right here is y equals one half x and it has a much

slower slope right there so there we go there’s the plot of f and g

and you can see how they’re symmetric through the line y equals zero sorry y

equals x which is just basically going right through the middle there

all right so now let’s look at our next example

so here we got two functions right here this function is going to take an input

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and it’s going to multiply by -4 and then it’s going to add three

so to undo that we’re then we’re going to first multiply by x

uh negative one add three and then divide by four right there so

um that’s one way to write it another way to write this is um do x plus three

and then divide by minus four so that those two things are the same right there

so because if you’re wondering like informally like how did you do that you

first multiply by minus four and over here i’m last going to divide by minus

four and then over here at last i add three

right because when you input an x you first do this and then you add the three

and then over here i’m first going to uh actually it would be minus three

wouldn’t it i’m going to first subtract three

um and then i’m going to divide by minus four so yeah these two things are the

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same here and we can just see that by just simply in the numerator just

factoring out a minus sign right there so that’d be x minus 3

over 4 and then just move the minus sign to the bottom right there so if you’re

looking at this and just learning this the first time this would probably be a

better way to think about this but either way these are the same

and so yeah i’m just going to define this function f2

and it’ll be 3 times -4 x and then g2 will be 3 minus x and then

actually let’s move the parenthesis right there it won’t matter but i’m just

going to do it anyways here and then now let’s just shift enter that

so here we’re plotting f2 and g2 and so we can see

that’s just going to be symmetric right there through the y equals x line now

you can see right here this point right here

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is zero something and then you got the exact uh opposite point over here in

yellow which is going to be some intersection right there

yeah so this looks good and okay so let’s go on to

so this one has got negative slope going right through there

and this one right here um also has negative slope but it’s just not as steep

all right so now let’s look at uh something like this right here so this

one i didn’t use the plot functions and the reason why is

because the problem with the plot functions definition that we used earlier

the problem with it of doing something simple like this

is that it uses the same inputs for both functions

but in fact you don’t have the same domain generally speaking

so to have the most flexibility i just took out the insides of the function and

i put it right here so and also i want to set the aspect ratio of the

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plot and so this will make it seem more the aspect ratio be the same so a circle

will look like a circle in other words instead of an ellipse but in any case

yeah i’m still going to customize my axes i’m going to set the aspect rate

ratio to be the same so in other words this distance to get to 30

is the same distance to get the 30 over here

in any case i’m going to for the x1s for this function right here the domain

is you know we’re going to start at 4 and go large

and for this function right here right so for this one right here i need four

and great i need greater than or equal to four for this function right here

with this function right here i need greater than or equal to zero now the

reason why is because remember the domain of this function is

the range of this function and the range of this function is the

domain of this function so that’s why you switch them like that

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and so i’m yeah i’m going to say the x’s for the

for the first function right here um will will be on this domain right here

and the x’s for this one right here will start at zero so that’s why i have a

zero here and a four here um and so and i also incremented the increment uh

all right so i also changed the increment right there to be a little bit

finer right there all right so we’re going to graph yeah x

x 2 squared so in other words x squared plus 4 so we got that right there

and i said x2 because those are my inputs

and here i said x1 because those are my inputs right there

all right and then we’re going to plot x2 y2 and we’re going to plot x1 y1 so

we’ve got two functions basically all right and then i’m also going to plot um

[Music] the identity but i don’t see it actually

plotted right here so let’s execute this and see what happens here

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so i don’t see the identity function showing up here so that’s okay so

i’ve told it to plot uh x3 is going from 0 to 30 and

so maybe it’s just some color that i can’t see

but it certainly doesn’t look like the y equals x for this for these x’s are

showing up here on the plot uh any case there is the function here

the first function right here is square root it’s been shifted to the right four

units so there we go it starts at four and

goes right there and then the y equals x

line should be coming right through here and if i do the inverse right here

then that will look like the parabola shifted up four units but we’re only

going to look at the right branch of it so there we go right there so there’s uh

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y uh x equals uh x square plus four but only

the right branch of it and so that these are inverses right here you can see how

they’re symmetric about the y equals x axis in other words four

zeros here and then zero fours right here

all right so very good so let’s look at one more maybe

so here we have square root also so 2x minus 3

and then again the outputs have to be 0 or greater which means the inputs over

here are 0 or greater so i’m going to restrict the domain so

i’m going to have two different domains so i’m going to use this stuff again

here which we just did right here right so i’m going to customize my axes so

they have nice arrows and the aspect ratio is going to be

equal and this time we’re going to go on the axes here um so to 10 and then

to five um yeah and so it’s going to go to 10

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and then this one goes up a little bit higher right there so so they both so

this distance right here to 10 is the same distance right there to 10

right there but this but this goes all the way up to 14 right there

all right so we got the x ones for the first function right here

and they go from 1.5 so i got 1.5 by looking at 2x minus 3

is greater equal to 0 because we have to take the square root of it yeah and so

just moving the 3 over and dividing by 2 1.5 right so that domain right here is

1.5 and we’re going to 10 and that just looks like the square root

function right there but it’s been shifted a little bit

and so there we are right there and so yeah you can see the line y equals x and

how these two functions are symmetric about the y equals x line right there so

for the second function we’re just going to do x squared

so that’ll be x 2 square because i’m using x 2s for my domain so x 2 squared

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and then plus 3 and then all that divided by 2 right there

and so then we’ll plot x2 y2 and then again here it didn’t show the x3 y3

it didn’t show it but i drew it in right here so that’s good right there

all right so there we go yeah so you know there’s how easy it is to use python

to sketch the function and its inverses or its inverse function also on the same

set of axes so yeah i hope you enjoyed this video hope you had a lot of fun in

the next episode we’re going to talk about how do you know if a function or

not has an inverse function and then the episode after that we’re

going to talk about how to actually find the inverse function and not just

informally by just sort of guessing and verifying i’ll show you a method that

you can use but first in the next episode i’ll show you how that you

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absolutely know whether or not a function has an inverse

all right so i want to say thank you for watching until i see you next time have

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