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in this episode you’ll learn how to find the inverse of a one-to-one function

we’ll go through the process step by step and you’ll see how easy it is

let’s do some math [Music] hi everyone welcome back um in this

episode we’re going to talk about finding the inverse of a function

now before we get started with that i wanted to mention that

in these videos here we’ve already talked about composition of functions

we’ve already talked about what inverse functions are and how to verify

and in the last episode we talked about what one-to-one functions are and we

looked at lots of examples so in this video we’re definitely going to

go beyond these three videos here these three episodes here so i recommend that

you check them out and give them a view the link is below in the description

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so what we’re going to do is first you know ask the question what why is

the inverse function unique so we never really addressed that in these three

videos here and so i just wanted to kind of talk about

why the inverse function is unique so let’s say we have a function here

let’s call it f and let’s say we have inverse functions g and h are inverses of

f so f is given and we and and we’re uh saying g and h are inverses of f now um

because we often use the word the with the inverse function

so if i take two inverses of f how do we know that they’re equal to

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each other and that it’s there’s only one inverse of it

so how do we show that two functions are equal so what we’re going to try to do

is to show that g of x is equal to h of x

and we’re not going to put any condition upon the x

and so this is going to be true for all x in the

domain that we’re going to be interested

in so i just want to kind of give you an idea of how something like that would

work now for this video right here we know

what it means to be an inverse function right here and from this one we know

what composition of functions is so to say that g is an inverse of f

what that means is that f composed of g this one is equal to x

and g composed of f is equal to x right here so these are

these two functions right here the composition are both the identity function

and we’re also assuming that h is also an inverse of f

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so f composed of h of x is equal to x and also h composed of f of x

is equal to x so using this right here we can show that g of x is equal to

h of x but exactly how how would we do that so to do this we can

take this x in here and try to use something in here and try to get out an h so

um you know how would we do that so i’m going to look at this h right here

and say this x and i’m going to substitute in all this right here for it

and so let’s see what happens right there if i do that

so if i take this x right here and i just substitute all this in here

for the x so this will be f composed of h right here of x

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now you might be asking why why did i choose that one why did i replace this x

with all of this i mean because we got the four things to use here to look at

um and so well i just want to pick this one right here because it’s going to

give us something useful so that’s a f with parentheses there yeah so this is

going to give us something useful something that we can look at here so

you’re welcome to try different combinations and in fact if you were to

start with h of x here you’d you’d probably do

use a different one of these but in any case um what happens if we replace this

x right here um right here with all of this right in here and then now

we’re going to get this is the function g composed of f composed of h of x

that’s just by definition um and um if we want to we can um [Music]

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re use associativity so this will be g composed of f composed of h

and all that with an x here and now we’re going to use this one right here

which says this is the identity function right here so

this right here is going to be g composed of f composed of h of x right here

and so one more step here this right here is h of x

and the reason why is because g composed of f

if you give it any input you get out that

same input right there so if i give it h of x

it goes into this function right here you just get out h of x right there

so that uses these two right here um [Music] that just kind of gives you

an informal i don’t assume you have any background in writing mathematical

proofs at this point this is just a beginning precalculus series but this

kind of gives you an idea of why you would say the inverse function is unique

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if you take two different inverse functions you can actually show that

they’re equal just by using the um definitions of what

it means to be inverse functions right there so if this gives you some kind of

idea right here of why this works right here um i use associativity and we

didn’t really prove that yet but in any case g is equal to h no matter

what the input is and so those functions are basically the same function right

there those are the same function all right so that kind of gives you an

idea of why we’re going to say the inverse function

is unique you know why we use the word v all right but anyways let’s get on to

some examples because uh in this video what we’re really concentrating on is

uh how can you find the inverse of a function so that’s the

that’s the topic of this video that was just a little aside there that that i

hope you enjoyed but anyways how can i find the inverse

of a function right that’s the question in this video here so uh let’s work out

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some examples here so in the previous episodes we talked

about finding the inverse but we only did so informally so let’s look at a

more formal procedure now and so what i’m going to say is we have a function

right here and i’m going to give us a list of steps

that we can follow so the first step is to use x’s and y’s so i’m going to say

this is a y right here so that’s the first step is to set the function or

set it equal to y and you know anytime you have a function you can do that

so now what i’m going to do is i’m going to switch x’s and switch y’s wherever i

see a y i’m going to put an x and wherever i see an x i’m going to put a y

so this will be x equals and this will be 3y

minus 5. okay so that’s the second step first step set equal to y

second step switch x’s and y’s and now the third step will be to solve for y

so we’re going to solve this back for y again after switching now we’re going to

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solve for y so to solve this for y i’m going to add the 5 over to both sides

i’m going to add the 5 over and then i’m going to simply divide by 3.

and this is the inverse function right here so i’m just going to name it so our

original function was f so i’m going to use f inverse right here and it’s just

going to be x plus 5 over 3 which is what you would expect the

inverse to be because if you’re going to give an input you’re going to multiply

it by 3 first and then you’re going to subtract 5. so we’re going to undo those

operations in the reverse order so here i’m going to add 5 first and then i’m

going to divide by 3. so these are inverses of each other

and we can verify that by checking these two

equations here checking this right here is the identity function

and by checking this right here is the identity function so we can check both

of those and we can verify and i’ll do something like that here in

a minute in case you missed that episode previously

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so we could verify both of these and we would figure out that this this is the

inverse so this is what i haven’t talked

about so far in this series set it equal to y

switch x’s and y’s and then solve for y so this is the basic three steps right

there so in fact let’s just do one more quick one here let’s say f of x now

is equal to x to the third plus three all right so from the previous episode

we can graph this we can sketch the graph pretty quickly it’s just an x to

the third and then shift it up three then we can look at the graph and by the

previous episode we know what one to one function means

and we can see that it passes the horizontal line test so we can say hey

this function has an inverse function right so covered all that in the

previous episode check it out so yeah this is a one-to-one function it

must have an inverse and so now what’s new in this episode is how do you find

that inverse what is the step-by-step procedure well it’s the same every time

00:10

you set it equal to y we switch x’s and y’s so wherever you

see a y put an x wherever you see an x put a y and now we solve for

the y we solve for it and because we know this is one to one function

one two one one to one function the inverse has to exist so f inverse exists

passes horizontal line test all these three things mean the same

so when i set this equal to y that’s really just for uh ability to

manipulate algebraically right there’s really no

uh advantage to that other than just algebraic manipulation we switch x’s and

y’s that means we switched inverse relation

so now we have the inverse relationship but because we know all these three

things are true whichever way you want to think about

them we know the inverse must exist so that means we must be able to solve this

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for y and we can in fact we can move the

3 over and then we can take cube root of both sides so when we move the 3 over

and then when we take cube root of both sides we have solved for y

and so now we have found our inverse function so this is the original

function with an f so i’ll use f inverse of x

and this will be cube root of x minus three and so there’s the inverse function

right there so again it’s just a simple matter of setting equal to y

switching the x’s and y’s and then solving for y now solving for y is

definitely the hardest step and so yeah let’s practice some more

so let’s look at um some more here and this time not only are we going to

find the inverse function but we’re also going to verify so we’re going to check

this out here so this will be a good review here alright so here we go

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let’s look at this function let’s call it g of x is equal to let’s go with

cube root of x plus five and yeah let’s just go with that

function right there so cube root of x plus 5.

so let’s find the inverse function and then we’ll verify the inverse function

so to ver to find it we’re going to set it equal to y

and now we’re going to switch the x’s and y’s

and now i’m going to solve this for y now solving for y because this is cube

root so i’m going to raise both sides to the third power

and then i’m going to move the 5 over and so there is the new inverse function

right there so this was g is where we started so this one right

here is g inverse why is it g inverse because we switched

the x’s and y’s right there there’s where we did the inverse right there

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all right so we set it equal to y switch x’s and y’s

solve for y and we have our new inverse function right there

okay so now let’s do the verifying part now that we found it now let’s verify so

i need to check these two things right here g composed of g inverse of x

equals x now we’re going to show some work

we have to end with x if we end with x we know we got it right we verified it

so let’s see what happens here so let’s use the definition of

composition so this is g of g inverse of x

and we know what g inverse of x is this inside part right here because we just

found it so this inside part right here g inverse

of x g inverse of x is this right here so this will be g of

g of x to the third minus five so i’m going to take x to the third

minus five and wrap that up and think about that as a whole thing that we’re

00:14

going to plug into g so what does g do it does cube root of so cube root of the

input and all of this is the input so x to the third minus five that’s the input

and then so it does the input plus five so the input plus five there we go so g

is the cube root of the input plus five so now let’s simplify this so this will

be cube root of and this is x to the third minus five plus five so just cube

root and then now that you see that this is just x

all right so we verified one of them so now let’s verify that

g inverse composed of g also works so this will be g inverse of g of x

so this will be g inverse of g of x we know is the cube root of x plus five

and so this now needs to go all of this needs

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to go into g inverse so g inverse is going to cube something so it’s going to

be the input all of it cubed and then minus 5.

so these undo each other cube root and cubed and we get x plus five

and then minus five and that’s just equal to x

so we verified it this right here is equal to x

and this right here is equal to x and so there we go

we verified we found the inverse function and we verified the inverse functions

all right so [Music] um let’s do some more but i want to do

some that maybe are a little bit more challenging

let’s get some space right here [Music] so let’s do some more here

so let’s look at these two functions here and see if we can get them

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so i’m going to go to [Music] 8x minus 4 i’m going to put the function

right here and we might need more space so let me see what we can do about that

let’s see if we can just go to there right there all right so there’s the

function right there let’s find the inverse function now

one of the things you may want to do is say uh find the inverse function right

well how do you even know there is an inverse function

so if you could graph this and see if it passes the horizontal line test

then you would know it has an inverse function so being able to graph this

is a is a nice thing to know how to do so you but if you don’t know how to do

it you could you know use a graphing program or a computer

or calculate a graphing calculator but in any case you look at a graph of this

and you realize oh it passes the horizontal line test right so it is one

00:17

to one and so the inverse does exist don’t just think that just because you

make up a function that the inverse has to exist it has to pass horizontal line

test then i know that it exists so i know what the graph this looks like

it does pass horizontal line test so i’m going to go and do the next step which

is to find it so don’t think you can find it first you have to know that it

exists all right so i know that the f inverse exists so i’m going

to find it now so i’m going to set this equal to y

and now i’m going to switch the x’s and y’s wherever i see a y i’m going to put

an x and wherever i see an x i’m going to put a y

there we go and now we’re going to solve this for y

now to solve this for y we’re going to need to multiply through here so i’m

going to say x times 4y plus 5 and then this is 8y minus 4.

00:18

and so this will be 4xy 4xy plus 5x equals 8y minus 4. so

we don’t have all the y’s on one side yet right so let’s move the 8y over here

and let’s move the 5x over here so i’m going to say this is 4xy

and then minus 8y and then say this is minus 5x minus 4.

now out of these two we’re going to factor out the y so i get 4x minus 8

and minus 5x minus 4. and now the last step is i’m going to

take this right here and divide it underneath so we’re going to get

minus 5x minus 4 over four x minus eight and now there’s three negatives so

maybe we can write it more positively you know

factor out a minus one on top factor a minus one out on bottom and cancel the

00:19

minus ones and then we get five x plus four over um eight minus four x

so that would be a nice way to write it right there so we found the inverse

function right here we’ll go put it up here so f f inverse is 5x plus 4

and then 8 minus 4x there we go oops can’t quite see that so let’s go

with 5x plus 4 and then 8 minus 4x all right very good so here’s the function

and here’s its inverse and now let’s verify now technically you

don’t need to verify unless someone just asks you to do it but i’m going to

go ahead and verify this right here and see this right here

um yeah let me get rid of that there we go so let’s verify here so

00:20

i’m going to do f of f inverse of x and if this is right if this worked

this should get x it may take a lot of work

but we should get x so dot dot lots of work we should get equal to x when we’re

done let’s see if this works so this will be f of f inverse of x

maybe i can go one more over here uh so this will be f so what’s the

inside part of the inverse f inverse of x it’s all of this so this will be 5x

plus 4 over 8 minus 4x and then let’s change the size here there we go

so f of f inverse so f of f inverse and now i need to take all of this and

substitute it into f so let’s do that say about right here

so think of all of this as all rolled up and put into an input so what does f do

when you give it an input i can put it here and i can put it here

00:21

so i need to put all of this right here and right here so here we go so 8 times

all of this 5x plus 4 and then 8 minus 4x so 8 times x minus 4

and then 4 times the x the input which is all of this so 5x plus 4

all over 8 minus 4x so 4x plus 5. so 4x plus 5. there we go

so all this should be equal to x right so obviously it’s equal to x right

all right let’s see what we can do um notice here the denominator is 8 minus

4x the denominator is 8 minus 4x but these two don’t have a denominator of 8

minus 4x you can think of this as 4 over 1. so here’s what we’re going to do

00:22

8 times 5x plus 4 minus 4 times 8 minus 4x all of that over 8 minus 4x

all of that’s over and then here we have 4 times this part and then 5x plus 4

minus 5 plus 5 so here i’m thinking about this as 5 over 1

and i’m just multiplying by 8 minus 4x top and bottom so i have a common

denominator so this 5 is going to change into this so this will be 8 minus 4x

and then i have 8 minus 4x on the bottom and an 8 minus 4x on the bottom so i

just use common denominator here all right so here we go so these cancel now

and so let’s see what we’re going to get on top here we’re going to get a 40x

and then a plus 32 and then a minus 32 [Music] and then a 16x perfect

00:23

now here we’re going to get a 20x and then a plus 16 and then a plus 40.

and then a minus 20 x all right so let’s see what happens here

so these two add up to zero so on top i’m going to get a 56x

and in the denominator so these right here cancel

the 20x and the minus 20x they add up to zero so this gives us a 56

and now the 56 cancels and voila we get our x perfect

so we verified that f composed with f inverse that that’s the identity function

whatever you plug in x is what you get out x

all right so we verified one of the two we need to do

so now let’s go do the other one so let me erase this real quick all right good

00:24

so now we need to do the f inverse of f at x

and so now f is going to be applied to x first

so f of x goes first so now i’m going to use this one in here first so this will

be f inverse of all of this 8x minus 4 over 4x plus 5. very good

all right so now we’re going to take all of this right here wrap it all up as a

big o x and put it into f inverse so this is my input all of this

so what does f inverse do to an input it multiplies a five and adds four all

right so we’re gonna do five and then all of this eight x minus four

over four x plus five plus 4 all right we did the top here and now 8

minus 4 times the input 8 minus 4 times the input which is 8x minus 4

00:25

over 4x plus 5. and so as you see we’re going to do the

same thing that we did on the last one a second ago

because what we’re going to need is a common denominator of 4x plus 5 on this

one so i’m going to think of this 4 as a 4 over one and and then multiply

top and bottom by four x plus five and so what we get here is i’ll do this

on the next step here so we’re gonna get five times eight x minus four

plus four times four x plus 5 all over 4x plus 5 and then all this is over

8 times 8x minus four over four x plus five so it’ll be eight times all of that

and then so minus wait sorry uh what am i doing here okay okay that’s not right

00:26

so sorry about that so this is going to be 5 times that oh

that’s over the denominator okay yeah sorry sorry we’re getting common

denominator so yeah this will be 8 times 4x plus 5 and then minus 4 times

8x minus 4 and then all over 4x plus 5. all right so

now we can cancel these right here and you know actually this compound fraction

i’m going to rewrite it because whenever you divide you’re just multiplying by

the reciprocal which is 4x plus 5 over all this right here

which is 8 times 4x plus 5 minus 4 times eight x minus four

all right so now i can erase this here and have enough room

00:27

so you see the four x plus fives cancel let’s see what we’re going to get here

we’re gonna get a 40x minus 20 and then a plus 16x plus 20.

and so that canceled with that and so then on the denominator we’re going to

get 32x and then plus 40 and then minus 32x

and then a minus 4 times -4 so plus 16. all right so now let’s see what’s

happening here the 20s add up to zero and so we’re going to get 56 x

and down here the 32 x’s add up to zero and we get 56 so we get 56x over 56

again which is just x all right so this this composition right

here is also the identity function and so we did it we verified so we found

00:28

the inverse right here and then we verified using this and the other part

that we erased there so that’s good all right so now um let’s see here we

have this part right here which is this g of x right here um so i’m going to

we’re going to find the inverse first let’s find the inverse of this right here

so here’s uh g of x right here so i’m going to call this equal to y so 6x plus 4

2x plus 6 here now look at all those twos

i don’t know would it be easier to just work with smaller numbers 3x plus 2

cancel a 2 everywhere x plus 3 right so let’s just look at that function there

is this the same function right cancel a 2 everywhere in any case

00:29

i’m going to switch x’s and y’s now so three y plus two and then y plus three

there we go and now i’m going to solve this for y

so to do that i’m going to multiply y plus three

and then we get three y plus two multiply both sides by y plus 3 let’s

say and then xy plus 3x equals 3y plus 2 very good

so now i’m going to move the 3y over here so i’m going to say xy minus 3y

and minus 3x plus 2 so we’re going to set it equal to y

switch x’s and y’s and then solve for y we almost have it solved for y i’m going

to factor out the y x minus 3 minus 3 x plus 2

and then now i’m going to divide so we’re going to get y equals minus 3x plus 2

over x minus 3. and now there’s only two negatives and two positives so maybe

00:30

we’ll just leave it the way it is all right so there’s g

and so here’s g and so here’s g inverse now maybe you want to put the

twos back in everywhere and say minus 6x plus 4 and then 2x minus 6. but i mean

it doesn’t really matter in fact i’m just going to say over here that g of x

is equal to 3x plus 2 and then x plus 3 and g inverse is the minus 3x plus 2

and then the x minus 3. there we go and so let’s verify now

find the inverse and verify and so let’s verify

and this time i’ll try to pick up the pace a little bit

um so i’m going to do g of g inverse of x

00:31

and so this will be g of g inverse of x and so this will be g of so g inverse is

this one right here minus 3x plus 2 over x minus 3 [Music]

and so now i’m going to put all that into g so here’s g right here so 3

and then all of this minus 3x plus 2 over x minus 3 and then plus 2

and then all over x which is minus 3x plus 2 over x minus 3.

so there’s the x and then plus 3. okay so now let’s get a common denominator

in fact i’m going to get a common denominator and cancel

all at the same time so i’m going to say 3 times the minus 3x plus 2

and then plus two times x minus three all over so that’s um plus

00:32

two times x minus three and then all over okay so all over

so i’m gonna get a common denominator and we’ve already canceled the x minus

threes so i’m going to get a minus three x plus two right here for that part

right there plus three times the top and bottom x minus three so x minus

three right here right there we go so on top we’re getting minus nine x plus six

and then plus two x minus six and then minus three x plus two

and then times three x and then minus nine all right so the sixes add up to zero

and so we’re going to get minus seven x on top

and then here with the minus three x the

three x’s add up to zero and we’re going to get minus seven in the denominator

and voila we got the x now i’d like to just reiterate that it

says to find it and verify so finding it was

crucial because this is a one-to-one function

00:33

and you know this function right here you can look at the graph and see that

it passes horizontal line test so it is one to one and so we found the inverse

and this right here is the inverse and it says to verify so once we if it says

to verify then we can verify but we don’t really need to verify because the

process of finding the inverse always works once you know it it does exist

then it will always work so verifying it is just because they asked us to do it

so we did half of the verifying now let’s do the second half

so the second half is when g inverse gets applied to g of x

so this will be g of g inverse of g of x and now let’s substitute in here what

did g of x is so that’s the 3x plus 2 over x plus 3. [Music]

00:34

so now all of this goes into g inverse i’m going to get minus 3 times the input

so minus 3 times the input plus 2. all over the input -3 so the input right here

and then the minus 3 there so again i’m going to get the common

denominator of x plus three so we need to multiply top and bottom by x x plus

three and multiply top and bottom you know think of that as top and bottom

right so by x plus three so we’re gonna get minus nine x and then minus six

and then two times the x plus three so that’ll be plus two x and then plus six

and so now this will be three x plus two and then minus 3 times the

x plus 3 so that would be minus 3 times x

and then minus 3 times -3 so that would be minus 9.

00:35

and so these add up to zero and i get minus seven x

and these add up to zero and i get minus seven here and so in the end we get out

an x wow that’s nice all right so there we go i hope that helps you

able to find the inverse and we also you know practice the composition some

more and practice verifying and so i hope that you enjoyed this video

and i look forward to seeing you in the next one take it easy bye bye

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