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in this episode you’ll learn what the inverse function is and how to verify

whether or not two functions are inverses of each other let’s do some math

[Music] hi everyone welcome back i’m dave

uh let’s start talking about what is an inverse relation

and so i’m going to do that by um kind of looking back on what we talked about

when we first talked about what a function is

so let’s recall what a relation is so for example

we can say this right here is a relation so i’ll put in some ordered pairs here

so here’s a set of ordered pairs it’s got three ordered pairs

and i’m naming the set s and we can talk about this being a relation

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so is a relation simply because it’s a set of ordered pairs so um we have one

can be the input into s and 2 would be the output minus 1 would be the input 3

would be the output and 4 would be the input and 5 would be the output

so associated with the relation we have the inputs and we have the outputs

now to talk about inverse relation so here’s the notation we would use for

inverse relation and so what we’re going to do is we’re

going to switch the inputs and outputs so this ordered pair becomes the ordered

pair 2 1 and this ordered pair becomes 3 minus 1

and this ordered pair becomes 5 4. and so this is also a relation

now we use this notation right here not to mean like one over or like it’s

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an exponent but it just means it’s the inverse that we switched the x’s and y’s

the inputs and outputs so these are inverses of each other

inverse relations of each other because if i do the inverse of this inverse i’ll

get back to the original right here so we can say s is a relation and s inverse

is the inverse relation of s so we could say s inverse is the inverse relation

of s and we can repeat that process we could say s inverse s inverse is the

inverse relation of f inverse and so this is just s

because i would just switch and then i would switch back

right so that’s equal to s so s is the inverse relation of s inverse and s

inverse is the inverse relation of s now nowhere in here am i talking about a

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function though it’s just all about relations so this episode is about inverse

functions so let’s talk about that now so this is an inverse this is a relation

the question can be asked is it also a function

so for each input we have to have a unique output and that actually works

doesn’t it because if i input one i get only one output if i input minus one i

only get one output so if i added something like this to the relation here

so i’ll say here what if we added the pair minus one five

so this is still a relation is a relation but notice that this right here is

not a function right so this would be the point of switching

x’s and y’s this would be minus 1 1. so this is also a relation

i forgot to put parentheses there when i

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closed about before but in any case this is s now and this is s inverse there’s

still relations and these sentences still hold

but now when i ask the question is this a function is s a function now i have to

say no because if i input minus one i get two different outputs i don’t get a

unique output i get three is an output and a five is an output

so this is not a relation uh this is not a function right here so not a function

now what happens if i take that point back out

so let’s take that back out now and now let’s ask the question again it’s still

relation but now let’s ask is this relation a function

now we can see this is a function and so we can ask the question

is this function does this function have an inverse function

and so now i’ll look at the inverse and look at the inverse relation right here

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and i ask the question is the inverse also a function we know the original one

is a function but now i’m asking is the inverse a function so input two i only

output one thing i input three i only output one thing

i input five and i only output one thing so this is also a function

so not only is this right here an inverse relation of s

but it’s also an inverse function of s it’s the inverse function of s

now what happens if we put this point right here in here let’s say we have 3 5.

here we go so now is s a relation yes it’s a set of ordered pairs

now is this a relation a function and i’ll have to check four things i’ll

have to check four inputs remember you have to check every input so i check i

get only one i get only one output i get

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only one output i get only one output it doesn’t matter that four and four and

three share the same output what matters

is does every input have only one output so this right here is a function and so

now i’m going to go is a function and so now i’m going to go and try to find the

inverse and ask if it’s a function or not so now i’ll have to put in here 5 3

to find the inverse so here’s s and here’s s inverse and so now i’ll ask the

question is it a relation of course yes is it a function

is the inverse a function and now we’ll have to say no for this example here if

we added this point in here 3 5 when you switch it around and when you switch

this one around now we can say 5 has a unique output no 5 has only

five has two different outputs so this is a relation but not a function

but not a function so here’s an example of a relation that is a function

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so here’s an example of a function that does not have an inverse function

its inverse is just a relation so i hope that helps

but now we’re not going to just work with a finite number of points one two

three four points you know how can we work with infinitely many points how can

we work when we have function rules so um let me try to go with a

definition now of what an inverse function is now so let’s uh get this out

here real quick so what is an inverse function so we’re going to start with two

functions right here let f and g be functions now there should be an s right

there let f and g be functions and so now we’re going to say that

these two right here are inverse functions of each other and to do that

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we’re going to define what that means there’s the yes if

so in the last episode we talked about composition so

i’ll write the composition notation out here that we talked about in the last

episode so this is the composition notation right here

so yeah before i go on let me just mention here that this episode is part

of the series functions and their graphs step-by-step tutorials for beginners

link below in the description for the full series but in the last episode we

talked about the composition of functions and this is the definition of

it right here so what does it have to do with inverses well if you input an x

and g does something to that x maybe it squares it or cubes it or subtracts

three whatever g does to that x then f will undo

what g does in other words whatever input you get in

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you get that out also so if i input a 2 that 2 will go into g g will do

something to the 2 and then f will undo what g did and you get out back to 2.

so you have if that condition holds and this condition holds in other words

it doesn’t matter which way you go if x goes into g first then f will undo it or

if x goes into f first then g will undo what f does

so you have to get the identity function

out in both of these cases right here so and again in the previous episode we

talked about what the composition is we we worked out lots of examples with

this is the composition right here so if this composition right here is just the

identity function just you put in you get out you put in you get out and this

has to happen for every x in the domain of g

right because x is going into g and this right here has to happen for every x in

the domain of f all right so if those two conditions hold then what

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we have f and g are called inverse functions of each other and we have

special notation for that which i’ll show you in a moment

so let’s just do a quick example right here so let’s say we have the domain of f

so let’s just call this domain of f and let’s say we have an x right here

and so let’s say over here we have the range of f range of f

and let’s say we have an output over here f of x and f is going to be

x plus four so what function do you think will undo x plus four

so we’re going to have a function right here called g

to undo the x plus 4 it’s going to do the x minus 4.

so these functions undo each other for example if i input a 7 i’ll do 7 plus 4

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and g will say oh no you don’t take and take away that 4 and you get back to the

- so we can we can check that we can say f composed of g of x will be

f of g of x which is so g of x is x minus four so i’ll put in here x minus four

and now i take x plus four all into f so x minus four and then plus at four

and so the fours undo each other i don’t need this here

and so this would just be x and then we can check this one right

here also g composed of x uh sorry g composed of f

of x so this will be g last f first so what is f of x f goes first so it’s x

plus 4 and now i put x plus 4 into g so x plus 4 and then minus four

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so that’s the input and then minus four and the four is still undo each other

and they add up to zero so x plus zero in other words x so

both of these work right here and so yeah we just have the domain

and then we map and then we come back over here and so this right

here is the domain of f but you could also say i’ll make it a little bit

bigger you can also say that this is the domain of g and

so i’ll say domain of f and then i’ll say a little bit bigger here domain of f

and this will be the range of g and so here i’ll just put a little x here

so yeah the domain of f range of g range of f domain of g and so we can go

and and map this right here to an f of x

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and then we can take that f of x and map it right back to that g

and that would be the composition right there one composition and then you have

the other composition also all right so yeah let’s look at another example here

here we go let’s let’s take a look at this right here

so which of the following functions is the inverse function of this right here

so what would you guess so would you guess a or b or the function g

what is that you think is the inverse or do you think h is the inverse

so think about how to undo this if someone put an x in here how would

you undo this so for example if i input a 3 what would i get here

or let’s say i input a 1. if i input a 1 here first thing i’m

going to do is subtract the 5 and then i’m going to divide

and then multiply by 2. so how would i undo those operations how

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would i do them backwards and undo them so do you think you’re going to take an

x first subtract a 5 that doesn’t seem to be undoing this right here i think

i think um actually that i think this um should be a 5 right here actually

and so let’s make that a 5 right there so let’s see which one it is here let’s

let’s try g first here so i’m going to try g just to see i don’t think it’s g it

doesn’t seem very intuitive to me that’s g because to g here just looked like it

just flipped it up upside down right and that’s not really what

inverse means inverse means undo it it doesn’t mean flip it right that’s

completely two different things for example in the last episode we said

add four subtract four that’s an undo operation that’s not a flip operation

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right so i don’t think it’s g or part a so let’s check though what

happens if we do f composed of g of x if we get out an x right if we do work

work work we get out an x i don’t know maybe it is g

let’s see if work work work will give us the x so here we go f of g of x

and what is the g of x here it’s x minus five over two

and now if i do x minus five all of that

into f i’m going to get 2 over the input which is x minus 5 over 2

and then minus 5. now is all that equal to an x

well it’s maybe it’s a little bit hard to say

maybe let’s you know simplify it and see

maybe maybe it’s an x who knows so let’s just say here this is

um 2 over x minus 5 over 2 and then let’s go with 10 over 2.

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instead of a 5 here let’s say it’s 10 over 2. now i have a common denominator

here so 2 over so this will be x minus 5 minus 10 and then all over two

and then so this is two over or let’s say two times two over one

times and then now let’s flip this right here and say

so two is in the denominator of the denominator so let’s put 2 here and then

x minus 15 and so this is 4 over x minus 15 and

that is certainly not equal to x right there yeah that’s not an x right there

so the answer cannot be g right here a and of course if you’re only given two

choices you would have to say h of course maybe your answer is c none of

the above anyways let’s see if h is the right answer let’s just check that

so here we go let’s check the composition right here so here we go f

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of and i’m going to use an h here for part b so f so this has to be true

in order to be in uh in it for them to be inverses of each other so let’s check

if this is true or not so i need to do work work work and i need to get to x

let’s see if that works out or not so h is going to go first so f of h of x

and i’m going to take what the h of x is and substitute it in right here so this

will be f of and what is the h 2 over x minus 5. so 2 over x minus five

and you know two over x and then all of that minus five

okay so now let’s take all of this wrap it up as one little thing and put it in

as an x so i’m going to get 2 over 2 over and then the input so all of this

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so 2 over x minus 5 so 2 over all of that and then minus 5. so

here does all this simplify to an x and the answer is i don’t think so

because i have a minus five and a minus five so

i actually want this to be a positive don’t i if i could put a positive here

then i think all that will subtract off so let’s actually change this to a plus

five so let’s see if we can do that real quick let’s go with plus five

if i have a plus five here for my h now this h of x right here is going to

be two over x plus 5 so now i’m going to take 2 over x plus 5

and substitute it in and now i’m going to get 2 over

and then the input is going to be this right here all of that for the x

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all right so now the fives add up to zero and i get two over two over x

in other words this is two over one in the numerator

times and then flip you get x over two so yeah you get x

so that works now hold on a second that may not be the inverse they may not be

inverse functions yet because what we haven’t checked is the

f composed h composed of f so what this one shows is that

whatever input you give to h f will undo that and get back to the x

but that doesn’t mean this one right here works so now i’m going to send the

x into f first and then we have to check that h will undo whatever the f does

so let’s see let’s send the x into f and we have a rule for that

so that is this all this right here so this will be h of 2 over x minus 5.

and so now i’m going to take all of this and plug it into the h and what does h

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do 2 over the input so 2 over the input and all of this is the input

so two over input plus five so two over input plus five and now

uh does that simplify well let’s see think of this complex fraction right

here two let’s think about it as two over one times the reciprocal here

x minus five over two and then we still have a plus five

and so let’s go down here so the twos cancel so we’re going to get x minus 5

plus 5 and so yep we get an x so in fact these two functions right

here are inverses of each other f and h are inverses or

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f and h are inverse functions all right so let’s um

look at finding some more just by not doing all this but just by

thinking about informally how to figure out what the inverse

function is so let’s try to do something like that

so let’s see the next example right here find the inverse of the function um

informally let’s think about it informally let’s get rid of this real quick

all right so here we go so part a i’m going to give you some functions here

now in an upcoming episode i’m going to show you exactly

step by step how to find the inverse but i think that before you actually try to

find that recipe to just kind of figure out how to do it

on your own how to think about it because you don’t want to think formally

every single time so i’ll give you a formal method um coming up here in

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another episode with this episode let’s just try to do it informally and i think

that’s very instructive because you really want to have some kind of

intuition in terms about functions and inverses so if someone gives you f of x

equals 6x what do you think would be the inverse function

and by the way here’s the notation for inverse function so if your function is

named f we’ll use f with a minus 1 exponent but actually it’s not an

exponent it’s just notation that replaces that f is the inverse function of f

oops of f so so this right here f inverse is the inverse function of f okay so

what do you think would be the inverse function

what would be the undoing of multiplying something by 6

well i would think it would be dividing something by six i think these are

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inverse functions of each other i think if you put something in here divided by

six and then send that over to here and multiply it by six

you’ll get back to where you started conversely if you start with something

over here and multiply it by 6 and then take that result and put it over here

and divide it by 6 i think you’ll get back to the x so informally these are

inverses of each other what about x plus six

what do you think would be the inverse function of x plus six

so we did one earlier where we said x plus four and the inverse we showed was

x minus four well it’s not going to be really any different adding 6 to

something how do you undo that you subtract 6 from something so if i give

an input i subtract 6. let’s try something like a combination

or how about let’s just do make sure we get it x minus three what would be the

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inverse x plus three let’s try another one f of x is two x minus seven

f inverse is so here i’m going to uh put an input in and the first thing

i’m going to do is divi um sorry multiply by 2

and then i subtract 7. so we need to not only do undo those operations but we

need to undo them in the right order so here i’m multiplying by two first so

what i’m going to do last is divide by two

now after i multiply by two i’m going to subtract seven so what’s undoing

subtraction it’s adding so i’m going to add 7 first and then divide and this is

going to undo that this is going to multiply it by 2 and

then subtract 7. so these undo each other and then what do you think think about

if you have another power um like x to the third what do you think

would undo an x to the third power so if i send something in

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as an input and i cube it i say like a four what’s four cubed four times four

times four so what would be the inverse it would be taking cube root of it

so these will undo each other this will raise it to the power three

this will take the root cube root and so yeah these will undo each other

so just informally these are some examples right here of

how you want to think about inverse function here all right so let’s do um

a one or two more examples here let’s go here and um

yep so we did those examples right there

and now let’s look at this example right here so let’s erase this right here and

let’s practice and making sure that we get this right here verifying an

inverse function because this has a technical meaning right here we say verify

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so i’m given the function f and g and i say yeah they are inverses of each

other so i’m going to again i’m going to show you how to find the inverse in an

upcoming video but right now what we want to practice on is verifying in

other words someone told you hey those are inverse functions of each other and

you’re like uh let me verify if you’re right so i’m going to verify this right

here and see if we’re right so i need to check f composite of g of x

and i’m going to see if i get an x out of all that so here we go f of g of x

and then this will be the g of x will be this right here so it’ll be a minus and

then a 2x plus 6 over 7. so i’m going to take all of this right

here and substitute it in right there so i’m going to get minus 7 over 2

times the input which is all this so minus 7 over 2 times the x and then

minus 3. and let’s see what we get for all this

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it looks like the 7s are going to cancel the minus 2s are going to cancel

well wait a minute there’s a 6 there isn’t there so what we need to do is to can

factor out the minus 2 don’t we so i’ll just write it like down here so

this is a minus 2x plus 6 over 7 is the same thing as minus 2 times 2

over 7 times and then what is it x minus 3 right

so we get minus two x and then plus six um

a minus needs to go for both so i think it’s going to be plus three actually

there we go so when i look at minus seven over two times this the

twos cancel the sevens cancel and so what we end up with is the x plus three

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and then we still have the minus three there

and so then this turns out to be an x so just a little scratch work down here

hope hope it explains my thinking but basically the 7’s cancel it’s a minus

times a minus and the 2’s cancel but in order to cancel 2 you really need to

factor out a 2 first here so this will be 2 times x plus 3 and then we cancel

the twos but you still get an x plus 3 left all right so that’s just scratch

work there so there’s there’s an x right there and now to verify we need to

verify with f going first so this will be g of f of x

and so what is f of x it’s all of this right here so this would be minus seven

over two x and then -3 and so i need to take all of this

and send it to g so i need to put it there so i’m going to get a minus sign

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out front and then the two times the whole input here

so two times the whole input plus six and then all of that but not the minus

sign so all of that gets over seven so let’s see can we simplify this

so the twos here cancel and so this will be a minus over seven

and so the twos cancel here well if i multiply the two times the

first one the twos will cancel and so i’m going to get minus 7x

and when i multiply the 2 times the minus 3 we’re going to get -6 right there

okay so and then we still get the plus six well

okay i almost forgot that part all right

so yeah the sixes add up to zero and the sevens cancel

and so this will be minus x and then we get a positive x out so i

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didn’t touch that i left it right there and the 7s cancel leaving us with a

minus x right there all right so there we go there’s uh

f and g and that’s how we verify it right there if you’d like to see some

other examples let me know in the comments below and we

can work out some more examples um i hope you have a great day

and i look forward to seeing you in the next episode see you then

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