Let’s think about what

functions really do, and then we’ll think about the idea of

an inverse of a function. So let’s start with a pretty

straightforward function. Let’s say f of x is

equal to 2x plus 4. And so if I take f of 2, f of 2

is going to be equal to 2 times 2 plus 4, which is 4

plus 4, which is 8. I could take f of 3, which

is 2 times 3 plus 4, which is equal to 10. 6 plus 4. So let’s think about it

in a little bit more of an abstract sense. So there’s a set of things that

I can input into this function. You might already be

familiar with that notion. It’s the domain. The set of all of the things

that I can input into that function, that is the domain. And in that domain, 2 is

sitting there, you have 3 over there, pretty much you could

input any real number into this function. So this is going to be all

real, but we’re making it a nice contained set here just

to help you visualize it. Now, when you apply the

function, let’s think about it means to take f of 2. We’re inputting a number, 2,

and then the function is outputting the number 8. It is mapping us from 2 to 8. So let’s make another set here

of all of the possible values that my function can take on. And we can call that the range. There are more formal ways to

talk about this, and there’s a much more rigorous discussion

of this later on, especially in the linear algebra playlist,

but this is all the different values I can take on. So if I take the number 2 from

our domain, I input it into the function, we’re getting

mapped to the number 8. So let’s let me draw that out. So we’re going from 2 to

the number 8 right there. And it’s being done

by the function. The function is

doing that mapping. That function is mapping

us from 2 to 8. This right here, that

is equal to f of 2. Same idea. You start with 3, 3 is being

mapped by the function to 10. It’s creating an association. The function is mapping

us from 3 to 10. Now, this raises an

interesting question. Is there a way to get back from

8 to the 2, or is there a way to go back from

the 10 to the 3? Or is there some

other function? Is there some other function,

we can call that the inverse of f, that’ll take us back? Is there some other

function that’ll take us from 10 back to 3? We’ll call that the inverse

of f, and we’ll use that as notation, and it’ll take

us back from 10 to 3. Is there a way to do that? Will that same inverse of f,

will it take us back from– if we apply 8 to it– will

that take us back to 2? Now, all this seems very

abstract and difficult. What you’ll find is it’s

actually very easy to solve for this inverse of f, and I think

once we solve for it, it’ll make it clear what

I’m talking about. That the function takes you

from 2 to 8, the inverse will take us back from 8 to 2. So to think about that, let’s

just define– let’s just say y is equal to f of x. So y is equal to f of x,

is equal to 2x plus 4. So I can write just y is equal

to 2x plus 4, and this once again, this is our function. You give me an x,

it’ll give me a y. But we want to go the

other way around. We want to give you

a y and get an x. So all we have to do is

solve for x in terms of y. So let’s do that. If we subtract 4 from both

sides of this equation– let me switch colors– if we subtract

4 from both sides of this equation, we get y minus 4 is

equal to 2x, and then if we divide both sides of this

equation by 2, we get y over 2 minus 2– 4 divided by 2

is 2– is equal to x. Or if we just want to write it

that way, we can just swap the sides, we get x is equal to

1/2y– same thing as y over 2– minus 2. So what we have here is

a function of y that gives us an x, which is

exactly what we wanted. We want a function of these

values that map back to an x. So we can call this– we could

say that this is equal to– I’ll do it in the same color–

this is equal to f inverse as a function of y. Or let me just write it

a little bit cleaner. We could say f inverse as a

function of y– so we can have 10 or 8– so now the range is

now the domain for f inverse. f inverse as a function of y

is equal to 1/2y minus 2. So all we did is we started

with our original function, y is equal to 2x plus 4, we

solved for– over here, we’ve solved for y in terms of x–

then we just do a little bit of algebra, solve for x in terms

of y, and we say that that is our inverse as a function of y. Which is right over here. And then, if we, you know, you

can say this is– you could replace the y with an a, a b,

an x, whatever you want to do, so then we can just

rename the y as x. So if you put an x into this

function, you would get f inverse of x is equal

to 1/2x minus 2. So all you do, you solve for x,

and then you swap the y and the x, if you want to

do it that way. That’s the easiest way

to think about it. And one thing I want to point

out is what happens when you graph the function

and the inverse. So let me just do a

little quick and dirty graph right here. And then I’ll do a bunch of

examples of actually solving for inverses, but I really

just wanted to give you the general idea. Function takes you from the

domain to the range, the inverse will take you from that

point back to the original value, if it exists. So if I were to graph these–

just let me draw a little coordinate axis right here,

draw a little bit of a coordinate axis right there. This first function, 2x plus 4,

its y intercept is going to be 1, 2, 3, 4, just like that, and

then its slope will look like this. It has a slope of 2, so it will

look something like– its graph will look– let me make it a

little bit neater than that– it’ll look something like that. That’s what that

function looks like. What does this

function look like? What does the inverse function

look like, as a function of x? Remember we solved for x,

and then we swapped the x and the y, essentially. We could say now that y is

equal to f inverse of x. So we have a y-intercept

of negative 2, 1, 2, and now the slope is 1/2. The slope looks like this. Let me see if I can draw it. The slope looks– or the line

looks something like that. And what’s the

relationship here? I mean, you know, these look

kind of related, it looks like they’re reflected

about something. It’ll be a little bit more

clear what they’re reflected about if we draw the

line y is equal to x. So the line y equals

x looks like that. I’ll do it as a dotted line. And you could see, you have

the function and its inverse, they’re reflected about

the line y is equal to x. And hopefully, that

makes sense here. Because over here, on

this line, let’s take an easy example. Our function, when you take

0– so f of 0 is equal to 4. Our function is mapping 0 to 4. The inverse function, if

you take f inverse of 4, f inverse of 4 is equal to 0. Or the inverse function is

mapping us from 4 to 0. Which is exactly

what we expected. The function takes us from the

x to the y world, and then we swap it, we were swapping

the x and the y. We would take the inverse. And that’s why it’s reflected

around y equals x. So this example that I just

showed you right here, function takes you from 0 to 4– maybe I

should do that in the function color– so the function takes

you from 0 to 4, that’s the function f of 0 is 4, you see

that right there, so it goes from 0 to 4, and then

the inverse takes us back from 4 to 0. So f inverse takes us

back from 4 to 0. You saw that right there. When you evaluate 4 here,

1/2 times 4 minus 2 is 0. The next couple of videos we’ll

do a bunch of examples so you really understand how to solve

these and are able to do the exercises on our

application for this.

@chizzy555 Would be Algebra II or pre-calc

@chizzy555 Maybe some trades. I could imagine the inverse trig functions would be pretty useful if you're building something.

Interesting!

that just blew my mind

I see you've updated your sketch pad. Never thought these videos could get any better than they already were. Thanks, Sal, your a lifesaver.

loving the new writing Sal. Much neater!

Who the hell hired my professor? and why am i paying 1,000 dollars to take this class when i can just come on here and learn more in ten minutes.

This makes me want my teacher to die >:D

5:47 I like it quick and dirty too >:D

you explain better than my math teacher ever could in his life! THANK YOU! My head does not swim in confusion anymore thanks to your detailed lessons!

imagine if he taught a class on teaching, then maybe education in America would get better…

why do i even bother going to my maths lectures.

Linkin Park

sal,u are great..!

made a you tube account just to thank you..!

Makes perfect sense. Before, I didn't know what solving for the equation did. Now, thanks to Sal, I do!

Trust me, i'm an engineer !

There are huge problems with the public school system as well as universities.

i would never have made it through half of last year without these videos. thanks so much man.

is f(n) the same as f(x)? thanks to anyone who answers!

how is y/2 the same as 1/2y?

what software are they using? :3

thank you very much

I learned this in my algebra 2 class if that helps.

160000 views and only 250 likes?

coming here when my teacher fails (that means i'm always here)

why cant my schools just play your vids in classes???

I turned 17 today

And i would like to say i love khan academy

Teachers in our school just complicate it as much as possible to mash our brains…but you Khan Academy are something that brought help to me and many other people…I am making progress in math only because of you and not because of my teachers…they are very bad,thank you,keep going.

Why do the functions are reflecting over the line y=x?

It's funny how I learn so easily and so much from you vs all my years of schooling. "Make things as simple as possible, just not any simpler" and you my friend definitely get the job done. Thanks a heap load! And please do more

THANK YOU KAHN ACADEMY YOURE MY HERO IF I GAVE YOU A DOLLAR FOR EVERY TIME YOU HAVE MADE EVERYTHING CLEAR I WOULD BE IN REALLY BAD DEBT

Woaw man!! Just woaw!! I'm studying Biomedicine at my university, and my professors have tried to teach this to us for a while now, nobody gets it. Then i watched your movie and you explained it sooooo easily in just 9 minutes. THANK YOU! <3

In 5:58 you say that you can go back to the domain from the range "If possible". Is there an instance in which it is not possible?

one perhaps dumb question but I'd like to ask if the inverse of a function really is changing the dependent variable to the independent

I mean if you have say

y = 2x + 1

x = 1/2y – 1/2

if you graph the f(x) you get the same as if you graph the f(y), it is only when you change the y to x and x to y in the second function (the inverse) that you get the correct graph.

Does it only become the inverse function when the switching of the y and x is done? I mean if you solve for x with respect to y you will still obviously get the same graph.

Thanks!

7:37 is confusing

i cant.

I understood it until 3:59, completely lost now.

thankyou so much!

You're the best!

I'm a seventh grader being forced to watch this help me but it's kinda cool

THANKS SO MUCH I ALWAYS COME TO YALL WHEN I NEED MATH HELP YOU'RE THE BEST!!!!

Im a freshman watching this for a honors junior class. Im gonna dislike the video to negate the view.

Very helpful

As always, this video was a life saver 🙂

It was a really awesome video but to be honest i still don't understand😭😭😢😢😵

can i ask you why we switch places in the end? pls someone help me. Idk why we switch places.

شكرا جزيلا على هذا الشرح المبسط والمفهوم.

MATH IS CANCER!!!!!!!!!

this is so simple! i have no idea why my sir is complicating it by interchanging the values XD

sir u divide 2 on both sides but your answer is wrong 4:08

Do we need this in life lmao. 😂😂😂

How did you get half 1/2

its just amazing how this video is uploaded way back in 2010 but is still very useful today and will be for the years to come. Thank you Khan Academy!!

my math teacher, why didn't you tell me this. My life would have been so much easier.

I dont like the notation for inverse functions. I thougt f exponent -1 means 1/f. irritating

what that range figure looks like😁

Ok, when u understand mathematics then it's the sexiest subject ever, but when u don't get it, u hate it like hell. reason being that straight forward stuff that u could easily get they play around with it to make it complicated so you could get confused

Now y= 2X + 4 … and therefore x= 1/2y – 2 …. why don't we just stick to that.. why bring the term of F to the power of freaking -1 then a "y" between ( ) and after all that it equals "x" which we already know its equation..

Sir please tell me that what is the condition to be the function is invertable

If only all my teachers were this good…

I wish you'd actually write out the subtracting and dividing on both sides of the equation instead of just jumping to the answer. It makes it hard to follow along.

Simplifying things so much.has been a great help to understand concept easily.thnk u

Why there's Thai caption?

It's funny how after 2 weeks of garbage from my teachers mouth I was lost and learned nothing but after 9 minutes I understand the whole thing

This video was super helpful. Your ability to explain things makes it unbelievably easy to comprehend and conceptualize these otherwise confusing concepts. It's bizarre that I could spend about an hour in a class with an instructor going over this material and still left not knowing much more than I did at the beginning of that hour, yet I watch this video that's less than 10 minutes, and I totally get it now. However, what I can't wrap my brain around is how 69 individuals found this video unhelpful and gave it a thumbs down.

at 6:50 did you go the wrong way for 2x shouldn't go to the right and not the left? cause left is -2x

Good job, I completely understood but unfortunately not what I’m looking for.

YOU'RE MY G BRO UR CARRYING ME THROUGH SAT MATHS 2 . SEND ME UR PAYPAL IF I BECOME RICH THANKS TO U IM SENDING U SO MUCH CASH BRO

But why do we want to find an inverse?

Thanks my teacher literally did 2 problems the whole period

the goat

My god, this is amazing.

wut

KHAN ACADEMY, KEEP IT UP!!!

THE BEST LEARNING CHANNEL.

I REALLY LOVE IT FROM BOTTOM OF MY HEART!

EXCELLENT job!

Bruh Algebra 2 teachers CANT TEACH