Introduction to function inverses | Functions and their graphs | Algebra II | Khan Academy

Introduction to function inverses | Functions and their graphs | Algebra II | Khan Academy


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.

74 thoughts on “Introduction to function inverses | Functions and their graphs | Algebra II | Khan Academy

  • July 16, 2010 at 6:29 pm
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    @chizzy555 Would be Algebra II or pre-calc

    Reply
  • July 16, 2010 at 6:46 pm
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    @chizzy555 Maybe some trades. I could imagine the inverse trig functions would be pretty useful if you're building something.

    Reply
  • July 16, 2010 at 11:59 pm
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    Interesting!

    Reply
  • July 24, 2010 at 9:22 am
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    that just blew my mind

    Reply
  • August 11, 2010 at 12:54 pm
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    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.

    Reply
  • November 23, 2010 at 5:52 am
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    loving the new writing Sal. Much neater!

    Reply
  • June 22, 2011 at 11:25 pm
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    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.

    Reply
  • August 25, 2011 at 2:38 pm
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    This makes me want my teacher to die >:D

    Reply
  • December 20, 2011 at 5:44 pm
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    5:47 I like it quick and dirty too >:D

    Reply
  • January 6, 2012 at 4:09 am
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    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!

    Reply
  • January 17, 2012 at 12:09 am
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    imagine if he taught a class on teaching, then maybe education in America would get better…

    Reply
  • April 22, 2012 at 9:57 am
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    why do i even bother going to my maths lectures.

    Reply
  • June 20, 2012 at 3:10 am
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    Linkin Park

    Reply
  • July 3, 2012 at 4:04 pm
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    sal,u are great..!
    made a you tube account just to thank you..!

    Reply
  • August 31, 2012 at 2:30 am
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    Makes perfect sense. Before, I didn't know what solving for the equation did. Now, thanks to Sal, I do!

    Reply
  • October 4, 2012 at 6:04 pm
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    Trust me, i'm an engineer !

    Reply
  • November 21, 2012 at 12:03 am
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    There are huge problems with the public school system as well as universities.

    Reply
  • November 23, 2012 at 10:56 pm
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    i would never have made it through half of last year without these videos. thanks so much man.

    Reply
  • November 27, 2012 at 1:49 am
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    is f(n) the same as f(x)? thanks to anyone who answers!

    Reply
  • November 28, 2012 at 3:21 am
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    how is y/2 the same as 1/2y?

    Reply
  • December 6, 2012 at 10:13 pm
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    what software are they using? :3

    Reply
  • February 25, 2013 at 4:01 am
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    thank you very much

    Reply
  • June 14, 2013 at 4:00 pm
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    I learned this in my algebra 2 class if that helps.

    Reply
  • August 20, 2013 at 3:00 am
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    160000 views and only 250 likes?

    Reply
  • September 3, 2013 at 7:37 am
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    coming here when my teacher fails (that means i'm always here)

    Reply
  • September 25, 2013 at 9:46 pm
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    why cant my schools just play your vids in classes???

    Reply
  • November 6, 2013 at 8:35 pm
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    I turned 17 today

    Reply
  • November 6, 2013 at 8:41 pm
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    And i would like to say i love khan academy

    Reply
  • December 14, 2013 at 11:33 am
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    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.

    Reply
  • March 19, 2014 at 4:18 pm
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    Why do the functions are reflecting over the line y=x?

    Reply
  • August 10, 2014 at 8:42 pm
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    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

    Reply
  • August 17, 2014 at 1:28 pm
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    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

    Reply
  • September 14, 2014 at 9:13 am
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    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

    Reply
  • September 20, 2014 at 4:15 pm
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    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?

    Reply
  • November 10, 2014 at 2:06 am
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    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!

    Reply
  • March 10, 2015 at 10:30 pm
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    7:37 is confusing
    i cant.

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  • May 7, 2015 at 6:37 pm
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    I understood it until 3:59, completely lost now.

    Reply
  • May 16, 2015 at 7:10 am
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    thankyou so much!

    Reply
  • September 4, 2015 at 1:12 pm
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    You're the best!

    Reply
  • March 23, 2016 at 4:37 pm
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    I'm a seventh grader being forced to watch this help me but it's kinda cool

    Reply
  • September 7, 2016 at 1:28 am
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    THANKS SO MUCH I ALWAYS COME TO YALL WHEN I NEED MATH HELP YOU'RE THE BEST!!!!

    Reply
  • September 18, 2016 at 8:17 pm
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    Im a freshman watching this for a honors junior class. Im gonna dislike the video to negate the view.

    Reply
  • September 24, 2016 at 4:33 pm
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    Very helpful

    Reply
  • October 8, 2016 at 7:36 pm
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    As always, this video was a life saver 🙂

    Reply
  • December 23, 2016 at 3:18 pm
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    It was a really awesome video but to be honest i still don't understand😭😭😢😢😵

    Reply
  • April 2, 2017 at 3:23 pm
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    can i ask you why we switch places in the end? pls someone help me. Idk why we switch places.

    Reply
  • April 9, 2017 at 1:12 pm
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    شكرا جزيلا على هذا الشرح المبسط والمفهوم.

    Reply
  • April 19, 2017 at 10:23 pm
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    MATH IS CANCER!!!!!!!!!

    Reply
  • June 24, 2017 at 3:47 pm
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    this is so simple! i have no idea why my sir is complicating it by interchanging the values XD

    Reply
  • August 22, 2017 at 4:31 am
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    sir u divide 2 on both sides but your answer is wrong 4:08

    Reply
  • September 12, 2017 at 12:40 am
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    Do we need this in life lmao. 😂😂😂

    Reply
  • September 15, 2017 at 10:28 pm
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    How did you get half 1/2

    Reply
  • October 17, 2017 at 12:56 pm
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    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!!

    Reply
  • October 19, 2017 at 7:33 pm
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    my math teacher, why didn't you tell me this. My life would have been so much easier.

    Reply
  • December 15, 2017 at 10:24 pm
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    I dont like the notation for inverse functions. I thougt f exponent -1 means 1/f. irritating

    Reply
  • March 31, 2018 at 6:17 am
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    what that range figure looks like😁

    Reply
  • April 5, 2018 at 8:52 pm
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    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..

    Reply
  • April 14, 2018 at 1:11 am
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    Sir please tell me that what is the condition to be the function is invertable

    Reply
  • August 13, 2018 at 5:43 pm
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    If only all my teachers were this good…

    Reply
  • August 29, 2018 at 6:37 pm
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    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.

    Reply
  • September 1, 2018 at 12:23 am
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    Simplifying things so much.has been a great help to understand concept easily.thnk u

    Reply
  • September 24, 2018 at 4:44 am
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    Why there's Thai caption?

    Reply
  • October 4, 2018 at 3:49 am
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    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

    Reply
  • October 12, 2018 at 9:12 am
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    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.

    Reply
  • January 11, 2019 at 8:45 pm
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    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

    Reply
  • May 1, 2019 at 12:09 am
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    Good job, I completely understood but unfortunately not what I’m looking for.

    Reply
  • August 13, 2019 at 6:03 pm
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    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

    Reply
  • August 26, 2019 at 7:35 pm
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    But why do we want to find an inverse?

    Reply
  • September 12, 2019 at 3:33 pm
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    Thanks my teacher literally did 2 problems the whole period

    Reply
  • September 23, 2019 at 3:44 am
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    the goat

    Reply
  • October 2, 2019 at 12:02 am
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    My god, this is amazing.

    Reply
  • October 9, 2019 at 4:14 am
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    wut

    Reply
  • October 13, 2019 at 9:52 am
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    KHAN ACADEMY, KEEP IT UP!!!
    THE BEST LEARNING CHANNEL.
    I REALLY LOVE IT FROM BOTTOM OF MY HEART!
    EXCELLENT job!

    Reply
  • November 18, 2019 at 2:03 am
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    Bruh Algebra 2 teachers CANT TEACH

    Reply

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