How to Find the Inverse of a Function (NancyPi)

Published on September 9, 2018 by

MIT grad shows how to find the inverse function of any function, if it exists. The inverse function is the reverse of your original function. It undoes whatever your function did. If your function takes x and gives you y, then the inverse function takes that y and gives you back x. Nancy formerly of MathBFF explains the steps.

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How to find inverse functions: There are three steps to finding the inverse function, if it exists. The first step is to replace the f(x) with just the variable y. Second, swap the x and y variables everywhere they appear in the equation. Third, solve for y again so that you have just "y=" on one side of the equation. If this relation is a function, you can then replace the y with the "f inverse x" notation, or f^-1(x). To know if it is a function, use the Vertical Line Test or consider the form of the equation. Remember that in a function equation, for every x you input into the equation, there can only be one corresponding y value. For an example problem, jump to 00:43 in the video.

For more of my math videos for algebra, algebra 2, precalculus, and more, check out:

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  • Amour Learning 4 years ago

    Hey NancyPi! Thank you for making awesome math videos. This one about inverse functions really made sense. Your channel has inspired me to make my own youtube math tutorials to help students. Keep up the great work!

  • syed amanat 4 years ago

    Even though i didnt watch it, i’ll hit the like button because Math education support by Nancy is what most students need.

  • amty 4 years ago


  • Haram 4 years ago

    I love u, and u make me love math <3

  • ALL CAPS 4 years ago

    Thank you! <3

  • Carlton Davis 4 years ago

    Have you ever done a video on the precise definition of a limit?

  • محبي الله 4 years ago


    • BigMan Ting 4 years ago

      Why u saying that say assalam alaikum

    • BigMan Ting 4 years ago

      Tut tut

  • محبي الله 4 years ago

    I am a university student from the State of Palestine and follow your channel first and take advantage of your explanation.

    • Хидежи Охира ! 4 years ago


    • Roy Lewis 4 years ago

      You are a wise person to seek knowledge from a truly gifted teacher. We wish you the very best in your mathematical studies!
      RL Buffalo, NY USA

    • dean tom Dugma 4 years ago

      whats the name of the university?

  • Leonardo Camargo 4 years ago

    Good video. I had forgotten this stuff.

  • Elena Jikidze 4 years ago

    I looooove your voice

  • Ikhtiar Hossain 4 years ago

    just when I need this !

  • Kush Patel 4 years ago

    i actually needed this ,3

  • Ricardo Mazeto 4 years ago

    I have two questions. 1 – What if the function is something different than x+2/x? 2 – Couldn’t the function have two solutions like the quadratic equation?

    • Romee Cadiz 4 years ago

      Ricardo Mazeto to find the inverse of any function you just switch x and y then solve for x

  • Ron Ramnarinesingh 4 years ago

    Great videos keep up the good work

  • OrphanPaper 4 years ago

    what invert would click thumbs down

    • Windy City 4 years ago

      some dumbfunction probably

  • Maria das Santos 4 years ago

    Great clarity.

  • Raymond H 4 years ago


  • Amy Roberts 4 years ago

    Hey can you plz make a video on alevel math trigonometry

  • Amy Roberts 4 years ago

    Can you plz wear more makeup in your videos. Not that you aren’t pretty it’s just that it’s nice to have some eye candy you know

    • Poopiroopi 4 years ago

      Amy Roberts shut the fuck up amy

    • VR Prasad 4 years ago

      Tf lmao

  • Nathan McLaren 4 years ago

    Thank you for your videos!!! Its really helping me, I am a part time engineering student.. This really helps me understand maths much better

  • FatGuyonTheCouch 4 years ago

    Make linear algebra vids please. Thanks Nancy!

  • PrincessSakuno 4 years ago

    Thank you <3

  • Mwila Arthur 4 years ago

    Hey Nancy please make a binary operations tutorial video please

  • Nemo Clanc 4 years ago

    When a function isn’t one, when there are multiple y values for one x value, then why is sqrt(x) even a function? For x=4 I’ve got y=2; -2.

    • JumpingBamboo 4 years ago

      sqrt(x) is understood to be the principal square root. That is, only the positive square root. y=+-sqrt(x) is not a function to begin with so that’s a non-issue.

      What you’re mixing up is solving for x in a radical equation. It is true that if x^2=4 that x=2 or x=-2.

    • JumpingBamboo 4 years ago

      Also, sqrt(x) is never negative in the real number system. You wrote that sqrt(4)=2 or sqrt(4)=-2, in which the latter is not true.

    • Veritas et Aequitas 3 years ago

      Correct. A calculator only chooses to use the positive version of the rule. You can ONLY discard it if the logic or other math makes it impossible to happen, e.g. negative apples in a story problem. In much math, you also need to track the other possible solution, representing the minus sign as sqrt(-1) by writing “i”.

  • Isaiah Moss 4 years ago

    Thank you so much. I am good at math but been out of school for a while. Your explanation excellent.

  • Chaminda Silva 4 years ago

    Thanks for teaching me how to find the inverse function of any function.

  • Uğur Ali Kaplan 4 years ago

    Thanks for the video. One example of why we would need inverse functions is security. There are two way to hide what you wrote, one is using an encryption function and another is using an hashing function.

    Encryption functions have inverses, so a cryptic mail sent by you can be decrypted using a decryption function. This decryption function is the inverse of the encryption function used by the original sender.

    Hashing functions on the other hand, have no inverse functions and therefore more secure. The thing is, receiver of the mail can’t decrypt your mail so it doesn’t make sense to use it in that context. One useful area it is used is password databases. For example, when you sign up for Youtube, your password will be hashed. So “hunter2”, your password will result in a text like “6y3478twed” etc. Now, when an attacker gets the database, s/he has no way to access your account. Because if attacker were to type “6y3478twed”, it will be hashed again and when compared to hashed password in the database, it will be different. Since there is no inverse function, attacker can’t get the original text neither.

    There is only one way to access your account then. If, as an attacker I have hashed versions of the common passwords, I can search for the passwords I know in the database. So, use uncommon passwords.

    • Kishore Subramani 4 years ago

      Yes this is a good example for application of inverse function

    • Veritas et Aequitas 3 years ago

      This by simple and fairly obvious application is also how passwords are “brute force” hacked: the attacker hashes the passwords see if one of them matches. Instead of a dictionary, there are programs that trivialize generating them procedurally, 11111, 11112, 11113, etc., Or by combining dictionary terms or symbol sets. If this is confusing, think of default router passwords like “StrongNest735”. Those are just 2 words from word lists and a random number, adjective+noun+###. Yes, this is how weak router passwords are. If you have their word lists or can find adj. And noun lists online, it becomes almost trivial.

      The solution? Nonstandard passwords and LONG ones. As computing is faster, especially hashing on graphics cards being exponentially faster than a CPU, the “r@nd0mP@$$w0rD” method becomes trivial if it’s short. The best password? Something very long is now key.

      Except with the advent of quantum computing, if you’re a high enough target for someone to use one, it can be broken almost instantly regardless of length. Have fun knowing that the government can do this and has actually forced RSA to build in a flaw to their “trusted” encryption that allows the gov to break any RSA they want trivially without quantum computing.

      Yup. RSA themselves even warned against using their own encryption for this reason. Of course they were under NDA so the “government” portion was tongue-in-cheek, but it’s easy to catch their hint.

    • Avishek Bhattacharya 3 years ago

      Well, the concept of inverse functions are also applied when developing predictive models of real world problems, like highly non-linear (read as complex) systems of manufacturing defects, organ failure in a human body, or even shut-downs in a processing plant, to name a few…..Long story short, we talk about inverse functions in situations wherein we have a prescribed/known effect, and we’re then required to figure out precisely what are the root-causes (and by how much). Such systems are typically characterised by a known cause-effect relationship – but only based on empirical data. Some of us might be familiar with Reverse Mapping features of Artificial Neural Networks, just to give an example where this concept is applied.

    • Seth Ellis 3 years ago

      In addition to encryption and coding, when we see an effect and make a conjecture about possible causes we do “inverse thinking.” Our justice system depend on it. Companies regularly reverse engineer products which is inverse thinking. In math we constantly deconstruct in order to understand new topics. As a math teacher, I believe learning inverse functions help us understand cause and effect relationships and how functions work. It’s part of logic just as analysis and synthesis have a hand in hand relationship in thinking. The question Nancy brings ups is pertinent, should we teach this kind of algorithm in algebra class? We should always teach thinking with every topic, but is this topic really necessary? Maybe it will help with solving equations using inverse matrices later? Maybe it helps us consider how plus or minus root x is similar to x²? It is a least interesting and noteworthy that inverse functions have a reflection relationship across the y=x line that’s what I emphasize in these lessons. I have them graph with technology to visually check their work.

    • Noor e Hidayat 3 years ago


  • Siddiq Alwi 4 years ago

    Ur a smokeshow

  • manuel zimmerschied 4 years ago

    Can you do integration by parts cal 2? Just covered this yesterday and definitely could use a video over that please 😃.

  • orp0piru 4 years ago

    How does the original function’s limitation (x cannot equal zero) limit the usage of the inverse function?

  • LISSA MERO-LOPEZ 4 years ago

    how do you know what direction the lines are going on the graph ?

    • VR Prasad 4 years ago

      They go in both the directions

    • Veritas et Aequitas 3 years ago

      Memorizing the general shape of each type of function (squared, div, sine, etc.), Then choosing points for each term in the equation, defining the intervals where the lines cross or.touch the graph (odd vs even powers), choosing a test value for each interval, and then testing in a table whether it goes above or below the x axis… You may not be able to draw it to scale, but you can draw the general shape, when it crosses positive/negative, etc.

      Or graphing on a calculator, computer, or experimentally by choosing values of X and plotting them by solving the equation for each.
      That is how graphing is done on a calculator/computer too. Each pixel on the screen is a calculation of the function with an x value.

  • Anirudh 4 years ago

    omg thank you for starting off with explaining what inverse of a function actually means. Had a hard time understanding what it meant. Great video!

  • Mundane Monday 4 years ago

    Why is it x-1 instead of just x?

    • Remy Fritz 4 years ago

      cause its inverse

  • Sophia B. 4 years ago

    A true savior. Some one I can actually understand

  • Cari 4 years ago

    Still confused though..

    • mbs 07 3 years ago

      Dere was a shortcut technique too tho which I forgot 🙈

  • Mister Meaty 4 years ago

    1 a.m. and i have been studying for my final since 10 a.m when i woke up thanks for the help

  • Edrees Hamidi 4 years ago

    thnk you nancy for making such videos
    love from Afghanistan💌

  • رنوشة العراقي 4 years ago

    فديتج جنننننننن شرحهااااااا

  • رنوشة العراقي 4 years ago


  • Shubham Jain 4 years ago

    Make a video on differential equations

  • kSpades 4 years ago

    How do you make the inverse function come back to normal with no ^-1

  • mià bahr 4 years ago

    thank you omg

  • Mahmoud Ashraf 4 years ago

    it’s difficult for me

  • Roshan Sharma 4 years ago

    I just watch her teach cause she’s hot and smart

    • Jason Cohen 4 years ago

      you forgot about hot

  • muhammd mahedi 4 years ago

    You are my best teacher . U make math easier. Thanks a lot

  • Phoenix 4 years ago

    You write everything opposite right?

  • amykaye daugherty 4 years ago

    Thank you for all your videos and hard work

  • Sniper Pro nerf mods 4 years ago

    Why the hell do you talk so slow

    • RiPLiXX 4 years ago

      Why the hell u so slow

    • Sniper Pro nerf mods 4 years ago

      @RiPLiXX my teacher made me take notes on the video, casserly btw

  • Petri J 4 years ago

    How did you get y(x-1) from xy-y?

    • Hari Bhatia 4 years ago

      Petri J y is taken as common

  • Parminder Singh Raj 4 years ago

    thanks for the video!!!!

  • ThatGuyBasher 4 years ago

    1000th like lol

  • Mubaarig Koosaar 4 years ago

    Good lesson

  • Sumayya F. Syed 4 years ago

    I understand how to do it, I just don’t understand what the point of this function is. And how do we consider y=f(x) but then it’s also y=f^-1(x), y should be y=y, but it’s also two opposite functions??

  • BILTON31 4 years ago

    THank so much Nancy. You have opened my future in this topic and where you explained about domains. .

  • Edmund Castellanos 4 years ago

    Thank you, NancyPi 😀

  • TheSanctifieddiva 4 years ago

    💆‍♀️ me no understand; too much changes

  • Derrick Iqbal 4 years ago

    why isnt anyone talking about how beautiful this gal is!

    • Lil Sandal 3 years ago

      because no one cares how she looks. we just here for the math bro

    • Derrick Iqbal 3 years ago

      @Lil Sandal u gay?

      Yeah…most probably

    • Lil Sandal 3 years ago

      @Derrick Iqbal Ay bro I’m sorry, ik you wish I was, but things don’t always work out the way you want them to. You also sound pretty insecure about your sexuality, maybe you should learn to embrace it. I’m sure there’s a plethora of guys out there who would love to date you 🙂

    • Loopy Playz 3 years ago

      @zone kid No its because your a kid and we are not perverts

  • Prince G-MO 4 years ago

    Your vids are a godsend.

  • MS 4 years ago

    Demonstrated with inverse writing. Respect.

    • FunWithAiman 3 years ago

      Hah it’s appently mirrored using a lightboard from what I understand

    • Veritas et Aequitas 3 years ago

      It’s a clear panel and flipped horizontally. This means she’s right-handed.

  • Sergi hassan 4 years ago

    If she was my maths teacher I won’t skip a single class😚

    • Everything 4 years ago

      Sergi hassan 😂🤣😂

    • Loopy Playz 3 years ago


  • lame ellen 4 years ago

    i think you the best math teacher ever…

    • Geema Roy 3 years ago


  • Carrot T 4 years ago

    I ate a potato before watching this video.

  • Mohammed Kashif 4 years ago

    Can’t concentrate

  • Abdul-Raman Murtala 4 years ago

    I love you

  • Matthew Harrison 4 years ago

    This video really helped,but i really wish she would go over graphs more. Otherwise shes amazing.

  • Everything 4 years ago

    How can you do that??

  • Everything 4 years ago

    Writing backwards

  • leaking hamburger helper 4 years ago

    Next video: how to reverse letters

  • KC Truenorth 4 years ago

    what grade is this for?

  • HALE BURNSIDE MD 4 years ago

    i took PRE-Calc in high school and had a very BAD attitude about it…I thought they should have been teaching Calculus in High school. my friend in Florida was taking calculus at his high school , thus i thought this class was like a place holder , a nothing class because the school teacher could not teach calculus because he was too dumb or something .. i felt like i was in a remedial class… BUT THE FACT IS….. this class forms the basis for A LOT OF CALCULUS and when you factor out calculus equations , you have to know the steps you learn in PRE CLAC…I just wish my teacher would have told me this .. thus I want everyone who is learning this , learn this as well as you are able , it will make Calculus a breeze …. I took two years of Calculus and I was always thank full for my GREAT math teachers in high school..

  • Ryan Vo 4 years ago

    Thanks!!! ::::::::::::::))))))))))))

  • master _aj 3 years ago

    Thank you so much it help me a lot

  • AK Sports Creation 3 years ago

    So cute

  • Geema Roy 3 years ago

    Is this the range of a function

  • mayra Martinez 3 years ago

    Like this comment if you’re here for mr.gaugler’s class

  • Alan Goldstein 3 years ago

    According to Nancy….why do we want to do this? “I don’t know!” This is a very disappointing answer from a very good teacher.

    • Veritas et Aequitas 3 years ago

      “Because your teacher wants you to”, lol.

      I’m sure as an MIT grad, even if she isn’t a STEM major because she says it’s not her job, she could think of applications. Maybe she’s just being facetious.

  • Starmyzen 127 3 years ago

    Is this on SAT?

    • Meme Me 3 years ago

      Starmyzen 127 yes

  • truc luong 3 years ago

    no more vid?

  • Twisted Fate 3 years ago

    Thank you very much appreciated😤

  • Hitler was right. 3 years ago

    She doesn’t know how to teach. Disliked.

  • Akin Aykenar 3 years ago

    Should you add X not equal to 1? Because you’d be dividing by zero?

    • Veritas et Aequitas 3 years ago

      Yes, and during the solving she had x replaced with y, so at that time it would have read y≠0. She then changes it to f-¹(x). Be careful to keep track of what ends up where after moving things around and renaming them. This is a good reason to perhaps not rename them.

      If anyone wonders what happens at zero, that’s the vertical asymptote, where y (the value of the function based on the input x) becomes “unbounded”, going toward** (not to) infinity or -infinity, never quite reaching it. That is a “limit”..

      Very close to the point where several concepts of graphs, trig, and calculus finally come together…. Before going down a dozen other rabbit holes, lol.

  • The Next JoJo 3 years ago

    give her the best female ever award

  • Panayiotis Krialakos 3 years ago

    You make mathematics to look more beautiful..

    • اسامة الرفاعي 3 years ago

      😂yes right

  • Edward Mulenga 3 years ago

    Thank you Nancypi….. I have an exam tomorrow

  • syousef 3 years ago

    You videos are excellent for describing how the methods work. Absolutely first rate. I think these will really help my kids get on top of the methods. But your videos are also awful for discussing why you would want to apply the method, and why it works. It’s not really your fault. The math is taught so damn badly, and by that I mean precisely that why you’d want to do it and why it works are not generally taught when the topics are first brought up (if at all). “Becauise you’re told you have to” by your teacher or in a test is certainly a reason you want to do it – to demonstrate that you can and earn credit BUT it is NOT the reason the method exists or is taught.

  • abdu2s 3 years ago

    Hot AF

  • Gunnar Jensen 3 years ago

    If F(x) is a simple multiplication or sum of parts ??

  • Monzir Mohamed 3 years ago

    Like the vidio helped me with precalculus

  • Veritas et Aequitas 3 years ago

    First step: just replace x with y in the function, as long as it’s already in the form f(x)= ____

  • Veritas et Aequitas 3 years ago

    Be careful to remember semantics when stating the inverse function “doesn’t exist”. Some inverse still does, but it just technically is not a “function”.

  • Mrs. Health 3 years ago

    Who watches her video because she is cute?

  • Love You 3 years ago

    Why did the middle eastern find algebra??

    • David Amaer 3 years ago

      Aww thx ☺️

  • Kumar Eklavya 3 years ago

    Thanks for that mam. You are looking so sweet.

  • D FI 3 years ago

    1. swap y with x
    2. Solve the function for y

  • Nyar Mhamad 3 years ago


  • Sir Ivan Wide 3 years ago

    If ya need inverse of fx then on the rhs 2 div by x-1 if ya need inverse of fy then 2 div by y-1. 😁noob mode

  • kib Hab 3 years ago

    You are great Nancy

  • Deepak Jain 3 years ago


  • Anis Matar 3 years ago

    One comment about swapping x and y I think people do it so they can later on solve for y as they are used to since y actually represents old x but makes the dealing with equation easier.

  • ආකීල් AWM 3 years ago

    how to inverse.
    f(x)= -x²

  • Yash Rajput 3 years ago

    I love your teaching skill

  • Ben Vredevelt 3 years ago


    Thank you for your positive influence.



  • Geoffrey Elward 3 years ago

    So many thirsty simps out here

  • RunForestRun 3 years ago

    Why am I watching this at 4 AM. Idk
    But I learned something though

  • moh afl 3 years ago

    0:25 yezuku itha naan sonna yenna paithiam nu solluranval yekka nalla iru # just for marking us more mental

  • jullissa rose 3 years ago

    hi can you do another one with only f of x equal x minus 12 divide by 4

  • VICTOR WILLIAMS 3 years ago

    What a beauti

  • Isidoros Argyroudis 3 years ago

    thaaaank you. Ευχαριστώ κούκλα

  • PoeticSonic 3 years ago

    thank you it was very clear with a usable example rather than in other videos. again THANK U

  • Azhar Imam 3 years ago

    I love you

  • Singgih Solution 3 years ago

    Smart solution

  • David Kwon 3 years ago

    Thank you. This is so good to my son and daughter.

  • Micah Burnside 3 years ago

    Greatest “why” I’ve ever heard. “I don’t know but your teacher wants you to”. Perfect lol

    • Snow Bird 3 years ago

      she spitting facts xD

  • David-Paul 3 years ago

    Fabulous as usual save hours of reading.

  • musabbir arham 3 years ago

    I wish she could be my mathematics teacher 😢

  • Adhil the salbatic 3 years ago

    Me : why do I want to learn this?

    Nancy : I don’t know your teacher wants you to do this

    • piyush kumar 3 years ago

      I also

    • M G 3 years ago


    • Miaulin K 3 years ago

      Somehow or another it’s used (or gone back to) in calculus with derivatives. I’m in a section on it, but since I can’t remember how to do inverse stuff, that’s why I’m watching the video.

  • stuart paterson 3 years ago

    Watching this during Corona-lockdown… I’ve been reading a lot of popular physics so far and as it’s been going well, so i thought to myself i really must get my head around calculus. Knowing that @NancyPi is the absolute best i figured, start at the very beginning of her precalc series. Having watched this first video, i’m happy to report that I am none the wiser and now there is a nerve twitching on the side of my head 🙂

  • Parvathy B Nair 3 years ago

    Tnx mam

  • MS SUNSHINE 3 years ago

    How abt if there’s an exponent

  • Bishal Baral 3 years ago


  • Byron Hashimoto 3 years ago


  • Onita Buffum 3 years ago


  • Jonathan Lee 3 years ago

    lol. “Because your teacher asked you to”

  • Noor Muhammad 3 years ago

    nancy which type of software do you use for writing

  • Nick System 3 years ago

    Love that intro haha

  • irh dqo 3 years ago

    Why don’t the y’s cancel out so it’s x/y=2 instead of xy=y+2?

  • Tolu Maku 3 years ago


  • Gunnar Jensen 3 years ago

    Beautiful teacher..

Comments are closed.