# How to Find the Inverse of a Function (NancyPi)

128,212
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.

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: http://nancypi.com

Category

Advertise here. Telegram, Signal - Call/message +1-868-308-4028.

• 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!

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

• NANCY

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

• Thank you! <3

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

• Hi

• Why u saying that say assalam alaikum

• Tut tut

• Peace

• 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

• whats the name of the university?

• Good video. I had forgotten this stuff.

• just when I need this !

• i actually needed this ,3

• 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?

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

• Great videos keep up the good work

• what invert would click thumbs down

• some dumbfunction probably

• Great clarity.

• Y

• Hey can you plz make a video on alevel math trigonometry

• 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

• Amy Roberts shut the fuck up amy

• Tf lmao

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

• Make linear algebra vids please. Thanks Nancy!

• Thank you <3

• 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.

• 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.

• 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.

• 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”.

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

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

• 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.

• Yes this is a good example for application of inverse function

• 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.

• 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.

• 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.

• epic

• Ur a smokeshow

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

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

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

• They go in both the directions

• 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.

• 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!

• Why is it x-1 instead of just x?

• cause its inverse

• A true savior. Some one I can actually understand

• Still confused though..

• Dere was a shortcut technique too tho which I forgot 🙈

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

• thnk you nancy for making such videos
love from Afghanistan💌

• فديتج جنننننننن شرحهااااااا

• 😍😘😘😘😘😘😘😘😘😘😘😘

• Make a video on differential equations

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

• thank you omg

• it’s difficult for me

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

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

• You write everything opposite right?

• Thank you for all your videos and hard work

• Why the hell do you talk so slow

• Why the hell u so slow

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

• How did you get y(x-1) from xy-y?

• Petri J y is taken as common

• thanks for the video!!!!

• 1000th like lol

• Good lesson

• 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??

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

• Thank you, NancyPi 😀

• 💆‍♀️ me no understand; too much changes

• why isnt anyone talking about how beautiful this gal is!

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

• @Lil Sandal u gay?

Yeah…most probably

• @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 🙂

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

• Your vids are a godsend.

• Demonstrated with inverse writing. Respect.

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

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

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

• Sergi hassan 😂🤣😂

• Perv

• i think you the best math teacher ever…

• Yes

• I ate a potato before watching this video.

• Can’t concentrate

• I love you

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

• How can you do that??

• Writing backwards

• Next video: how to reverse letters

• what grade is this for?

• 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..

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

• Thank you so much it help me a lot

• So cute
Thanks…

• Is this the range of a function

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

• 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.

• “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.

• Is this on SAT?

• Starmyzen 127 yes

• no more vid?

• Thank you very much appreciated😤

• She doesn’t know how to teach. Disliked.

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

• 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.

• give her the best female ever award

• You make mathematics to look more beautiful..

• 😂yes right

• Thank you Nancypi….. I have an exam tomorrow

• 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.

• Hot AF

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

• Like the vidio helped me with precalculus

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

• 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”.

• Who watches her video because she is cute?

• Why did the middle eastern find algebra??

• Aww thx ☺️

• Thanks for that mam. You are looking so sweet.

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

• 👍👍👍👍👍👍👍👍👍👍👍👍👍👍👍👍👍👍👍👍

• 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

• You are great Nancy

• Thanks

• 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.

• miss.
how to inverse.
f(x)= -x²

• I love your teaching skill

• Nancy,

Thank you for your positive influence.

Love,

Benjamin

• So many thirsty simps out here

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

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

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

• What a beauti

• thaaaank you. Ευχαριστώ κούκλα

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

• I love you

• Smart solution

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

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

• she spitting facts xD

• Fabulous as usual save hours of reading.

• I wish she could be my mathematics teacher 😢

• Me : why do I want to learn this?

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

• I also

• Hahaha

• 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.

• 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 🙂

• Tnx mam

• How abt if there’s an exponent

• hello

• 03:35
04:42
0:12

• 02:31
03:38
01:04