[Music] in section 6.2 we're going to learn about inverse functions the first part of this is really just review getting some properties of inverse functions uh how they look what their behaviors are then we're going start taking derivatives of inverse functions it's really not a hard section I think you guys will will be okay with it uh but we do need to know learn some things about it firstly what is it take for a function to have an inverse X and A Y it's got to have an X and A Y right but it's got to be in a relationship it's very specific one to one that's right it's got to be one have you ever heard that term one: one before one: one means that in a function every input has one unique output that means if you plug in in a number and you get out two you get out two once you never get out again um in other words I can show this with the horizontal line test you ever familiar with the horizontal line test one of these functions is one to one one of them is not which one's one to one left or right that's one: one this is not out the horizontal line test passes here it fails here this says that you have multiple outputs of the same value that would fail being one: one that matters for inverses because an inverse is you switch X and Y if you switch X and Y inputs become outputs outputs become inputs let me explain why that's important if you have a non one to1 function here and you try to find an inverse it's going to go like this it's not even going to be a function anymore and that's a problem so what we say is that for a one:1 function we can find an inverse for for a one: one function f ofx it's absolutely true that if functions 1: one it will have an inverse so if a for a one: one function f ofx an inverse it's denoted F you've seen this before I'm sure F iners it's not F to the little negative one okay it's not how we say that uh it's not an exponent at all it's probably the worst notation all the mathematics I don't know why they have the negative one there it does look like an exponent but it's not one it's pronounced F inverse of X so for a one: one function f ofx an inverse F inverse of X can be [Music] found little side note here this is how you check whether two functions are inverses it's just a little simple composition I if you do a composition of f inverse on F or F on F inverse what an inverse does is it undoes a function and a function undoes its inverse I always say it's like a any you guys have kids nobody oh you do okay so it's like a little rotten kid all right so they're not rotten though are they okay good well if I have kids they're probably going to be rotten kids so it's like if you went around around cleaning your house and the kid right behind you and no no and put everything right back to where it went so they have their toys all messy you pick up toy they go and they put it right back okay that's what inverses do they undo everything that you just did so if I plug in X into my inverse it's going to do something f is going to clean it up put it right back where it was or vice versa F would do something and inverse puts it right back where it was what that means is that whatever I plug in it's going to go through the machine through the anti- machine and give you back exactly what you plugged in that means that if you compose two functions that are inverses your result will be X and that's either way that you go here and that's how you check let me give you an example on how you find inverses then we'll check it with the composition to show that I'm I'm right here that give you the right thing and then we'll talk about the graphs and we'll probably end it right there so let's find an inverse first thing I want to tell you it is one: one uh next thing let's find an inverse here's how you do that ultimately our goal on inverses is to switch the X and the y That's What inverses do to do that we have to have a y so typically what people will do is they'll change whatever the function name is into y so instead of f ofx we're going to have y equals your function next thing here's the place where you actually make the inverse everywhere you see a y you put an X everywhere you see an X you put a y you just change the variables here so what's that going to become mhm and what's this going to look like one over very good root 2 Yus yep that's right you okay with that so far the idea now is you have to solve for what just solve for what uh give me some ideas on how to solve for y the denominator with X sure we can just switch those the square root three cool we can do that let's 2 y - 3 = 1x^2 absolutely all right then what did you say add three cool we can do that so 2 y if we add three is this yeah if my math is right did you guys get the same thing just for funsies let's make this one fraction so if we made it one fraction 1 + 3 mhm 1 over we'd have to find a common denominator that would give us 3x^2 2x^2 so we'd have 1 + 3x^2 over 2x^2 you guys okay with that one at this point you found your inverse just call it the inverse so so the notation we use if we start with f we're going to have F inverse if we start with G we're going to have G inverse all you do is you put whatever the function name is the inverse notation and then rewrite it quick head now if you're okay with with that one it's really not calculus we're just doing some inverse uh because we're going to be doing some calcul calculus inverse later what I do want to show you is that if you do compose F inverse with f ofx and this is either way you want to go you can do F inverse here f ofx it doesn't matter you're going to X in either case here's what this would say it says what I want you to do I want you to take this function I want you put it everywhere you see X here and see what you get that be the easiest one oh man either way it's going to suck that would be our comp composition are you guys okay with the composition you see where everything's coming from we're taking FX FX we're putting it into our inverse function and we're checking to see that we've done all our work right we're checking to make sure that these are actually inverses do you see what I'm talking about so we'd have one sure plus three no problem this x becomes that function that's what we got this x becomes that function and that's what we got so if we continue with this I'm going have to move over here tell me what the squares do I know it's fast but I wanted to prove it really quick because we're on of time we just proved that those things are actually inverses just do a little bit of algebra it says anytime you compose two functions and you actually get x out of it then you know their inverses uh could you follow that one yeah did I get you fast with that one a little bit do you have questions on it we have about 10 seconds so pretty much just to prove yourself you plug in whatever yeah you you compose them and if your answer is X you know the inverses all right so we're right at the beginning of start of talking about inverses I remember last time that we actually found the inverse of one function the idea behind our inverse is is basically this uh for every one: one function you can find an inverse which is also one: one uh we'll talk about a couple properties right now of inverses I'll give you some examples on how to use this as far as calculus goes then we'll start talking about another topic so a few notes on the relationship between a function and its Inverse First thing anytime you have a function which has an inverse you can always identify the graph pretty easily if you have one or the other so for instance if I have a function let's call that F firstly is that function uh one: one y yeah all right so if it's one to one it has to have an inverse here's the relationship between a function and its inverse on a graph namely a function and its inverse graphically are just a reflection across the line Y X so if this is my function well the inverse because an inverse is found simply by switching the X and the Y coordinates a it's horrible well hopefully you get the idea I'm not the best artist in the world but do you get the idea that I'm trying to show you here it's just a reflection across the line Y X if this is a one:1 function this also has to be a one to one fun function if it's if this was not one: one this would not even be a function that would be a problem so if F feel okay with the idea of the graphs so a function and its inverse are just Reflections across the line yal X second secondly if f is continuous on a certain domain then F inverse will be continuous on a certain domain just keep this in mind the domain of F becomes the range range of f inverse so what I'm going to State this and I'll try to explain a little bit better for you the wording is a little bit weird but here I'll say it this way if f is continuous on a certain domain this doesn't mean continuous everywhere this means that where it's continuous I'll explain the rest of a minute where it's continuous so if f is continuous on a certain domain F inverse will also be continuous on a certain domain however I'll put this in parenthesis here the domain of f inverse is the range of f so so for instance let's say f is continuous somewhere okay it's continuous on its on a certain domain which means it's continuous on its range as well so where a function is continuous it just doesn't have any holes or gaps so what this say is that if f is continuous from X = 3 to X = pos2 well it's also continuous on the range from let's say this is like -2 -2 to pos5 that means F inverse will be continuous from what I say -2 -2 to POS 5 along the x-axis the range of f becomes the domain of f inverse so you could think about it this way if f is continuous along a certain range F inverse will be continuous along that certain domain okie dokie so continuity it doesn't try not think about about just a domain or range issue continuity just means it's continuous as in it has no jumps gaps holes ASM tootes or anything like thata right so what we're trying to say here is that where f is continuous on its range F inverse is continuous on that particular domain it's it stayed a little bit funny so if f is continuous on a certain domain F inverse will be also be continuous on a certain domain but this is the range of f here example small example because I can tell by looking your eyes so you're like what what you talking about here we go uh easy example let's say that here is our here's our function here's our function if this is the function f first Le is it one to one yeah it's got to have an inverse how you find inverses is you just reverse X and Y so if this is at -2 comma 7 we're going to also have a point on the inverse from where at what where cool and if we have a point at -3 1 where's our point on the to be okay my graphs are not to scale but let's make the points bigger and hopefully you get the idea so this just flip-flops right across the y x line well here's the deal what we'd say here is this is continuous along the x-axis from3 to -2 and continuous along the range so that's our domain and continuous along the range from 1 to 7 well that means here it's continuous along the domain of 1 to 7 and along the range from3 to -2 does that make sense you guys see how the range in the domain switches so don't get all hung up on the continuity basis says if this is continuous that's continuous that's the idea anyway also in the last thing thing we'll work most with right now if f is differentiable which means we can find a derivative at any point along the domain for which it's defined if f is differentiable f inverse is also differentiable question can you real quick just Mark where those points are at on the graph like with the box like number comma number so I can make it better on my notes cuz I I'll forget it go for it3 1 -273 okay do the same thing there that's 31 uh 13 7 -2 so a couple things that we know about the properties here a function and its inverse have relationship graphically of just a reflection across the line Y X are you guys okay with that one okay secondly basically basically if this is continuous this is continuous that's the idea if this is not continuous this is not continuous at some point that's the idea okay the range of f becomes the domain of f inverse that's another idea also if f is differentiable means we can find a derivative then F inverse is differentiable and here's actually how you do it if I want to find the derivative of an inverse function here's one way we can do this uh now if you're wondering why don't we just find the inverse and take the derivative of that sometimes it's not that easy to find an inverse explicitly sometimes it's it's hard to do that so we have this way of working around it uh where all we need to do is find the derivative of the actual function and use that to our advantage to find the derivative of the inverse here's how we would at a certain point here's how we show it so if I want to find the derivative of an inverse I can do it two ways I can find the derivative of the function first which would sorry the U the inverse of the function first then go for the derivative or I can do this I can find the derivative of the actual function compose F inverse into it which if you don't know how to find that explicitly you can't do that uh for every point but you can do it for certain number of Point like for like a a given point uh and I'll show you that example in just a minute so we can actually work around finding the derivative of the actual inverse itself um let me give you a couple examples on how we can how we can evaluate inverses at certain points uh given a function let me give you another piece of notation here too some books to do this some books say well you know what let's suppose g ofx equals F inverse then they do this they say G Prime of X those of you who like the prime notation like this G Prime of X would then equal 1/ fime with G of X composed into it that's the idea do you guys see that the notation is the saying the same thing here did you guys see that here's derivative of the inverse here's derivative of the inverse remember that's the inverse here the derivative is one over one over the derivative of f no problem with the inverse composed in it with the inverse composed on to it should H feel okay with with that would you like a couple examples CU right now it's pretty vague yeah okay let's start here just some Basics on how we can go about U evaluating the inverse at a certain point without actually finding the inverse I'll say that again how we can go about evaluating the inverse at a point without actually finding the inverse please pay attention here because the next part of this uh using derivatives is going to be based on this idea okay very similar to this idea so quick example some of your homework questions are worded just like this so let's suppose we have a function x cubus one and what our homework or what our test or whatever you're trying to do this how it's worded what I want to do is find find F inverse of a when aal1 like f inverse of a when aal1 I'm going to give you some steps on how to do this it's not a hard concept for the most part uh but here just just think of this for a second we're going to use one byond conditional statement that's absolute truth for inverses to make this thing work in our advantage here's the the statement we're going we're going to talk about F ofx equal a if and only if F inverse of a equals what do you know not negative one no x x that's right look look if I plug X into our function it gives me a then I should plug in a into my inverse and it'll give me back X remember inverses undo functions does that make sense to you so this would have to equal x so if and only if if and only if that well then here's what we're going to do step one set F ofx equal to Whatever A You are given check this out if we set F ofx equal to whatever a were given if I can find out what x is I know what F inverse of a is so set this equal to whatever a you're given what's our function in this case good let's set that inut to whatever a I'm given how much is the a so here on this step we do X - 1 = 1 you okay with that by the way this is a really simplistic example you'll see that in a second but I just want to get you into the pattern of this next solve for x how would I solve this for X plus one okay so we'd add one and then what's our answer going to be so if I solve this for X sure add one we're going to get X Cub = 0 or x = 0 step number three man we're just going to State what the inverse evaluated to a is so find F inverse at a so here's the deal you guys believe this yes this definition for for inverses basically says okay well if f ofx equals a then F inverse of a equals X switch to the X and the y That's How inverses work so it's going to undo whatever operation whatever function you have okay well we're going to use this say okay if F ofx = A then in this case X - 1 = 1 if we solve that for X check this out if we solve that for X this is the X we're talking about so when we set our function equal to a and we solve it for X we're basically saying okay well if this works in terms of a function then if I take X inverse of what's my a again I me then F inverse of1 has to equal the x that I just solved for equals that's what we're trying to find that's the idea let kind of working backwards we set our function equal to a we solve for x that X is the answer that you're looking for in this case does that make sense to you it's a little backwards work it's a little weird uh let's try it one more time and then we'll uh we'll move on this one to take a little bit more thinking so I want you to really focus on this uh I know some of you don't really like trigonometry all that much but we're going to deal with it right now here's our function so here's our function I want us to find find F inverse of a so same question when a equals what do I want uh one okay why don't you guys tell me what's the first step what would I what would I start by doing a very good so let's do that if we set F ofx equal to a and we solve for x that's our answer we can just solve for x so on our first step we're going to have what equal to what please very good notice what I'm not doing I'm not taking this and plug it in in that's not the idea I want to find out when a is plugged into F inverse what do we get so I'm not plugging this number in here I'm setting this equal to a so 3 piun x + sin x x = 1 now I'm giving you this example to show you that it's not always very simple algebra to solve this this was really simple algebra sometimes you actually really have to think about what's going on here so there's no algebraic way to solve that we're going to have to just think about what we can plug in to make this true what what do we put here to make that true start thinking I'll give you a second so think about it for a second see if you can find something remember it's got to be the same thing in both spots so same thing in both spots think of the angles that you know sign for I wouldn't be thinking of like 1/2 because you can't plug in 1/2 I'll be thinking like pi over something for two reasons firstly I got to get rid of a pi here does that make sense got to get rid of Pi otherwise I'm not going to get a one so a pi over something here and a pi over something here and see what you can get so that when you add it together it's going to give you one think for a minute [Applause] if you can't think of one off the top of your head start trying some start trying things like pi over two or pi over 3 or pi over 4 or pi over 6 start thinking of them let's try pi over 2 if I try pi over 2 what's sign of pi/ 2 one one p two here would pies would be gone I'd have 3+ 1 is 3+ 1 equal to 1 no no okay so Pi 2 doesn't work try things like pi over 3 all right so pi over 3 what's s of pi over 3 3un 3 over2 well there's no way that I'm going to have a root3 over2 and something doesn't have a square root in it and get a one so that can't be right pi over 4 also has aun2 over two right so that's not going to work how about pi over what do you think pi over six might work let's try that so if I do pi over 6 so this would be step number two showing our work here how much is s of Pi 6 what happens here that's kind of cool right we got eyes are gone one does it work okay so here's the idea we just found out the number so that when I plug it into my function it gives me one my a so how much is our x equal to in our case yeah our X in our case is pi/ 6 your H feel okay with that now let's just put it all together remember that F inverse of a equals x whatever X you just found here so what we show this as is f inverse of f inverse of what what's our a one equals that's the idea that's working with inverses okay quick show hands you feel okay with the idea of this I got to be honest with you sometimes it's really easy sometimes it's not sometimes takes a lot of thought here so but do you understand the concept of it yeah okay cool the last thing we're going to talk about in our section is how to find the derivative of an inverse evaluated at a point um I think I told you that sometimes it's really easy to find the inverse and just take the Der of it sometimes it's not so I'm going to show you a way that we can go around actually finding like without actually finding the inverse function to find a derivative evaluated at a point so here's what we're going to do we're going to start with this example we're going to find the derivative of G where gal F inverse for FX = X cub at the point 28 if you're wondering what in the world are you talking about Mr Leonard this is making no sense to me here's a plan what's a derivative mean in other words the average change that's not the average change instantaneous instantaneous change or slope of a curve at a point you ever heard that before I mean my students memorize say der is slope of curv point now set it so derivative is oh you're getting it one more time derivative is okay so what I'm looking for is the derivative of remember this is the inverse the derivative of the inverse or the slope of the inverse at that point are you guys with me on this so at this point let's let's let's just practice for a second this point is on the function on the inverse would it be 28 or 8 2 so we'd be finding the the slope of this curve inverse at 82 are you all with me on this that's the idea here that's what we're going to be doing I got some steps for you on how to do this every single time it's honestly it's not that hard step number one just verify that this point is actually on that function it's very easy just plug it in so verify that Point's actually on the function so step number one F of what would you plug in would you plug in you plug in plug in two make sure it gives you eight so F of 2 gives you 2 Cub gives you8 that's step number one just verify that the point is on the curve because we want to make sure that we're we're talking about for inverse the same scenario here on the inverse so 82 rather than 28 you guys okay with that so far all right number two number two is kind of a two-part thing I want us to find the derivative of the function so find frime of X and find G of whatever your y coordinate is because when I find my inverse they're going to switch so find G of 8 let's start with the um with the derivative firstly what's the derivative of f what are we going to get perfect how about G of 8 keep this in mind I know we're using G but G really stands for the inverse if F ofx has a 28 in it tell me something about the inverse what happens if I take the inverse are you guys okay that that's actually the inverse here yes no since this says inverse if I say hey plug in8 what what's that have to give me yeah the point says it okay the point says when you plug in two you're two to the function you're going to get out eight therefore the inverse says when I plug in eight I'm going to get out two I don't even care that you have a formula this says it for you so that's got to give you two show hands you feel okay with with that one okay now here's the cool part if we know that all we got to do is use that formula I just gave you so our formula said G Prime of X gives you 1 F inverse sorry fime evaluated at G of X here's what we're going to do we're not trying to find G Prime of X in terms of X we're trying to find it at an actual point so we're trying to look for G Prime of 8 G Prime of 8 says this it says you take one over the derivative of f evaluated at G of 8 now how much is G of 8 so we're going to find check this out please watch we're going to have frime of what how much is G of 8 two all you got to do is take that two plug it in here and figure it out one over 3 * 2^ 2 how much is that going to give us 1 over 3 3 * t^2 yes no or did I lose you I don't want to lose you but I want to make sure that you're you're okay on this we're going to do one more example on how to do this but I want to make sure that you guys are good to go so the idea is this G is going to stand for your inverse if I want to find the derivative of g at a certain point here's the way that we can go about it we don't actually have to find the inverse function of G to do it what we can do is all right well cool uh first let's make sure that 28 is actually in the function it is if 28 is in the function 82 is in the inverse function now you happy you understand that okay all right well then that's all this is saying is that 82 is in the inverse function all right cool well let's find the derivative of f very easy to do for us CU we're in calculus 2 and then here's how we can find it if you want G Prime of whatever point I'm talking about G Prime of 8 take one over the derivative evaluated at that number whatever that is so just take the derivative plug this into it and that's our that's the slope of this curve the inverse curve at the point 82 show P feel all right with that one okay it's a little weird right we're kind of going like the back door approach here we're not going directly at it we're going around it can we do one more example see if you really hang on to it yeah maybe you can try some of this so same idea I want to find the derivative of g at a point for this function so same start here let's just do it when f ofx is x^2 + 1 Cub at the8 okay quickly first we're supposed to do is what please ver let's verify it yeah plug in the one make sure it gives you eight for the function itself doesn't I don't think it does you plug in one pluged in one no oh yeah sorry get two did it work okay so if I plug in one you get eight you're good okay listen you can do the second step right now at least part of it do the Second Step second step is a two-parter uh what's name one thing you're going to do in the Second Step you have to find the derivative because that's what we're using here so find the derivative Also let's let's do this together right now okay I'll have you do the derivative in just a second but check this out I know I'm about to find the derivative but tell me this how much is G how much is G of 8 if you know F of one gives you 8 how much is G of 8 how much is that one perfect tell you what when you get down to this step right here after you find the derivative what number are you going to be plugging in one it's the same thing you found right there just plug in one so I want you to find the derivative right now of f what do we need for that derivative what are you going to use love the chain R hopefully you got that did you get that yes yeah okay cool you guys are practically done now we're going to use the formula that I gave you the formula says if you want to find G Prime at whatever this point is if you want to find G Prime of 8 what we're going to do is put 1 over fime of G of 8 that's the idea 1 over F Prime um how much was we we already talked about this how much was G of8 again so really all it says is okay okay make sure this works firstly find the derivative plug in that number plug in that number to your derivative put it over one put it under one and you're basically done have you guys plugged in one to this you guys see where to plug in the one we plug in one right here how much did it give you 24 24 so we' say G Prime of 8 = 1 24 here's what it says provided that you are actually able to find the inverse of this if you were to find the inverse take a derivative and plug in the number eight it's going to give you 1 24 it's finding the slope of your inverse at 81 that's what it's doing it's kind of cool right show if feel okay with that one that's it that's really all we're going to go for as far as inverses you guys have any questions at all about about this so far can you find an inverse if I give it to you can I find an inverse if I give you a function okay can you do examples like this one where they're pretty basic yeah this one you have to think about it's a little bit harder can you are you guys okay with these two cool question um so for that one where we just have to think about it will we ever be able to isolate that X and later on so it's always yeah you can't not here at least you can't get that explicitly defined you can't solve it for X it's all unit circle pretty much right yeah this one's a thinker because you have an algebra concept and you have a trigonometric concept you use inverse here but you're going to be stuck with an ex good questions all right we're going to move on the next thing we're going to talk about because we've already talked about natural log function it really seems logical that we're going to talk about exponential functions next because exponentials and logarithmic functions are hand in hand they're inverses of each other hey that's what we talked about inverses so we did logarithmic okay we did inverses now we can talk about exponentials because inverses sorry exponentials and logarithms are invers of one