so welcome back from the break we do have an exam coming up the way this course is laid out exam two typically happens right after spring break so if you go into unit two and you pull up the exam material you will see that there are dates here available and the time limit and tells us some information about what to get the comment to be had about this again is know how you're going to get it put into grade scope so I realize that some people have not tried the grade scope app if you haven't tried the grade scope app for loading your stuff on to grade scope after the exam that might be a good one so I'll probably send out an email with that information on it regarding how to send it what to look out for when you print the template some people haven't printed the templ correctly so don't open it as a Google doc open it as a PDF on your own device and print it from there the other thing to talk about on exam two is with regards to what material is covered so exam 2 covers everything out of unit two but as I was looking over some stuff I realized 3.6 is a horrible section maybe you already knew that but looking at over I went ahead and crossed off a bunch of problems so I left problems there to be graphed and then I I've got a couple of problems right in the front numbers 9 and 11 it's a slight tweak on how do we decide how to transform and shift certain graphs and these graphs happen to have vertical ASM tootes and the ASM tootes will move along with the graph left or right so I've really streamlined 3.6 because most of the problems that were assigned there do not line up well with what we are going to be testing on on the exam and for that reason I've stripped out a good chunk of them and that'll be easier for a lot of people to finish the assignment and get this done so yeah and I feel you for some of those comments what break right no joke it's been crazy isn't it so this will help streamline exactly what you need to focus on it'll help Target what you need to study and then once you get through 1.6 and these few problems out of 3.6 you should be pretty well prepared for this upcoming exam it's not horrible it it's you know just study the highlighted problems that are here and you should be in pretty good shape going into that exam okay couple of things review the thing to review highlighted problems here or just a few problems that represent what the section covered there's not really a particular review for the exam itself per se but the general gist of what we've covered that it'll be on the exam the big problem let me just get that one out of the way there is a quadratic ax^2 plus BX plus C and guess what you've got to find the vertex you've got to write it in vertex form you've got to find the X intercepts the Y intercepts you got to sketch a graph and make a note as to where those things are there's one long problem like that on the test you remember that one like Parts a through z or what a through G or whatever it was so be able to do that there is one of those on the exam the other types of problems that are there talk about zeros and multiplicity and how would the graph look with its end behavior does it bounce or cut through the graph figuring out the zeros of a function there's several ways sometimes you can just Factor things apart other times you need to do synthetic division or long division so those are also on the exam intermediate value theorem is on the exam we talked about that you test the end points to see whether it goes from negative to positive or positive to negative to determine whether or not it crosses a height of zero and then some other things are like finding the domain and vertical or horizontal Asm tootes as well as the last things which were the inequalities and we did some inequalities but if you go back over what we covered out of 3435 and just a few of the 1.6 problems here that are highlighted you should be in pretty good shape for most of that material and then there were a few others with square roots and rational powers like where you have that power of a fraction and you had to make each side go to the opposite powered fraction the reciprocal fraction so that's really the main focal point as far as stuff from unit one directly there's nothing really no unit two is kind of a standalone unit yep yeah okay so if you saw practice exams available practice exams are only available for certain classes that I teach usually those are classes that are not designed to go to another level of class uh statistics is most likely one of the classes that was there contemporary math is another class that's a that's a terminal class they stop at that and they don't go beyond it the classes that don't have reviews are classes that are leading to another math class right afterward so like this one is designed to get you to pre-calculus for that reason I provide highlighted problems as a revieww guide and then even even like Cal 2 which I'm teaching this semester they have no review or highlighted problems in the homework whatsoever yeah so it just depends on which class uh I'm teaching as to whether or not there's a review but this unfortunately is one of the classes that doesn't have a practice exam with it I I'm sorry about that we are having class on Wednesday we are having class on Wednesdays far as I know that is not one of our holidays the other things here to comment on because we're finishing up unit two and that exam is taking place we are now moving into new material that will be covered on homework assignments from unit three in particular 4.1 is start so thanks what we're about to embark on right now is designed to help you get the homework for next week and it's a light one because we have our exam but uh this next homework assignment excuse me which is due on uh April 2nd this one is a lighter one because you will you'll have more time to focus on the exam but we're going to go ahead and start moving into that material early in hopes that we can get through some of it and make it a little easier for you as you work on that so the material we're about to work on here is called inverse functions and for those that joined and didn't get a hold of the board the board here is G4 TL G4 TL is the code for the board some of this is going to feel familiar some of this is going to feel kind of awkward the idea with inverse functions is that some functions are invertible what does invertible mean invertible means they can be worked backwards and what we're going to discover is not all functions have that property of being an invertible function so they start us off with these definitions to kind of get us to think about well what is a function first of all and a function first of all is where each input is going to generate at most one output we did this a while back right functions can only generate one output for every input they can't generate two or more it would create a problem that's what a function is stick one number in it spits one number out but then there's an additional modifier that we would call one: one what is a one: one function well a one: one function still has that same each input generates at most it still has that flavor but look at the output it has to be considered a unique output a unique output once one number gets spit out for that function that height or whatever on the graph can't be spit out ever again oh look when we put the number seven into it function produced the value three that height three can only happen once in order for it to be a one toone function so that leads us to this concept of a horizontal line test if we're given the graph you can tell whether you've got a one: one function so the wording for this is going to state if there exists a horizontal line that hits a function more than once then it is not a one to one function if there exists anywhere on the graph if you could create anywhere on the graph a horizontal line where it hits your function in two or more places so it has to hit it more than once if that happens anywhere that thing is not a one toone function guarantee why because you just found two or more spots where it had the same height we said a one one function only has a unique height for each of its Heights which means if you've got a horizontal line that hits it the same height in two spots you've invalidated it being one to one it can't have the heights repeat so what this is going to bring us to this is probably the easiest part of this section in general the game of is it a one to one function is it a onetoone function so again what are we looking for well we set up above the thing we're looking for horizontal line can we create a horizontal line anywhere on this that hits it more than once if so then we conclude it is not a onetoone function can you create a horizontal line for this graph that hits it in more than one spot I don't know it depends on how good I am at drawing well barring that barring our artistic ability yeah sure right here now do you need it to be a thick horizontal line no of course not of course not you could make it a thin horizontal line right there and it hits one two three times if it hits it more than once that thing is not a Ono one function so again it's about asking the question can you make a horizontal line that hits it in more than one spot is there any place a horizontal line would hit it more than once if there is yeah you're done it's not a one to one function yeah how about another one here's one that we've seen in the past that looks awfully familiar but the question is is it a one toone function what do we think yes or no this looks like a one to one function yes maybe no yeah I would agree with you I think you're right why well think about every possible horizontal line that you draw every possible horizontal line that you're going to draw and of course the function keeps on going but every possible hor horizontal line Only Hits it in one spot for each horizontal line that you create you can't make a horizontal line hit that thing more than once so it turns out to be he a one to one function all right here's another one incidentally that last one is one that we're familiar with this last one that does that arching shape that looks an awful lot like a square root function all right how about this one what about this everybody knows this it's our favorite it is a straight line question is it a one to one function it is it is every single horizontal line you could create is only going to hit it one once and only once so it undoubtedly is a onetoone function you can't find a spot where it hits more than once I love this game Chris can can the third exam only have these problems on it I think I will pass I'm feeling [Laughter] strong all right here we go yeah had to again another one the graph this feels familiar we know what this looks like we've seen this way too many times right looks like our friend the smiley face a happy little smiley face all right so is the smiley face which is like ax^2 plus BX plus C it's a quadratic is it a one: one function it's not it's not now the reason it's not can be seen several places you just need to find somewhere you can draw a horizontal line oh and it hits it twice you're done you're done any place you can create a horizontal line that hits two or more times this thing is no longer a one to one function and again the question is why do we care about it being one: one well if a function is one to one we can go and find the inverse function and we'll talk about the inverse function in more details and more depth here in just a sec how about how about we take a graph like this one I like this one this one's nice it's fun let's do this graph and of course this thing keeps on going it doesn't stop is it a one to one function kind of weird because it's like this sideways smile right it's this got the head tilted I'll be honest this one is not the norm this one definitely is not the norm why not well it's not the norm because I feel like it fails something else that we've tried in the past there's been a test that we've done before and that one was a little different it wasn't the horizontal it was the vertical what was the vertical line test for the vertical line test helped to test to see if the graph was somebody remembers this spring break didn't take it all out of [Laughter] you a function exactly exactly so once upon a time there was a vertical line test and the vertical line test would do something like this and if the vertical line that you found if you could find a vertical line that hits it more than once then that means that the graph is not function it's not a function well Chris can it still be one to one no it really can't look if we go back up to our definitions here we said one: one is an adjective that describes a function so to say the thing as a oneto one function means that it has to be a function in the first place it can't just be one to one and not a function it doesn't work that way we're only worried about things that are onetoone functions so because this fails it being a function it forces us to not even consider it a candidate for being a onetoone function that was kind of a tricky one I'm sorry oh right out of the gate we get back from Spring Break and he's always he already thrown us curve balls so again the issue is it needs to be a function and then we go on to ask about its oneto Oneness so it needs to both pass the vertical line test and it needs to pass the horizontal line test here's kind of an interesting one that we can ask is it a one to one function this one again you've got the same kind of setup here we've got a graph we go put points on it and and this one we're just going to put points kind of all in here there we go that looks good kind of weird dots Chris where is the why is the graph not connected well nobody said the graph had to be connected you could just have dots question is is this a graph of a one toone function now think about that does it number one pass the vertical line test is it a function and we said in order to be a function every place you draw a vertical line can only hit one time if it hits more than once it's not a function and then if it passes that which it looked like this one passed I didn't see any vertical line hitting two dots then we do the same thing but with a horizontal line going so now we start putting horizontal lines across and asking is there any place we're going to hit two or more dots by drawing a horizontal line no every horizontal line we draw Only Hits one dot and one dot only so this one turns out to pass both the horizontal line test and the vertical line test which means this represents a one: one function that one's kind of crazy what would we need to do to make something like this with dots not be a one to one function we'd either have to have two or more dots that are on the same what there's a couple of options here I mean we could draw another graph out of this do similar things here we can put the same dot here on the horizontal axis put another dot over here in order to get this to break and no longer be a one to one function we have to do one of two things we could either draw a DOT at the same horizontal height that that's problematic that would be problematic right there why because this is going to invalidate the one to Oneness so that's one option we have that would be one option or the other option is is this would make it not a one: one function it's still a function but it's not one: one the other thing that we could do is we could move this now it's not invalidating the one to Oneness but if we stuck a DOT say right here that could also be a problem if we can draw a vertical line that goes through the exact same dot that we drew up so if this happens this line is telling us is not a function the other line if it hits a second dot would be telling us it's not a oneto one function so if it's not a function it can't be one: one if it's a function it might be one to one that's really the take take home with this one here so we'll leave it like this I I don't mind leaving it like this this one hits in two spots vertically so it invalidates the function this questions on that any questions any concerns regarding the graph here all right so the first things they have us deal with are the horizontal line tests just looking at these and determining is it a one toone function the other kind that they have after they do it with the graphs they set these up using a table and the table is a little so suppose they give you instead something that looks like this a table of XY values XY Pairs and we can see here what we have right x isg -3 Y is two so if you graph it right you'd go to the negative three and then you'd go to a height of two and draw a DOT this one's saying go left one up one don't go left or right but go up five two to the right down four three to the right up one but the real question is do we need to graph these do we need to draw a graph and put the dots where they go or can we look at this table for something that might just give it away give it away oh man there's a Red Hot Chili Pepper song going through someone's head Give It Away Give It Away give it away now all right here's the question what is it it mean to be a one to one function well we said one: one means each input right each input has one unique output the outputs have to be unique well are the X's or the Y's the outputs of a function usually normally when we've graphed these in the past or if you graph them in Dez mode the letter y is usually your output question do we see whether or not all of our y's are unique outputs are all of the yv values unique yv valued outputs oh yeah you see there's a problem Oh Yeah the code for the board for the uh for those that didn't just came in G4 TL G4 TL yeah so the issue is really when you have the same output in more than one location that's it duplicate outputs you have duplicate outputed y values the game's over it invalidates its possibility of being a one: one function because all one: one functions have unique outputs unique outputs so anywhere you see duplicate Heights if you see a duplicate height or a duplicate y value that tells us it is not a one: one function another one that you could have I don't know that this comes up in the book but it's possible to create something like this so maybe they put it in the book similar table similar table look at the values they are almost the same so the first row -32 the next row negative 1 1 then 05 2g4 and then the last row is a little different Nega one and seven are all the outputs unique are all the outputs unique they are all the Y values are unique values 2 1 54 and 7 but the thing that's getting us here and this will trip us up we're like ah they're all unique it's a onetoone function I don't think the book does it but they might look at the X's what wait but those are the same Chris that's not a problem well it's not that they're the same that causes the problem the thing that's going to cause an issue for us is more specifically there's an x value of negative one that spits out a height of one and then there's an x value of negative 1 that spits out a height of seven you just found one value for an input namely the value negative one as your input value but it's generating two different values of output and if that happens then it turns out we're not worried about one to Oneness anymore because why because by definition it's not a function why each input generates at most one output you just found an input of negative one that generates two different outputs if you graph it what would happen well you'll go to negative one and you'll put a dot up at height one and then you'll draw eventually later at the bottom you'll draw a DOT going left one and down seven or up seven excuse me so you'll have them vertically stacked where X is negative one but it inv it invalidates it or violates the vertical line test and that's what's going to cause the problem with this one yeah again that one may not be in the book they may be nice they may be nice to you but it is theoretically possible that somebody is being mean in the book and they decided to write it that way the key in point here is this if all X's are unique it's a function if all X's are unique then it's a function and then the separate statement after that is the then then if all the Y's are unique it's a onetoone function so what they're basically telling you is this when you go over to the table and you have a table of values what should you check for uniqueness first X's X's exactly if all the X's are unique that's the thumbs up saying okay now go check the Y's but if you check the x's and not all of them are unique that's kind of a red flag if that happens there's a chance it may not be a function so as long as all the X's are unique that guarantees it's a function then you move on and you check all the Y's if all of those are unique as well then you know you've got a one to one function and you're good to go good to go yeah any questions about that all right let's talk about a couple of ideas with what an inverse is and its notation so again why are we dealing with this one toone function business we're dealing with it because functions that are one: one have an inverse function so we'll talk about the properties in the notation for inverse functions the inverse function notationally is kind of weird most people are not accustomed to what it looks like why it's an F with a negative one as its superscript or power F to the1 of X but you don't read it that way if you to see this you read it verbally as F inverse of x f inverse of X it has X's presumably in it or it depends on X but you would read it as F inverse the the thing that people confuse the notation with and I get it I know why they do it they get it mixed up with another notation that's out there they com combine it with this notation right like this is just an example 5 to the1 you put five to the power of1 and you can rewrite it as 1 over five why because the negative one is the reciprocal it's called the reciprocal but that does not work here don't try it don't do it you'll be absolutely WR wrong why you're tempted to say f -1 is equal to 1 F by the same logic uh no no no no no no no no it's not I promise it's not equal to 1 over F why not well there's a slight difference between the way these are written think about this for just a moment the reciprocal this one is correct because we've already talked about it but the structure of it the number five oh listen to what you just called that you called it a number you called it a number f is not a number f is what the letter F presumably presumably F would be a function that's the major difference so functions to a power of Nega one those are not called reciprocals you can put a whatever you want here a caution a warning sign just remind yourself function to the negative 1 is called the inverse of f or whatever letter it is whatever letter it is if it were G to the negative one that is g inverse if it's capital T to the negative 1 that is T inverse so whatever the letter is the letter is assumed to be a function the function gets read as the inverse and it definitely is not going to do one divided by the thing they gave us that would only work with a very minute small handful of functions that exist in the world and they'd all be the same style so we're just not going to see that happen for the most part so that's the notation to be worried about the other thing is what makes an inverse an inverse well the properties can be written this way there's two formats one of them now now notice this looks scary but breathe breathe number one F of and then you stick F inverse inside it's spits out X and F inverse big set of parentheses little F ofx inside also spits out X both of these spit out X thank you okay get ready get ready here it comes I think the easiest way to remember this is by comparing what it would mean to have this in real life now there's other ways that we could describe this I I understand but I think this this will definitely serve as a as a a memorable moment for us so we're going to go with this okay here's the thought process what these are saying is that if you put an inverse function into the original everything cancels out and it gives you back X or if you put the original function inside the inverse everything cancels out and just leaves you with literally the letter X to give a thought process to this you do not need to write this down but here's kind of the thought behind it we are human beings right and I think most of us would agree with that there we go a little bit of hair all right human beings have input and they have output and I know there's different ways of describing this but I think this is going to make for a memorable moment this is going to be crazy just bear with me for a moment suppose our human wakes up and for breakfast decides to have a little apple okay there's our little apple that is the input sometime passes later and now the human is going to turn it into output what kind of output will the Apple become yeah yeah you're thinking it you're uhuh exactly exactly okay so let let's just imagine imagine for a moment there is output man where is the poop emoji when I need it there we go oh boy okay there is our output obviously not to scale now what if such an animal EX exist as the inverse human now before you think about what the inverse human looks like I need to explain to you the inverse human is a very strange animal the inverse human I don't know it's got these like gigantic rhinoceros uh legs all right in the back here it kind of works its way back it may even have a tail back here because it swims I'm sure it's fuzzy it smells like cotton candy all right and then I don't know it has some kind of a long trunk and while we're at it it definitely is going to need a horn on its head it can even breathe fire it's super impressive okay I I have no idea but we call this the inverse human all right here's the thing the inverse function takes as its input what ever the original function's output was okay you this is gross this is gross you see where it's going uhhuh so in other words our excrement our byproduct now becomes the food for our inverse hum but then the inverse human whatever animal this is also has its own output but here's what's just amazing and almost miraculous about this output the output from our inverse human has to produce literally the exact same input that came into the human in the first place So eventually when our inverse human function processes all this food so to speak it's going to produce as its output literally the original Apple that the person ate for breakfast in the morning I have never never heard anybody describe the inverse human or the inverse function in such a strange way but rest assured you will not forget what it means now there's also another side to this the opposite works one function's output is the input for the other that other function's output has to produce the ex same input that the other function once had they cancel each other out they cancel each other out okay this is way too visually disturbing I've got to erase these images This is Gonna Keep me up at night somewhere else scariness so what is the inverse function it undo whatever the original function original did all right here we go what's an example of this the way they start these problems they start by playing a game what game do they play they play the game are you my inverse are you my inverse I apologize if you have nightmares about this later based on our description of this somebody you know I've gotten into this discussion about the inverse before and somebody likened it to well why don't you just use air right we breathe air and then we produce right Co like um carbon dioxide and then a plant could take the carbon dioxide and produce air that that Chris that seems very similar right it's the same thing and I agree with you it pretty much is right the plant is the inverse function to humans as far as breathing goes and vice versa for hum but I I think our example will really just it Nails it much better it just will stay clear all right here we go this is the way it works with these functions like M ofx and N ofx they're gon to ask us are these inverses of each other do they undo one another I'm going to tell you right now there is a yes we can either prove or disprove it or we can kind of look at it and get an inclination as to whether or not we think they undo each other watch watch if you put a number in for this x say you replace that x with the number four what is the first thing that happens to the number four when we put it into the location of that X the very first thing that would happen to it according to the way the function is described it gets multiplied by seven after that what happens we subract three if you've got an inverse function it somehow has to undo all of that in fact it's going to have to undo it because it's doing it in the reverse order and it's doing the reverse operator think about this multiply by seven subtract three if you go over here and you look at this one what happens if you replace that X in the in ofx function with some number I don't know you turn that number into whatever well after that the first thing that would happen is you add three and then following that is where you divide by seven did you do the same steps but you reverse the order and you reverse the operator you did so this looks like it's probably the inverse where we're multiplying by seven and subtracting three this one is instead adding three and then dividing by seven but the way to show it the way to show that they counteract each other is to test it in two ways now theoretically if one of the ways works and you've done it accurately that should be sufficient to show it either is or isn't an inverse so in light of that you can either test one number one you could test M of n ofx or number two you could test n of M ofx what are we doing you're putting one function into the location of the x or X's of the other function that's how you're writing it so what you're doing with this for a moment this one is saying take the M function whatever's on the outside of those big parentheses that's what you're writing first write the M function first but don't write X instead write a giant set of parentheses where the X would occur so as you start to do this what you get is something that looks like seven times a giant set of parentheses minus three inside that giant set of parentheses that you just made you put the in X function so look I mean it's the seven oh but the X isn't X the X turned into parentheses minus three and then where the X once was that's where we're going to put the N ofx function inside the parentheses so we put that into the parenthesis and now we're set to go what happens well as we start to handle the arithmetic seven times the fraction there's two ways this could be done you could let the seven hit the top of each of those parts and give you a 7x plus 21 or because the entire fraction is divided by a seven You could argue well the seven on the outside is going to get divided by the seven in the bottom either way works either way works if you saw oh I'll let the seven hit the top stuff that's great let it do it let it do it you'll still get the right answer after you do that those sevens will disappear and now you're just left with x + three that was once on top the blue stuff but you've still got that minus three that's in green behind it and what's about to happen well a plus three and a minus three yeah technically they'll wash out each other plus three take away three they disappear and at the end of the day you end up just getting the letter x so what we're doing is we are officially trying to see this taking the N function little n and sticking it into the X location of the other function the M ofx function if we stick the N function where that X was do all of the components cancel out and just leave us with only x with only an X that's all it can leave us with if that happens what we can conclude is they must be inverses of each other so M ofx and N ofx are inverses they undo each other's actions now what if you had run it the opposite way what if you had done done what we said here on number two number two is a little different we should still if we got X in one of these we should get x out of it doing it the other way but this thing here n ofx that is n ofx we need to rewrite this entire thing except careful careful careful instead of X we are going to write the function m in its place and we need to have a set of parentheses if we don't have a set of parentheses there are cases that could lead us to we'll just say undesirable answers undesirable results so as we set this up here's how it looks so just try to be careful notice what's going on the N function says divide by seven okay divide by seven it says add three at the top right of our fraction we got the three it says here's X oh but X turns into a set of parentheses the X's location is replaced with a set of parentheses and then we Jam the M function into that set of parentheses as we do this now where do we get well it's fairly straightforward there's nothing out here multiplying the 7x minus 3 it's not like they had like a five out here or a negative five there's nothing out there so what that allows us to do is just rewrite this whole thing without parentheses but I'm glad we had them because if if if there had been multiplication involved and we forgot our parenthesis there's a really high chance we'd be wrong so because there's nothing touching the parentheses on the outside we can just get rid of them and now we've got 7x minus 3 + 3 oh well wait a minute wait wait wait the threes Min -3 + three those will cancel and now you're just left with 7x but it's still being divided by S and then the Sens are going to cancel each other out and even though the sevens do cancel each other out they're still going to be X and once again we said if you end up with just an X it shows they are in inverses some people have asked me well wait Chris what what if the X's cancel out they shouldn't if it's an inverse if the X's disappear totally it's probably not an inverse or you may have done the wrong arithmetic so usually the X's don't just cancel out or if they do and you've done everything 100% correct if the X's disappear it's not the case that they are inverses of each other definitely not yeah questions about that all right let's do this I know you're feeling it I definitely am too here's going to be our next problem we're going to work but we're not going to work it until we come back from a break so go ahead stretch it out and we'll come back here in about that five to seven minsh time frame and work on this next problem with P of x and q of X they want us to just simply go ahead and check is p of X the inverse of Q ofx is Q ofx the inverse of P of X and it's similar you're asking are they doing the opposite things in the opposite order the interesting thing about this is again you should be able to sense is this going to happen is this going to happen what do you mean well look the first function says here's X the very first thing you're doing is subtracting three and dividing by four so if you do that in the opposite order you'd be dealing with the four first does the other function deal with the four first well yeah the other function is multiplying by four so that's the opposite of division but the next thing is interesting this one says subtract three wait a minute this one over here said subtract three they shouldn't both be doing the same thing they should be doing opposite things and that's that's where this ends up becoming kind of an issue right how do we show whether they truly are inverses of each other now again you can start this off using either the method that we did before you had two options right option one or option two so if we go back here what you're technically doing is you're either testing is property one holding or is property two holding true you get to choose which function you want to stick it into I'm going to do this one as property one and we'll do it with the p function on the outside so if we do it like that using that property this problem would it would start off in this fashion P of but then on the inside we're trying to test is q an inverse of the P function and as we start this off and we try it what ends up happening okay now remember what do you trying to do here well you're trying to write the P of X function so write this first -3 but don't write X you're still dividing by four you'll still have a plus but instead of where X is put a giant set of parentheses why because where that X is that's where you're going to stick instead a 4X minus 3 that's where the location of the other function needs to appear needs to appear so as we get it set up it looks like this -3 plus a set of parentheses 4X minus 3 all over four now we start to ask questions okay well what do we do with this okay the the parentheses are great but fortunately it was just a plus it wasn't a minus it was a plus it didn't have a number in front so because it's just a plus we can just remove the parentheses had it been a minus though it would have reversed the signs on both of those pieces both signs would have gotten negated if it had been a negative but if it's a positive it's just saying keep it as a posi 4X keep it as theg -3 that was there keep it that way and look what happens now we can start to combine stuff in the top and don't let wishful thinking override intuitive logic oh I so want to get the fours to cancel I just want to get x out of that ah whoa Red Alert in order to get that to happen you'd have to get the threes out of your way first do the threes cancel each other out don't be saying I wish the threes canceled each other out whoa that's totally different do the threes cancel or counteract their contributions to each other um -3 minus three more well that's not going to really help because -3 minus three more is just going to give us -6 so now we're stuck are we able to get rid of the pieces and get the X's again don't force yourself to create new math we don't want mathematical proctology for an ideology that may not be true that's horrible thinking so what I would say is at this point it's not looking very promising that it's an inverse things should have canceled out more easily this is definitely not looking good can you go further you can you you could try to unravel this even further if you're really convinced that you can beat it up and make it work I mean the most you could do with this in a sense there's several ways to write I guess the way to that I might write it the one that's coming to my mind right now at least is okay well write it as 4X / 4 and then a minus 6 / by four you could do that you're just splitting this apart nothing major has happened you're just splitting the fraction into two pieces because they have a common denominator so that's really all that's going on and then you do get some cancellation oh yeah the fours will cancel that's great that's great what else cancels well technically this four and that six will cancel the six will turn into a three three right because 6 divided two would give us three the four on the bottom can also be divided by two but four divided two is just a two so we'll get two down there and no matter how much we try you get an X and then you [Music] get3 halfes again it is just telling us we're not getting an X all by itself we're getting extra stuff this shouldn't be here no matter how much we manipulate no matter how much we experiment no matter how much we try to get rid of things and get just X all by itself it ain't working it ain't working so sometimes you really do have to break out of the I wish or I want it to work and really ask is that right right -3 andg -3 add together they don't make zero so you you you kind of have to pay attention to these to make sure you're not doing convenient arithmetic and hopes that they're inverses so at the end of the day whichever way you end up looking at this no matter how we play with this [Music] one the answer is that P and Q are not inverses of each other well wait Chris what if I tried the other property you you just tried property number one What If instead we tried to put the Q of X on the outside and the P of X function on the inside well okay okay if you did that this is going to go inside of the X's location you're about to take the entire fraction that's right here and jam it right there where that X what would happen well the four on the outside is going to cancel with the four in the bottom those fours will go away but you're still going to have A3 and a positive X and A negative3 out here you're going to get a Nega -6 floating around that you can't get rid of so whether you do it using property number one or you do it with property number two either way you're not getting an X all by itself you're getting the and the whatever that and the part is the extras that are on the end that's also hanging on and it's stopping us from getting a pure X by itself where we can say ah now we can say they're inverses questions on that any questions okay here comes the long stuff how do you find the inverse function when you're given some original it's not bad the steps are not bad so here's how you do it how do you find the inverse if you were given an original function step number one you rewrite the original function as Y is equal to some stuff with all your x's in other words what are you changing well most original functions are things like f ofx p of x q ofx n of x m of X whatever it is it's a letter that notation we're saying represents a height y graphically so just call it y equals the function they gave you step number two turn Y into X and all X's into y's so y turns into X and all the X's turn into y's goal isolate y by itself once you get y all by itself you've got the inverse function the only thing you need to do at the very end is rewrite it using the correct notation which is that negative one stuff if the original function is G of X at the end of the day you're going to write G to the Nega one of X and that would be the inverse notation for the G function so the steps are not very hard you can I will say you can jump the steps a little but I'm not going to do it in lecture how would you jump your steps with these well the idea is to jump steps you could jump straight to step number two what does that mean well if you jump straight to step number two you don't have to write y equals the original let me give you an example suppose you're given this function here G ofx is equal to 8 - x over3 and you're asked to find or tasked to find its inverse well the first step says don't write G of X just write it as the letter y all the stuff on the right side is the same copy paste no big deal okay well what you could have done is you could have combined writing that with step number two and done it in all one Fell Swoop what do you mean well step number two simply said let the Y turn into an X and all X's turn into y's so from the step where right now which is Step number one we're going to turn this letter Y into an X and we're going to turn the X that's over here there's only one of them but we're going to turn it into y you could have just started with that immediately and you would have been in totally great shape and now the fun part so the left hand side just becomes X any X's that were in the function there was only this one up here turns into a y you could have jumped straight from this to that that would have been fine your goal isolate why there it is taunting you but you can't get me all by myself yeah I can yeah I can all right think about it the first number that we need to move is going to be what what's the very first number we should probably move the three exactly well the three is dividing the entire right hand inside of the equal sign or symbol so the first thing we're going to do is multiply that three over to the other side when we do we will simply get 3x and then on the right we'll get 8 minus y so just move that three over what are we doing technically this is Step number three we are isolating y so all we've done is taken the three and said multiply that x 3X move over and multiply the X the rest of this isn't bad the question is what would you do next I know what I would do what would you do the goal again is to get why what would you move next we already moved the three the next you could move well technically you could move the 3x you could move the eight or you could move the Y either of the two on the right or the 3x that's on the left could be moved I think what I would probably do I would probably just move the Y why well the Y is negative and what that means is we could add it to the other side so that would give us a positive y but we don't want this 3x and all to be over here we want it to go over to the right well it can come over to the right we just have to subtract the 3x if we do both of those at the same time the Y ends up on the left and ends up being positive the 3x moves over to the right and now ends up being negative and that's totally valid there's nothing wrong with that all you're doing is you're taking these and you're just subtracting and adding to the opposite sides now some people are concerned well wait if I subtract the 3x does it doesn't it have to go in front of the eight doesn't matter if You' put -3x + 8 that's fine too it's the same thing it's saying the 3x was negative and the eight was positive and they were adding together that's totally fine either way would be correct the only thing that's a little off right now you just found the inverse it's this 8 - 3x or -3x Plus 8 it's the wrong format what do you mean the wrong format well the original function was called the function G and if you're trying to find its inverse you should have similar notation so even though the 8us 3x is the right inverse function we need to rewrite it instead of y equals we're going to replace y with the original function's name which was G with the negative one as its power so this last step step four is just notationally we don't write y equals we instead write G to the1 or G inverse of X is equal to 8 - 3x so this is the official way to handle these when you're trying to solve for the inverse function this one you could even get a feel for whether you were on the right path immediately immediately why how do I know Chris I'm not totally sure what would happen well I mean just kind of vaguely feeling it out this function the original was dividing by a three you know in order to counteract that you better be multiplying by a three somewhere that's got to happen the other thing is you were adding an eight well in this particular case it still turns out you're adding the eight but why well because of the negation because of negation so negating allows us to negate what's going on every time and counteract but just those pieces give us a hint why how would you know you're wrong well if you had an eight times an X here that would have meant you were dividing by eight before so those are the kind of things that you're looking out for are we dividing or multiplying are we adding or subtracting because the opposite thing still has to do something something kind of similar any questions let's see here we got a few problems left to work on but I think we might make it through them all no questions yet all right here we go same type of idea given K ofx is equal to the cube root of x + 8 find the inverse you are allowed to jump straight to step number two if you would like but for the sake of the lecture and the notes we're going to take step one first take it slow all we do don't write K ofx write it as y equals y equals it's the only thing that changed then in the next step that's where the swapping happens left side becomes X any X's that appeared on the other side turn into y's X is now equal to the cube root of Y after that we've got to get y all by itself it's kind of a pain what do we need this is one of the topics that's on our unit 2 stuff not this you're not finding the inverse but solving this kind of a problem was stuff that we worked on here in the more recent sections towards the end of unit two how do you deal with this how do you get y all by itself what is the first thing that you have to do in order to get y isolated well it's Tethered to a plus eight and then it's Tethered to a cube root the question is which one do we handle first the answer is not the eight don't do the eight you need to deal with the cube root well what's the opposite of a cube root cubing oh yeah you remember [Music] these so you take each side and just stick the entire side in a set of parentheses and then each side is in its cocoon of parentheses and you put a three up in the top what the three is designed to do is it's designed to cancel with the cube root it's designed to cancel with the cube root now on the left you're left with X cubed is equal to y + 8 and of course all of this is being done in order to isolate in Step number three the letter Y by itself questions on that sometimes that's a step that we we forget about from here you want the Y all by itself well the good news is y is already positive and it is kind of by itself on the right hand side the only thing touching it is the positive 8 we'll just move it over to the left by making it negative so we'll get y Cub - 8 or X Cub - 8 excuse me is equal to Y the function itself that is the inverse is the X Cub minus 8 that's the inverse function so we know that the answer is X cubed minus 8 but notationally this is not supposed to be y equals it depends on what the function was that they gave us and if we go go back up and look function they gave us was called K of x k of X so it's K but to the1 of X that's the inverse that's the inverse function this one seems almost more intuitive than the last one at least to me why look at what this function tells us to do the original function says take a value add eight after you add eight take a cube root so take a number add an eight cube root the opposite of that and the opposite order do the opposite of a cube root the first thing you should be doing is cubing then after that don't add a instead subtract eight this one played out really nice why it didn't have a negative in front of the X it makes it look a little easier you could have probably just looked at that problem and not even necessarily gone through each of these steps individually and just argued oh well the opposite of the cube root is cubing the opposite of adding the eight is going to be to subtract the eight and still ended up with something that looks like this all right last couple of things here to address and I think I'm going to do these a little bit out of order just because this plays a little nicer we all know somewhere they're going to talk about word problems and when they do they give us a function and then they tell us what the numbers mean seriously if they give you a word problem this is all you need to write down why well the thing is you need to know the function you need to know what do the letters represent well here it looks like T is time in seconds and it looks like whatever this thing spits out as its output as its output s is the output represents the distance from home in miles it's the distance from home in miles so we kind of get it right so many seconds later we're so many miles away from home the question is what is the inverse function and what does it have as its input and output now the reason I like this one this feels Deja Vish what do you mean um wait look doesn't it it almost looks like the function we just got as our answer we got X Cub minus 8 this looks super similar T cubed + 7 it's almost the same exact thing well what did we notice about the function they gave us here well this was the function they gave us before so it had that Cube rooting to do the opposite of the cube and then it had the plus eight instead of minus8 guess what this one's going to have something that looks almost totally identical if we go and work the path we're going to end up having a cube root and instead of a plus seven we're going to have to subtract a seven so if we were to find the inverse we'd end up getting something like this and it's not a big surprise after the last problem we just worked look though closely the notation is s to the1 t t this is kind of weird why um because we know that inverses reverse the input and output of the original function what do you mean well look this function here has an input of time and when we put a time into the function the function is designed to spit out miles distance from the home in Miles how far away we are the inverse function is designed to do exactly the opposite so what we really probably should say and even though this is the right answer the input and output gets switched if the original takes time the inverse function spits out time if the original spit out distance the inverse takes in distance so I think probably the better way to write this in the case of a word problem is to rewrite it like this wait what did you just do Chris well okay the T that was here is no longer a t it's now the letter s why because this is the inverse function but we said the inverse function should take in the opposite thing that the other one spit out the other one spit out distance called s so now the s that we once had up above is now going to be our input and this function when you put in some distance as s this thing is going to generate the letter T so it spits out time so we've flip-flopped our inputs and our outputs and the way you might address that or remember that off to the side is by saying since the original input was time and output was distance the inverse will now have an input of distance and an output of time which one of these two formats are correct technically they're both correct both of those answers are correct but with a word problem I think this is the preferred just because it in it's illustrating that we've reversed what our input and our output are that's really the main characteristic of it questions on that all right and now the wonderful finale of finale problems the ones that we always love and just can't get enough of the hard ones all right here we go the good news is it's not a word problem bad news is it has multiple parts so the first one that they're going to ask us to deal with here is to graph f ofx is equal to x^2 + one and this is going to be on the interval where X is less than or equal to zero now let's consider this for a moment from the from a graphing standpoint x squar what does X squ look like well once upon a time we talked about it it looks you do your graph like this and then it comes down and it touches the origin and it goes up right but now we've got to add one to it so it's one Higher it's one Higher so the graph of this ends up looking like the normal one that we would draw but it's now up here at a height of one and it would go up to the right and it would go up to the left so we took X squar we've just shifted it up one we've gone up one with it but there's one other thing that they've swapped on us careful it only exists for X less than or equal to zero why are they doing that Chris well it's because we can't get an inverse function for this if you draw a horizontal line through it it hits it twice it's not invertible so what they're saying is you only want to draw this when X is zero or less meaning erase the entire right hand side so that's that's what their little comment is of X is less than or equal to zero and now we've got something that is a one: one function it is one: one wait why is it one: one Chris well it's one: one because now if we were to draw a horizontal line through all possible spots this will work out but remember that only works because they've restricted the X values it is one to one on the interval where X is less than or equal to zero if we add the other part then no this isn't going to be a one to one function what gets tricky about these is the following right now our domain for our inputs goes from negative Infinity up to zero and the heights range from a height of one to Infinity so if they were to ask us about domain and range we can put it domain represents those X values negative Infinity up to zero and the range represents the Heights from one including the height one on up to Infinity where this gets tough is when you've got to find the inverse and the reason it gets tricky and this does happen it happens in calc 1 it happens in Cal 2 it happens in Cal 3 where you deal with a problem of this nature the trick to this when you go to find the inverse you start by saying Y is equal to the original x^2 + 1 function you swap out the x's and the Y's which is totally fine that would be step two or maybe you would jump to it immediately if you're already in a rush which works that's not an issue the tough part is getting why isolated all by itself that's where life gets a little bit dicey or crazy or scary because this isolation step we're able to get the one on the other side leaving us with Y SAR on the right the problem becomes how do we get rid of the squaring and the answer is yeah take a square root I felt it I felt it somebody said just take the square root I agree here's the problem when you create a square root what else do you have to create plus minus oh boy there's the issue so in this particular one the problem is there's a plus minus situation going on there is a plus minus situation going on and what ends up happening with this is that the plus minus situation is really trying to discuss our values of our inputs or our outputs if you will so the thought process is this if you just play with this and keep it running I find it's almost easier just to deal with this and run with it you end up having the inverse function is the square root problem that you run into is is it the positive or negative version of the square root well if you think about what's going on with this you're swapping your domain and your ranges out so think of it this way our domain was negative numbers up here the domain was negative numbers up to zero and the range represents positive numbers from one to Infinity the domain and the range get switched for our inverse and other words the inverse function will have the opposite for its domain and its range those two two things will get flip-flopped which means if the domain for the original function was negative all of our inputed X's were negative then that means that the inverse of f has to be able to spit those out why well think about this for a moment well we're basically saying is the domain for f becomes the range for f inverse the domain for f becomes the range for inverse the values that we inputed on our function were negative values negative Infinity to zero that means F inverse has to produce as its range it has to spit out negative numbers which means we're going to need the negative version of our square root why because of these because the domain up here represents negative numbers in the original the range has got to produce all those negative numbers on this one and then similarly the other argument is the range for f which represented numbers from one to Infinity will represent the domain for f inverse so the domain and range get swapped out for these meaning that you're going to end up getting negative Infinity to 0o and the other one will be one to Infinity yeah so it's kind of analogous to what happened on the previous part what do I mean the previous part up here you remember we put in time we get out distance the inverse function its input is distance its output is time you're just switching inputs and outputs if your input is negative the inverse functions range is a bunch of negative outputs that's that's the thought process behind it and that pretty much is a great place to stop that wraps up all of 4.1 and should be in relatively good shape in terms of getting through most of it I think that'll get you through almost 98% of it if you have any questions I'm happy to help you otherwise have a great day we will have class Wednesday and we will keep on moving forward on the lecture material