alright guys as big as while the inverse functions okay so inverse inverse functions so we’re gonna do a quick little introduction going to quick little experiment alright so I need you to pick a number pick any number any number at all hopefully something kind of small okay so I need you to with that number add add to ok so once you add it to I want you to multiply by 10 ok once you’ve x 10 want you to divide by 5 ok so let’s say again you pick any number but I want to pick one number one pick let’s say 11 so my number was eleven eleven plus two is 13 plus 10 I mean times 10 is 130 / 50 cs10 says 26 so my number the final number i got was 26 so it we did it so if i told you my number and I told you the steps I had can you figure out the number i started with that should be yes you know just like we did in the warm-up we have to work backwards if you know the ending and we know the changes that we did we should be able to figure out the beginning part ok so the North for us to find the beginning so we have our ending so we have our beginning number right then we do stuff to it and we get the end okay if we have the end and we know the stuff then we should know the beginning ok so another for us to do this we need to work backwards button refers to work backwards s would be a specific order of working backwards which we can’t start from here right leg and put it on your shoes you put your pants on your socks on Daniel shoes you just can’t take off your socks before I take off your shoes or take off your pants before you take off your shoes right so there’s a certain order you need to do this in order to get to the result you want to do so we need to work backwards so if i started with 26 ok we’re going to work backwards ok so working backwards what does that mean does that mean are we going to divide by 5 well no we have to work backwards so in order for us to work backwards we need to do the opposite ok so working backwards means do the opposite so due to the opposite but a which way whoa we have to do it backwards from from last step to the first step so your first step is going to be your last step so we have 26 our last step was to divide so we need to multiply by 5 we multiply by 5 we get 130 ok the next step said the step right before dividing was to multiply so again we need to do the opposite so what is the opposite of multiplication hopefully you’re saying division so now we have to do division so it’s 130 / 10 this was 13 and my first step was added by 2 which is now going to be my last step so that was add 2 so the opposite of addition is subtraction so we’re going to subtract two so now we’re number 11 this is correct so we work backwards doing the opposite things right putting on your shoes obviously put it on your shoe is taking off your

shoes the opposite of dividing and multiplication officer multiplication is division obviously of addition and subtraction right you’re working out backwards and doing the opposite of what you did okay this essentially is called the inverse working backwards at doing the opposite is an inverse ok so our inverse always gets us back to our original to our beginning okay so we can do any weird crazy crazy amount of steps we work backwards and then we’re gonna get our original day okay so let’s write an equation with this so when we say pick a number we say that is what usually we say this is X right so then we add two let’s bring it right here we add two plus two x 10 and then divide by 5 so that was our equation for the for the regular not the inverse the inverse is your final product let’s say this is why right so now we have Y so Y times 5 so 5 y divided by 10 and then minus 2 and this is going to give us our X good so we found our inverse functions are left side is our regular function all right side is going to give us our is our inverse function it’s gonna give us our X ok so let’s do this a little bit more mathy ok so let’s say our equation is 3x minus 10 so we want to find the inverse so what is the first thing we are doing to the X or the first thing we’re doing to the X is 3 times eggs and then we subtract 10 okay working backwards so our last thing we did was doing 10 so working backwards remember we have to do the opposite instead of subtracting 10 we add 10 and then we divide by 3 and that should give us that she kills our X okay that should give us our X so working backwards we get our things so let’s try this try it needs you know just try it zero right 3 times 0 0 minus n is negative 10 let’s plug in negative 10 Sam plus negative 10 is 0 0 / 3 is 0 x is 0 so then it works out we do get our original function inside so now let’s write this in real mathematical form if this is f of X right the inverse rewrite inverse that’s f within with a negative one on top like this so this is not me reciprocal this does not mean add to the negative 1 power to the X that’s not right this you pronounce this as f in Reverse okay so just so we talked about it a little bit now we’re going to write it and romantical terms so if you have the inverse we know let’s get us back to our starting point the mathematical way to write this is so in order for you to know it is the inverse F of the inverse function is going to give you X so that means if you plug in your ear inverse function in to hear that is going to give you X or as well as if you plug in F and so you’re in verse you get X as well ok so we do this way is also going

to give you X okay so um so this is pretty basic it so get users working backwards working backwards and doing the opposite scald your inverse okay so there’s a few I want property that we definitely need to look at in order to be a to be right um so so once again so f is a function with domain of a and a range of be F inverse is a function with domain be and range of hey so our domain and ranges are going to be flipped okay if F has a domain of all real numbers then F inverse is going to have a range of all real numbers if F has a range of 0 to infinity then the F inverse is gonna have a domain of zero to infinity okay this is one really key thing this right here is really really really really really really important okay can’t stress how important it is for you to know that so the inverses are going to be flipped okay for example let’s take let’s say f of x is the square root of x okay so we know this function we know that domain for this function is 0 to infinity and the range is from 0 to infinity oh you know what let’s change this let’s say X minus 1 so that square root of x minus 1 okay so that means the range becomes 1 to infinity our domain because 1 to infinity okay so our F inverse so we know this kind of the domain of the F inverse it’s going to be 0 to infinity and the range is going to be 1 to infinity okay so now we need to work backwards so it is the square root of x minus 1 so first you have to subtract 1 and then take the square root right so subtract 1 then square root so f inverse we’re going to take we’re going to do we’re gonna work backwards and do the opposite so what is the opposite of a square root well that’s x squared and what’s the opposite of minus one well that’s going to be plus one okay so we’re going to square everything so we have our number y right so first we have to square this and then add one so good so our function is x squared plus 1 so now our domain is from 0 to infinity and 1 to infinity so if we graph this let’s make this quick graph paper craft we’re going to graph the square root of x minus 1 we’re going to do that in blue so this is one here that’s one there so this is square root of x minus one so now we’re gonna draw draw x squared plus 1 so x squared plus one we know how to draw this involves you know referring functions so it’s from one and up but now we have to pay attention it says our domain is from 0

to infinity 0 to infinity means just this quadrant here so we’re going to go up okay so yes I know that we usually draw x squared plus 1 as a parabola right it’s one hundred percent correct we do draw it this way but for this our parabola is the domain will be negative infinity to positive infinity now we don’t want that our domain is from 0 to infinity so we do not use this part okay we only use this part now so this is the inverse this is the way it works now let’s do it the opposite way let’s see if that works okay so keep this graph on the same paper we’re going to turn the page I’m going to turn the page but you can do the next graph of and the same sheet okay so now let’s say f of X is x squared plus 1 so now our domain for this one is all real numbers right in our range is 120 1 to infinity so now our inverse function we know has to have a range of I’m sorry a domain of 1 to infinity and a range of negative infinity to infinity okay so what’s happening to the X so first we square it and then we add one so we have to work backwards so first we subtracted and the opposite of of a square is tickets for a room okay so now let’s graph both of them if you have a graphing calculator you can go ahead and do this as well ones up here so this is x squared plus one so now we’re going to graph x the square root of x minus 1 so now this is the problem right rerun into right here this says our domain is from 1 to infinity which one hundred percent correct this is from from one to infinity or now our range our range says negative infinity to infinity here we only have 0 to infinity as to range right remember range means the wise so it goes from 0 remember these communities of keeps going up keeps going to the side so here is where we fall into one little pitfall one of the tricky parts of of the inverse functions okay now remember in order for you to be called a function rite aid function if you have to pass the vertical line test remember the vertical line test pretty much means for one input or one eggs there can only be one output okay so for each X there’s only one why you have one x + 2 y’s you cannot do that okay so so again vertical line test again let me say you have a circle you do the vertical line for x you can have this one or that one so this is not a function not a function okay this is not a function so now back up back to the graph okay so we have the square root of x minus 1 the domain is 1 to infinity the ranges from negative integer spin it so this cannot happen right because our range is not satisfied now so there’s one little technicality that we need we need to know about so when you take a square

root of a of a square you don’t know what the side was right because if you say because remember 2 squared is 4 can you take the square root of four you can get to but also negative 2 squared that is also for you take the square root of four that’s too but wait a minute here I’ll plugged in to and here i plugged in negative to them okay so do you know which one I use if I just give you four no you do not right so since we do not know every time you take the square root of a number it can be that’s called plus or minus okay so the square root of four is a plus or minus two so this means it can be a two or a negative to not remember to write a plus in front because we already assume that is going to be positive okay you already know it’s positive we don’t stay positive 2 we should say too okay so we took the square root of here so we need to also add a oops fun color a plus and a minus okay we take a plus or a minus so this is a positive square root now this is a negative square root okay so now is our domain satisfied yes it’s those what happened so now our domain dissatisfied still it is from 1 to infinity and I for my needs to hell is better and now our range is also satisfied now okay because this goes up and down and to the right this goes down and to the right as well okay so now both of our things are satisfied but is the inverse a function is this is that a function well let’s do the vertical line test if we do the vertical line test against this means we draw a vertical line and they can only intersect once so we do we draw a vertical line oh come on now we draw a vertical line if intersects here and it intersects here so is this a function no this is not a function okay so x squared plus 1 the inverse is plus or minus X minus 1 but it is not an inverse function okay so we can draw it we can clean or see it is an inverse because it has the same range it has an appropriate arrangement of your domain remember they just flip-flopped right but it is not a function okay now we go back to original one our domain so now this is a function right do the vertical line test oh no sex once and this is a function so that intersects at once as well so they’re both functions okay with this with this domain with this domain the inverse function the inverse is a function but now with this domain of negative fit into infinity this is not a function it’s a little bit confusing but now let’s see why okay so now we have the vertical line test so there has to be a horizontal line test horizontal line stations meaning is just a one-to-one function okay so a horizontal line test if it intersects emitter to sex and a vertical line more than twice then it is one to one not once you want sorry it is not one to one so when a function is not one to one is inverse is not going to be a function so let’s write that in test

if F is not one oops uh cannot slow one it’s not one to one then F inverse is not a function alright so again if F is not one to one then F inverse is not a function okay now that doesn’t mean we cannot graph it is we’ve clearly graph the other ones so now the short good way Oh graphing so to be the inverse norm for your graph it it needs to be reflected across the Y is equal to X mine so this is the Y this is the Y is equal to X line ok so we graph x squared we reflect it across the Y is equal to X mine and we get this point ok so every time you reflect the inverse or Y Z 2 eggs you get the graph now again and you can grab it it doesn’t means a function right because the vertical we do the the horizontal line test so this is not one to one so this is not too not not one to one we know if it fails the horizontal we know is going to follow the vertical line test as well for the inverse now let’s go back to the other graph ok x is equal to one I mean y is equal to X thats like that so you’ve reflected over the why you see what’s your next line we get that all right but my settings is really really bad but if you get your calculator you’re gonna see that it is X plus 21 let’s do this in the calculator real quick so it’s still graphing no not this mode this mode oh come on one internet so decimals calculator this is a pretty cool little calculator system you can use it in there in your phone so the square root of x minus 10 you know what’s do that graft it wrong ah hopefully you’re shouting that in your in your in your computer at home well bet I just graphed it wrong so this is actually X plus 1 right because we moved to the right so X minus 1 should be here so if you caught that mistake early on please let me know if you didn’t then don’t lie to me ok so you reflect it already y 0 to X line so this reflected over means you is going to switch your x and y points ok so we graph it here if we draw the y 0 to X line you can see that ok so let’s look at negative square root of x minus 1 won’t you grabbed the wrong this is wrong Oh sighs that why’s that sorry do not understand this hmm just make it a

negative thing all right so probably this cannot do that so that’s weird all right whatever so again if you reflect this over here like we did on this you can see this till the inverse okay so just like I said when you reflect it when you reflect over the X or the Y is equal to the XX line oops you are going to switch your x and y’s so this is X comma Y then for the inverse you’re going to graph y comma X ok so it’s do a quick example this is this is really really important uh this is the best way to graph things okay so let’s do a quick chart for our universe okay remember we said this is the square root of x and this is x squared so uh we start with 0 4 & 9 so this is 0 2 & 3 so now this means there we go from 0 2 & 3 we scar all these we get them backwards okay right so your inputs and your outputs for the F inverse becomes your outputs and that’s going to give you your inputs right so it’s going back from the beginning we have your imp yeah you put your output you work backwards and do the opposite and you get this input again okay so if we graph these so 0 0 1 2 3 4 5 6 7 8 9 1 2 3 2 0 0 1 1 4 2 93 this is the square root of x now with x squared the F inverse 0 0 2 and 4 4 5 6 7 8 9 3 and 9 sorry that’s not pretty but he understand okay we switch the X and Y point ok so to graph in reverse to graph the inverse we have to switch our x and y point right just like you switch your domain your range you’re going to switch your ex and your wats right your domain is your ex’s so then they become your range or you’re wise for the inverse your wise is a range of your f you switches that becomes your domain or your ex of the inverse okay so in order to graph the inverse you should switch them just like we did here this was f of X this is the inverse F inverse we switched our wise they became X’s and our exes became twice okay in class we’re going to do it the mathematical way so as long as you know you’re switching your ex and you’re wise you know how to check for 121 you’re pretty much going to get get all the stuff right and pick it up right away ok so again the main important thing for now so graphing it is just to grab when you graph it in reverse you’re just going to switch your x and y points okay you just clip it all right that’s pretty much it see you in class