welcome to the start of precalculus or college algebra if you're in one of those two classes or you just want to study this to get ready for calculus which is awesome this is going to give you a lot of background information on what makes calculus possible and the math behind doing a lot of the calculus that you might have been exposed to or forgotten frankly because we all forget things and that's okay that's what this course is intended to do is to teach it to you in a way that you remember or remediate you in a way that you can understand so to start off with precalculus or college algebra we need to understand what we're going to be dealing with so the whole premise of this this class is that we're going to be dealing with functions almost exclusively in college algebra and therefore precalculus so we're going to discuss in this video what a function is what makes a function a function how to determine whether something is a function and then we'll start exploring how to how to evaluate functions how to find the domain and the range of functions in another couple videos so let's start with that most basic question if we're going to be dealing with functions of function and mathematically its it's really kind of a are most simple sort of an idea that we can graph a function is something is this relationship that takes one input and just maps it mathematically to one output that's it you plug in one number you get one number out that's all that function really means now I always tell my students face-to-face that some sometimes we know just enough information to make it sort of dangerous and we can sort of project other things onto a definition I'm not talking about one-to-one functions and we're going to explore that a little bit but keep this in mind that a function it's the most basic sense and a one-to-one function are different things so when we say a function don't add to this definition it's just a relationship that map's one input to one output that's it so we plug in one number we get out one number now when we when we're talking about this we're talking about our inputs and outputs we're functions for this course our inputs are almost exclusively X X values and our outputs are almost exclusively Y values or f of X the function that's based on that X we're going to talk a little bit about the independent variable and we'll talk about the dependent variable as we move on so so in this lesson the the vocabulary that I want to get you to know is function what that means what our input similar outputs what is the domain and the range and then this idea of independent and dependent variables so what we've learned right now is that a function is a very basic relationship it takes one number in it gives you one number or one thing one input end and one output so our inputs are almost always x-values they can be different but typically X's while our outputs our Y values or f of X values we'll talk about that notation just a bit I guess the biggest note that I want to get you to get across right here that you you this sticks with you is this right here in a function relationship that functional relationship and input never gives more than one output so if we uh if we start plugging in numbers and we go alright well that's uh let's plug in the number four and sometimes this relationship gives you seven and sometimes it gives you negative two it'd be very hard for us to graph that because it wouldn't be dependable that the output wouldn't depend on the input the same way every time because it'd be really really difficult to to graph and to deal with it difficult to count on that so can we do we have these non functions that we can graph yeah we do typically we use parametric equations or some or other idea to graph them sometimes different coordinates like polar coordinates which we learn about it's like a calculus two level class but for right now and we refers to graph things well and easily we need functions we need it so that one output gives you one output and if it never gives you more than one output so long story short if you plug in 1x you better only get out one Y or one output one f of X value so a couple of notation issues that we're gonna deal with number one we typically see functions as either Y or f of X so when you see them when you see y equals something in terms of X or f of X equals something in terms of X they mean the same thing they both stand for this function notation now we want to talk a little bit about what X means what that is and what Y is or what f of X is so when you see that variable X that input value whether it's X or something else the thing that you're plugging into that input it's called your independent variable so in our case when it says y equals some function of X or we're going to talk my function of X that X right there that's what we're inputting that X variable that's an independent variable your input is always your independent variable Y well what's that what's it mean well independent is kind of like what we want to be we want to be independent people we want to make our own choices we want to sell actualize we want to determine what we're gonna do we want to plug in our other stuff and that's what the independent variable gets to do so we get to plug into the independent variable we say I can put anything I want that makes sense we'll talk about domain the things you can plug in in just a little bit but but really we choose the independent variable first and then the output the Y the dependent variable well it depends on what we've chosen you don't pick your output and then find the input that works for that not typically typically we choose our x-value we choose what we want to plug in we choose our in our input which is our independent variable X and then our output is based on that so when we look at our functions y is in terms of X or a function that's based on whatever this variable is that's your independent variable so in our case right here what we're talking about or empathize or X values are inputted independent variable is X so X we call that our independent variable because you get to choose it now once you chose number once it's been chosen for you once you've plugged that in put an input or evaluated for that X variable for your independent variable whatever that is that input the function is going to turn something out the functions going to take that one value and it's going to give you one value out we call that value either the Y value or we call it an f of X value or a G of X or whatever the function is we call it that value so Y or if we're using function notation f of X pause right here for a second please extrapolate from this if your functions name is an F if it's like G or H that's still an output value so if we have a function with a different name and we're going to talk about why we have different names for functions in just a little while this lesson is just to get your feet wet on what the functions are as well as introduction but extrapolate from this so Y values and f of X values they mean the same thing but if our function is a different name we might use G of X or H of X - H of X depending on what that function is named right there that might give you a clue as to why we have function notation is it everything is called Y if you're like everyone the world become being called Jim you know hey Jim everyone yeah you go ah shoot man we need different names that's one reason why we have function notation so you can call one function f and one function G one function H or whatever so back to this if X is our independent variable that you're plugging numbers into and your function is turning out an output that's either Y or a function name of X depending on your infinite variable this depends so this is this is uh this is not a chosen value this is what you're given your one output dependent on what you just plugged in so this one you choose this one you don't that's why it's called the dependent variable now what's nice about this is together they make up an ordered pair so when you deal with a function and you plug in a number and you get something out well I will in a minute our input values are X's our output values are Y or f of X and this is going to give us an ordered pair something that we can graph almost all in this class I'm going to relate back to graphing and show you that this makes sense graphically so when we're moving forward we know right now hopefully the from what I said before that we're gonna be graphing functions and without a functional relationship one it because you want output it's very difficult to graph so I've already covered the base like hey we need functions cuz we can really only grab functions right now so what what's that allow us to do if you have a function relationship your independent variable your X your input is going to be the first piece of information on an ordered pair so it's gonna be the first thing that horizontal axis whatever that is X or T something your output is either Y or we could have X comma some function notation those are going to be the two standardly accepted ordered pairs that we get input and output it's all the religious to it so just a little 10-second recap to make sure you got it one if it gives you one output in a function if that's not the case you don't have a function and then put never gives you more than one output in a function relationship how we write it is either y equals something with X's in it or f of X equals something with X's in it something with an independent variable something you get to pick to choose in to put in there and then your function gives you one output we call the X or independent variable and Y are f of X the dependent variable because it depends on what you've just chosen let's go through one quick example to make sure that you're you really seeing it that you're seeing what a function does so you might have a job maybe you don't have a job a lot of our jobs in society work on hours so the hours that you put in determines the pay that you get out you guys know that so if you work more hours you typical hopefully get more pay well in this case you think right now that what what are your inputs and what is your output do you input $500 and then get to work 25 hours if that's true come work for me I could've create stuff for you all right you pay you 500 bucks I'll take support 25 hours it doesn't work that way you put in the time you get out the money so you put in 25 hours you get out something so I'll make a little relationship here and we're gonna determine whether this is a function or not so some guy works 25 hours and makes $500 another person works 53 hours and only makes $310 at first glance you might be going well that sound that's not right well do people have different jobs I do make the same rate as everybody else for doing different jobs let's know even in the same job sometimes someone might have more seniority and make more money based on whatever whatever the pay structure is so don't be confused that this doesn't make sense because it can make sense like that someone else works 30 hours and makes $490 and another person works 40 hours and makes $490 what I want to do right now is number one again to determine the inputs and outputs and you have you that first because we're going to check next is does every input give you exactly one output so those are only things you check here to make sure that you have a function don't add to that definition so first our inputs I mean we do be thinking let's see what what's happening here what are we doing first and then what are we getting out of it we're putting in our hours first art inputs are twenty five fifty three thirty and forty in just a moment because there really should be no suspense in math because you're learning right so in just a moment we're going to call those inputs the set of values that we're inputting into a function we're going to call that the domain the stuff we get out of it is called a range so domain is a set of our inputs of a find that in a minute ranges a set of our outputs so so one more time we're looking at what we do first to get out something later our inputs are what we plug in first domaine this set of X values right here influence our outputs are 500 310 and 490 that's our range now here's the only thing you got a check to determine whether this thing is a function look at your input and see if it gives you one outlet so if I work 25 hours do I make one value of money yeah make 500 bucks if I work 53 hours do I make one value of money yeah I make 310 dollars if I were and remember these can be different people that's why they can have different rates if I work 30 hours can I make one value money yeah 490 if I work 40 hours or some of those 40 hours do they make one value of money yes 490 this right here is a function relationship please don't get confused with the one-to-one function like I said sometimes we know just enough to misinterpret things what a one-to-one function is is when you plug in one number and you get out one unique number so in this case this wouldn't be one to one because you get 490 twice as two different outputs that's not the same ideas that being an odd function all the function needs is I plug in one number I get a lot number I don't care if we get the same number out every single time we could plug in this like different inputs and get the same number on a horizontal line forever so you plug in five and get out ten plug in seven and get up ten plug in 15 get up ten plug in a million get out 10 so we can do that in functional relationships it's okay in a function to get out the same number twice that's okay the only thing I promise the only thing is one of them gives you one up with this input when I put this input one output this input one output this input one output didn't matter for the same think about you maybe maybe using somebody else if they're making different rates of money could this guy work 30 hours and make that much money sure did this guy work more and still make the same amount of money if they have different rates that makes perfect sense that's totally fun this guy's just making a little bit more money per hour than this person is is that making sense you stop right now and think if that makes sense that this is certainly a one input is giving you one output so right now what we have that's a function relationship and our inputs our 2553 30 and 40 our outputs are 500 310 494 and we're going to rename those in just a bit domain and range now I'm going to change it we're gonna see if this is still a function so let's say that we do all of this we keep it exactly the same and I add one thing all right tell you what I'm gonna do this right there and so this guy's pretty happy right he's working 25 hours he's getting 500 bucks that's dependable this guy is working 53 hours making $310 no problem this guy's working 40 hours getting $490 he knows what he's gonna get imagine being this person and I'm not talking about overtime I'm talking about where your pay you but you've not agreed to a pay rate change and so you work 30 hours and your boss comes to you and says hey great great deal man I'm gonna give you four hundred ninety dollars day you like whoo that's awesome and then the next day counts you goes Tate or the next week he tells you is tell you what you worked 30 hours but I just feel like giving you $310 and you like they say I just feel like it you know like I'm not I'm not really happy with that why well because you hadn't an agreed-upon rate right you agreed to work 30 hours to make that much money what if he just fee or she just feels like giving you that much money and what if you can't depend on it and you never know well that's not a very good working relationship is it because you don't know what you're going to get out for your work that's exactly what a function has a problem with if you have more than one output for the same it doesn't know what you're going to get out of it it's the same exact thing it's working for you right the functions do some work it needs to know what you're getting out so this right here this is fine this is fine having two different inputs got the same up but that's fine this right here having one input go to two different outputs that is a no-go that's a non-starter for a function we're not going to deal with that in this class right here this is a this would be a problem this would need something to redefine the scenario so to make it to make it right and so later on we can do that in higher classes right now you might say well this is a condition like here you're making whatever per hour and here you're making something different per hour and it depends on the scenario but as it stands right now this would be a non function so we have inputs we have outputs one input gives you one output no problem we kind of went through this and said when do functions happen what do they look like what's a non function non function is when this case is not met when one employee gives you more than one output that's a big problem for us so with functions we have what's called the domain and arrange our domain is a set of all the inputs or all the values that the independent variable can take and get a real number out let me say that again for functions we're not talking about imaginary numbers right now we're not gonna be graphing those on a real coordinate plane okay so what we're talking about for the domain is the set of these things that give you one of these things out that's a real number in other words the domain is a set of all the inputs that you can have for a function so right here if we just want to real Abel this domain the domain is going to be the set of the inputs it's just 25 53 30 and 40 that's what we're choosing to work that's the hours that we put in the range is the set of the outputs or our wise or the whatever dependent variable is in our case if we just want to real able that some range our outfits are the range so the range of numbers that we get is a 500 390 and sort of 310 490 that's what we're talking about when we have the idea of a domain and range domain is just a set of inputs that you can put into the function and get a real number out when we start talking about imaginaries we're gonna we're not gonna be really talking about functions that are graphing for a while but for graphical functions things that were functions were graphing here and we grabbed those on a real horn and claimed our inputs is called the domain the outputs are called the range so inputs X's we jump those together and juggle those together group those together in the domain idea and all the Y's or f of X G of X what your function is that's your range I hope that makes sense what we're gonna do is we're gonna come back with a few examples I'm going to show you how to determine when a relationship even an algebraic relationship represents a function and when it doesn't so be back in just a second all right let's continue so I have a few examples on the board obviously and we're going to determine whether these relationships even algebraic are functions or not and there's some really easy ways to tell sometimes it's I'm gonna make it so that's pretty obvious to you so you look at it good yeah that's that's not gonna work the key here is that with functions one input gives you one output and not more than that let's look at these two first you see a feel for it so right now what I want to do is look at the relationship s and look at the relationship R and I want you I need you to be able to pick out all your inputs and all your outputs so we're looking at s remembered inputs values are X values our independent variable values so our inputs here are negative 2 negative 1 0 and 1 do any of those repeat at all so negative 2 negative 1 0 1 there's no repetition this can't pop down even care about the outputs not at all because right now checking functions is does 1 input give you one output if an input is not repeated well it can't possibly have a different output so when I say negative 2 what's the output for that it's 16 there's no other case where it's not 16 negative 1 gives us an output of 4 0 gives us an output of 3 1 gives us an output of for every input gives us 1 output now our output values here the 16 4 3 & 4 oh wait yeah the 4 is repeated but that doesn't matter this would fail to be a one-to-one function but it's certainly a function there's it's just a very basic definition one input gives you one output if we're looking at something that just has a series of points then sorry a sequence of points then if our inputs are all different we can't possibly have the same input mapped to two outputs so for s this is certainly a function how are things do it on your own so just a little bit thing through I need you to be thinking about what the inputs are right now we should be checking those so in your head you should be reading through those inputs going ok it's this this this and this maybe look at the outputs read through those four outputs and determine what those are and then make a decision whether R represents a function or some other type of relationship should be thinking about in your head right now what I look it through this and I see negative 2 3 5 and negative 2 but wait a minute theta 2 here's map 2 5 negative 2 here's about two six one input it's the same number is giving us two different outputs that's a problem this is certainly a non function because of these points right here negative 2 5 and they 86 no I've been asked the question before what if that was a 5 would it be a function it'd be very awkward to write the same point twice you wouldn't see it but yeah that would be because this negative 2 5 and this numbers of 5 negative 2 would still give you 5 so we look at that a little bit but you're really not gonna see that the point here is that every input is not giving you just one output in these cases these ones are fine these two not so much this would be a non function hope that's making sense to you now unfortunately we heard all these kids stuff like that we typically get functions that are in algebraic form naturally so and how in the world are gonna determine that well there's a couple ways to go through it and once we make it to this these two examples it's going to be pretty clear so here's the idea if one input is supposed to only give you one output you can start by just plugging in a number to your independent variable like X so if you take and plug in something like 2 negative 7 times 2 is negative 14 plus 5 is negative 9 ok that's one number you can try that and that's a really good indicator that you're going to be plugging in one number all the time you get one number out does it work all the time no it doesn't because sometimes we can have non functions where certain inputs do only give us one output but others give us more than one so it's better to kind of understand what's going on here there's two things that you really look for and they're really the same thing that they come in two different forms number one if you look at these four examples these three are all solved for Y which looks pretty nice and that's the best way for us to determine whether we have a function not so normally what we're going to do is choose to solve our algebraic equations for y and we'll have a much better shot determining whether it's a function so this one looks really good this one looks fine this one looks well at least it's software why it looks at crap but it's at least they're solved for y this one not so much so part of this is reviewing how to solve things for a variable especially when we get to you this one right there I really need you to focus on that one so when we're solving for a variable where I'm doing the equation everything that's connected by addition subtraction first and then we're going to do anything that's multiplied or divided so we would do things like subtract 2x now I'll do from both sides and we get negative 4y on the right hand side negative 2x plus 8 then to get rid of the negative 4 notice how it's connected by multiplication we're going to divide you just need to make sure that you divide every single term by negative 4 when you're trying to undo things connected by addition subtraction you're trying to get zeros that's what we did here we subtracted 2x because get zero what because zero minus 4y can give us that single term negative 4y completely eliminates a term when you're dividing or multiplying you're trying to get ones what because if I get negative 4 divided by negative 4 giving us 1 1 times y it's an identity that multiplicative identity of 1 is going to give us back the Y so we're going to be dividing by 4 actually maybe of 4 and we have to do it on all three terms so negative 4 divided by negative 4 that's positive 1 that's we're looking for because 1 times y is y when we're adding and subtracting you're trying to make zeros in equations when you're multiplying or dividing and trying to make positive ones on the right hand side there's some sign changes a negative over a negative is a positive that's going to be 1/2 X positive over a negative is a negative so we think and make it a term we're going to translate that to a minus -2 so let's see we've subtracted 2x no problem we've divided by negative 4 that's going to give us positive 1/2 X minus 2 now all four of these equations are solved for Y they're in what we like it's not function notation because we don't have a name of a function but it's really close it's how we typically like to see functions written with the dependent variable equal to something in terms of the independent variable y equals something in terms of X here's what we're really looking for we're looking for a notation issue that's going to cause us to have two inputs when we plug in one number this doesn't happen there's nothing there that's going to give us two numbers for everything we plug in there's nothing here even though that's a power to that's not what we're looking for if you plug in a number this and you square it isn't that just one number plug in like three well it's nine okay plug in negative three notice you're switching inputs it's still positive nine but that's two different inputs you just plugged in I'm not saying plug in two numbers you get the same number out I'm saying if you plug in one number is it giving you one number out and answers yes plug in any single number here whether it's three or three or seven or positive eighty-four I don't care it's going to give you one output now you see what I'm talking about plug in the number here gives you one thing plug in number here gives you one thing there's no notation issues that's uniformly going to give you two outputs for every one input here the same thing you plug in one number you're going to get one in route there's nothing notationally that's going to give you two outlets this right here that's a problem and in general man that's what you're looking for something like a plus and minus when you plug in one number and saying oh yeah I forgot I said yeah how about zero C 3 minus zero is 3 square root of 3 is like one point seven something so this is a number oh and then I take the positive version and I take the negative version that's two outputs but I only plugged in one number this is what you're looking for a lot of the time to determine whether you have a function or not so function yes function yeah function you have yourself for wide to make sure it was easier check yes function no this has a notation issue that's going to cause us to have more than one output for everyone but that's the big deal here this is a non function my plus right there why have your attention don't miss this sometimes sometimes people like to not understand fully and misinterpret I'm saying this with the square root is a function it's not the square root that then messes us up here it's the plus and minus in front of it so that the square root here the square root of 3 is the square root of 3 or if I plug in 0 but the square root of 3 is the squared of 3 but then I take the plus in my it the positive negative that's the notation issue that we're talking about you guys see the difference in that this is fine this is for sure a function plug in one number you're going to get one number it's the positive the negative it says for every choice you're making I'm taking it positive and I'm taking a negative from that this is the issues that were generally looking for so everything here is all good except for this one this guy right here is the only non function that we have and I hope that I've made that really clear for you and that because of the plus and minus that's that's a thing that's it that's a problem now think think about it think of it right now when do you have to put that on your paper do you remember like when you're solving stuff oftentimes we'll have square roots given to us and square roots given to us it's fine you don't put a plus and minus in front of arbitrarily you just take it for what it is but when you're solving stuff and you get down to here you go alright I have y squared equals x and you know how how in the world do I get rid of power to I'm trying to solve for y and you go oh yeah I match the power to the root so a square power we need a square root and you put a square root around both sides because what you do to one side you have to do the other this is fantastic because a square root cancels a square power their inverse functions that's a power to that's a root two and you know why on the right hand side I get the square root of x and your teacher goes and I go no just tell me the black Mouth heart is hurting right now because that's not right you see in order to maintain the idea that when you square a number whether it's positive or negative you get a positive number out of it we have to have a plus and minus in front of that square root so here's the idea what I always tell my students is this when you're the one who puts the square root on the paper which is stems from I'm solving something from a power too when you're the one who does that takes a square root on both sides you have to put the plus and minus long story short the scrim is given to you you don't need it if you're solving for a square root of solving something that's squared if you take a square root you have to put that plus a minus that's a big deal not for a cube root not for an autumn but for even roots yes now can you can you extrapolate from that can you look through this in these last three examples and think about when we're gonna have an on function or not so take a look at it but remember we're gonna do we're gonna look at these two go hey I haven't solved for y oh this one is solve for y we're gonna solve all of them for y and we're gonna look at our result so think about it right now if you go through the process of solving this for y are you going to have to take a square root if the answers yes then you're gonna have to have a plus and minus and right there it's gonna tell you yeah that's not a function right here when you solve for y are you gonna have to take a square root and the answer is no no there's no y squared there I'm not gonna have to take a square root therefore I can't have a plus and minus that is gonna be a function how at this one think through it for yourself and go through the process and your head of solving that for what when you get down to the end are you going to have to take a square root the answer is yeah I'm going to so if I'm gonna have to take a square root on both sides I'm gonna have a plus minus that's not gonna be a function let's go through and see that in action so when we're looking through it alright man now now that I know about square roots and I know that when I take them I have a plus and minus which is yielding two outputs for every input this is probably not gonna work and we're going to practice your fun so that you get good at solving for four variables but we're gonna subtract 2x squared on both sides by the way the reason why we're doing this right now is to make certain that we're not making little bitty errors when we're solving things for instance we cannot take a square root right now without getting rid of that of three in front of y squared don't do it I see it all the time if you'll try and take a square root it doesn't work because that power to is associated only with the Y and not with the power three if there's no parentheses and therefore you can't distribute that power to onto that coefficient so we divide by three and give something kind of ugly really who cares if it's a fraction we're just seeing if it's a function or not and we end with aha there it is y squared negative two-thirds x squared plus one third is that solve for y no no software y means I have Y by itself no powers no no coefficients no nothing around it so when we solve for y right now we're gonna have to take a square root we have to match the count of the root we have to do to both sides and as soon as you put a square root on your paper wasn't there before you're the one who did it you needed plus and minus you don't need it on both sides and that's redundant because we would have the ability to get both the positive negative whatever do it once or twice on the left-hand side we need just Y on the right-hand side we have a square root we have a whole bunch of garbage inside but right here this that's the problem this right there is telling us that we do not have a function it's not a function because every time we plug in one number it would automatically give us two out systemically that's the issue I hope that's making sense right now I think I'd like you to do this one on your own so if you can if you want to pause the video and try that give it a try right now so see if you can just look at and tell me it's not a function and then see algebraically what's gonna happen when you solve that for y so I'm gonna start it now but you should be pausing the video trying that so for us we can subtract x squared no problem you could of course add for y squared and subtract one that's the same thing we're trying to make zeros when you add or subtract so zero minus 4y squared is negative 4y squared on the right hand side we have negative x squared plus one before we take that square oh wait square root I know they have a plus a minus I know it's not a function right now but before we show that we're gonna divide everything by negative four when you divide we're trying to get positive one on the side that we're trying to solve for our variable so negative 4 negative 4 is positive 1 positive 1 times y squared gives us Y squared on the right hand side I'm negative divided by negative is a positive so you can write that as x squared over 4 or 1/4 x squared it really doesn't matter minus 1/4 so we're adding a negative now lastly we're gonna take a square root on both sides because that's not solved for y so when we do that square root both sides the whole thing I have to do a plus and minus and solve for y that's good but when we do it we have this this expression that's not a function this statement that's not function it's a relationship it will give you numbers when you plug something in learning so if I plug in something like one okay this is a great example if I plug in one right now or if I plug in negative one do you see how it would only give me one outlet do you see that so if I plug in 1 1 squared is 1 1 over 4 is 4 1/4 minus 1/4 is zero it's okay to take a square root of 0 to 0 and plus a minus 0 well wait a minute you can't have the positive in the positive 0 negatives are going to make sense it's just zero this is the what I'm talking about how you can kind of get screwed up if you just pick a number to plug in because if I plug in the number one or even negative one sunita one skirts one also then it's going to give me only one output and I might make the mistake that this is a function it's not it's systematically taking every other value besides that and giving you two outputs so don't fall into that trap that's why we talked about how you can't check a lot of times with plugging a number in but really we want to understand not just go through some process of plug a number in but really understand what's going on that's a non function because for almost every input it's gonna give us chocolates lastly that even though it looks weird is going to be a function it doesn't have a y squared there's no square root up there with a plus and minus in front you're not going to get one but I want to go through the process of solving for y to set us up for for a couple things when we get to Exponential's and when we get to solving inverses which is much much later we're gonna have to solve this sort of stuff for a variable light y so one idea behind doing that when you have y's on multiple terms here's the thought process stick with this because isn't this a big deal okay what I need you to do is get all of your terms with Y on one side get all of your terms without y on the other side and that's set up right here this has a why this has a y in these two terms don't so out of the four terms the two death lies are already grouped the two that don't are already group now why would want to do that well if you group all the terms that have Y's and all the terms that you don't have Y's are on the other side of an equation you can factor it so the idea behind solving for something and your variable you're looking for us on more than one term group your terms with that variable on one side and all the ones that don't have it on the other then factor that variable a factoring does it create something that's being multiplied to remember that we divide out and create a multiplication problem so if we group or wise factor and then we can divide so we've already grouped our wise since they both have a Y as a common factor we're gonna factor it that gives us just X plus 2 we're removing that Y by division and lastly because that's multiplying is that's what factor does gives you stuff that's multiplied together by going through the process of division the opposite of distribution because that's multiplication we can divide both sides by X plus 2 and then y equals 3x minus 1 over X plus 3 now you can try it plug in a couple of numbers if you want to it's gonna give you one number L but the main thing to notice is there's nothing awkward going on there's no there's no notation that's going to systematically give us more than one output for every input we plug in I hope that this has made sense for you I hope that right now on your head you have the following things down like you understand what a function is and you could tell me either algebraically with points or with even a real-life relationship what is a function what is not you need to be there you didn't know that there's some notation that systematically causes non functions the last thing that I want you to do is be able to solve equations for a variable especially stuff where we might need to factor or take a square root with a plus and minus you should have sort of a decent idea of domain so we need to kind of associate domain with inputs X values and range of outputs y values f of X well I'm gonna I'm going to explore that more in a couple videos so I hope that makes sense I'll be doing well I'll see you for the next video you