welcome to another video today we're going to explore operations of functions and when we talk about operations of functions I really want to focus on what happens with the domain most people can understand that you can add two functions subtract two functions multiply two functions and dividing functions not that hard but what happens if the domain is important you see sometimes math it looks like you can cancel out of the main problem you can never ever do that so when we talk about domain we jump right into it here when we talk about domain it's one of the most pessimistic attitudes that we have right so domain is like like the skeletons in the closet you can't ever get rid of them they're always there like it carries its baggage with it all the time so when we start to compose functions or add functions subtract punches multiply divide functions even if it looks like the domain of your result the resultant function it's getting better it's not what that means is that if we give two functions the first thing we do is find the domain for each of them this is very easy because I give you a nice one is the same and then if we add them subtract them multiply or divide them we are going to have both of those domain issues in every one of these cases and then what's even a little bit kind of worse about it is that we can even develop some new domain problems by doing these operations so that's what we're gonna be looking for in this video I'm going to show you how we can add subtract multiply divide functions will simplify them as much as we possibly can and then we'll discuss the domain so let's start right now whenever you're going to do operations with functions whether that's going to be addition subtraction will take you to division or if you're going to start composing functions the first step that we do is we find the domain from each of our functions individually now here they both got fractions and after watching last video you know about fractions you know that we're looking for inputs that give us a real number defined output which means for fractions denominators equaling zero or bad things we're going to set our denominator equal to zero understanding that when x equals positive 2/3 that's a problem for us and the same problem for both of these fractions I gave you the fractions like this with the same two numbers so that we can add them and subtract them easy easily I'm not trying to teach you that trying to teach you what happens with the domain here especially with this case so here our domain says you're good as long as X cannot equal 2/3 and sometimes we even abbreviate that say hey X such that X does not equal 2/3 X such that X does not equal 2/3 I'm implying that all real numbers work except for 2/3 because that would make my denominator equal to zero and that creates an undefined result for that particular input so that's going to make sense 2/3 as an input don't work because your outputs undefined now when we start adding functions together I need you to remember did you know that no matter what we get this right here is going to be the domain at least this right here is going to be in the domain of the resultant function we could get worse things from it but we can't ever get rid of these problems they stick with your functions forever no matter what you get now typically with addition subtraction and multiplication you're going to see the same domain problems come up again at the end so it's not such a big deal for those but for this one it is and we'll see why in a minute so f plus G of X if we add these functions together keep in mind that we need to comment in order to add fractions and we have that so we're just combining the numerators right now that looks like 6x plus 3 over 3x minus 2 yeah you can to factor the numerator but it's not going to help you simplify so I would leave it just like that because if we ever have to deal with later the first thing we're gonna do is probably distribute that numerator denominators should always be factored so not you should factor that now here we can't do it but that's something to think about and notice something your resultant function here has the same exact denominator obviously as what we started with that means that your domain is going to be the same even if these denominators had been different you would have found a common denominator once you have which means you'd have the same factors and those factors equaling zero would cause domain issues for you so if even if your denominators are different you'd find a common - number and these things would appear again whatever they are in that result so our domain says hey I'm going to be exactly like how you started the same thing st. and sorry I'm going to be exactly the same domain as what you started with for both of your functions combined whatever those are here they were the same they don't have to be that just be different denominators F - G of X is identical and except that we're subtracting but we're also going to see that when we subtract that 4x over 3x - 2 I have a 2 X minus a 4 keep in mind if I said more than one term I want to show parentheses I'd want to distribute that - because you'd be subtracting every part of that numerator and we should know that so minus 2x or negative 2x plus 3 over 3x minus 2 again there's no simplification that we do and we see that the domain that we had originally is going to be the same domain that we get when we subtract functions you know what if you think about it with multiplication think about what happens when you multiply fractions well you take the denominators and you multiply them together and they they basically become the new denominator for your fraction now if we simplify stuff we can accidentally start simplifying away one of those two main problems but if you think about what happens in the domain I'm gonna have one fraction with the problem in the denominator and another fraction in general problems denominator I'm going to multiply them that becomes my new denominator do you see how I'm just gonna keep combining my domain problems for addition and for subtraction and even for multiplication we're going to get the same domain here moved over to this that's gonna the same domain is what our result is is what we was sort of it so this 2x plus 3 over 3x minus 2 probably parentheses would be good here multiplying fractions says multiply your numerators we're going to distribute the 4x multiplier denominators we're not going to distribute at all your denominators shouldn't always be factored as much as possible so instead of distributing 3x minus 2 times 3x minus 2 we leave them factored for a variety of reasons one of them is because when we graph these things it's really nice and very appropriate to have your denominator factored it would create a vertical asymptote we haven't got there yet but I said that in last video it created a vertical asymptote at two two-thirds x equals two thirds this is this kind of force field you can't touch and then because that powers there because we're gonna have this group as 3 X minus 2 to the second power it's gonna tell some multiplicity of that we're gonna get to multiplicity later so we'll distribute numerators absolutely so that's going to be 8 x squared plus 12 X on the denominator leave your denominator factored so I don't care what these things are really the same or not you leave them as factors it's going to read help you with graphene I promise and we ever have to add fractions or simplify fractions or multiply them we can find LCD easier we can cancel out simplify easier we can do everything easier if we leave these factors now because we have 3x minus 2 times 2x minus 2 that's a repeated factor that's repeated multiplication that's an exponent that's how I want to see it and do you see that no matter what because you're going to keep these factors you're going to see the same exact denominators that you had here so the same exact domain issues that you had here those are going to repeat every year so our domain doesn't change our domain says yeah your domain for this is X's such that X cannot equal again 2/3 I want to reiterate this I know that I've been taught it to you but you know I teach at this point you know that what I'm trying to do is prepare your brains I'm giving you an advanced organizer for later on as to why you do this sometimes if you don't know why you do something it's really hard to accept that you have to we have to do this and leave it factored and write the exponent because later on this is going to give us a vertical asymptote on the graph at that value so a vertical line x equals a number is a verb 1 if X can't equal that vertical line well then you have something you can't touch can you do you don't you can you one of those you have something that you can't touch don't you well if you have something you can't touch then you have a vertical asymptote and that power says that you are going to be even even functions match up so we're going to look like this or look like this around the asymptotes that's why we leave it that way I'm going to flesh that out with you to a few more videos so a long story as short as I can make it domain is still really relevant for your functions we talk about it lots and lots moving forward find your domain first if you have to add subtract term old five functions you're going to get the same domain almost exclusively there are times when you don't but almost exclusive to get that if you divide not only you're going to get the same domain you could add some other issues to that that domain the domain problems you already have the key here though is that when you start dividing it's gonna look like you can cancel out some stuff it's gonna look like your domain doesn't have a problem anymore you still have the problem it's just hidden so I said this at the beginning and it's gonna make some sense as we go through but these issues this X can't equal 2/3 that issue because the resultant function starts here it is these things just wrapped up together because these are the those root if you will functions you can't have that 2/3 no matter what we get here I can promise you that our domain is going to exclude 2/3 from it it's not going to look that way it's gonna look like we cancelled out but it still has to be here so when we divide functions were really conscious of what goes on the numerator and what goes on the denominator says F divided by G so f is this 2x plus 3 over 3x minus 2 divided by you can even write a divided by sine if you want to 4x over 3x minus 2 now we know something about complex fractions we know that fraction divided by a fraction when we divide so fraction divided by fraction when we divide fractions we can reciprocate and multiply so instead of this will reciprocate the second fraction or the bottom fraction space all right so I have my first fraction a problem division says multiply by the reciprocal of the second fraction they go wait a minute well it's awesome because look boom oh I can cancel out my team I do my denominator I can cancel I'm one of those domain problems it certainly looks that way it certainly looks that way but because of where this came from because your original functions had a domain of negative 12 two-thirds is a bad number so X cannot equal 2/3 we actually write that down first we go through this and go right I know X can equal 2/3 and no matter what happens X can I equal 2/3 even if I serve Kansas about I know that where it came from the baggage that holds on to the skeleton in the closet that's you can't see anymore is you cannot allow this function equal 2/3 based on those root functions of functions we started with the other good news here is that you can add some problems to it so notice our resultant function then we have that 2x plus 3 over 4x and you think about your wait a minute wait that's a fraction with a different denomination and we know that dinars equaling zero are bad things so if we continue to go all right let's let's find that let's find the domain of this so well for X equaling zero wouldn't be good because that creates an undefined issue for an output and we know that the domain needs inputs you give us defined real numbers and I divide by 4 0 to PI over 4 is 0 you go okay yeah hey X equaling 0 would be not a great thing here we'd get 3 over zero that's undefined so I know that might exclude 0 if we ignore this and have that we do understand domain but we're forgetting to realize where the functions came from so when we go ahead and we do division how do properly assigned remain first when you start dividing make sure that you write that domain down you go okay whatever these are you're going to have it here no matter what and then we might even get additional domain restrictions that we have to add to that we have to exclude that from the domain so even though it looks like the the 3x minus 2 cancels out it does but the domain issue is still there based on where we came from I hope that makes sense we're gonna go ahead and do one more really quick one just to talk about square root to get that one more time and then we'll be done ok last one so we're gonna add these subtract these multiply these and divide these functions and we're focus on the domain so really if you want to stop this and go through here your first process should be this thing about find your domain first and then recognize that no matter what I do I'm gonna have these two domains in the result of every single one of these function operators and I might even add some to it so I'm going to have at least ones here and maybe add domain problems as we're doing these operations so let's let's go ahead let's do our do our domain oh man so we think about it last time we're gonna think about domain like this for a while so I look at the square root I think square roots are this idea that they wants to be positive so it tells you what it wants to be it says I want this inside this radicand to be positive or at least zero so we think about square roots and say I need real numbers here what makes it happen makes it happen if the inside is greater than equal to zero if not I get imaginary numbers that's an issue I subtract three and it says that I need X to be greater than or equal to negative three in order for this to give me real numbers out that is the domain of that function then we go to the denominators right here this function G of X say what about denominators in order for this input these inputs to give me a real number to find value out well it's not about negative or positive it's about defined values for fractions we know that if x equals zero I'm going to have my denominator let's give me an undefined value that's going to be a so any positive is fine here any negatives here fine here but we just cannot allow zero to be an input for that particular function so we're going to say that yeah X cannot equal 0 that needs to make sense before we go any further so when we look at square roots we think what's gonna give this or what inputs will make this function give us real number to find outlets for square roots you gotta keep me positive solve it and these are the numbers that keep it positive and give you real numbers out for fractions denominator cannot equal zero because if they do you get an undefined value so square root is positive denominators can t 0 these two domains are going to follow us for every single one of these and we might we might get some more restrictions out of it what can't happen even if you start canceling stuff out you are still going to have these so you never ever cancel the way domain restrictions so if you wanted to do this right now say hey what's my domain well I know that that's gonna happen no matter why I get here I know that because they never go away also do you do you realize that this is not they're relevant here we had a case last video where we didn't have to put this because it was it was included in that well or sorry it wasn't including that that interval since this says X can be any number bigger than negative 3 and 0 is in that interval we have to show explicitly the rate excluding it either this way or with interval notation which I showed in the last video what we can write it here to same thing and that might be a really good option for you to before even doing it is write down the domain it's not going to change the only thing is you might have to add some more to it that could be the case well let's go through adding subtracting multiplying dividing the first they're going to go very quickly so if we add these functions there's nothing to really combine so function says hey you're just gonna have that X plus three you're gonna add fine Rex and it's very clear to see that that's that's still the same exact domain or if you subtract them the same thing X plus three minus five Rex same exact domain or multiply them I probably put the 5x first because multiplications community if we can do that five over x times X plus three you can even make it one fraction five times X square root of x plus three over X and you see that we have the same exact domain is not going to change now the division we do have to be conscious of what's coming first because division is not commutative so we have to put the square root of x plus 3 on our numerator now we'd have to have 5 over X in the denominator wait a minute when we're when we're dividing fractions we multiply by the reciprocal so so this would be the same thing as the square root of x plus 3 times x over 5 times the reciprocal of 5 X so here's X so you can think about square root of x plus 3 over 1 square root of 3 over 1/5 X or x x over 5 you know this is fantastic I can simplify this just a little bit maybe put my X before that square root that's the resultant function of dividing F by G we divided square root by a fraction when you divide fractions you can multiply by the reciprocal no problem we would have X x squared vegetables 3 we'd have over 5 this is one of the reasons why that might be a really good option for you to do first do you see it do you see how the denominator changed for us the demanders said well there's no longer there's not an X there so if you were to just do this and then think about the domain you're probably gonna miss that don't miss that this is why the technique I'm giving you works really really well find the domains first if you want to write them down for every operation do that even for compositions that that sort of works really we take the domain of the inside function and the final function we combined it that way but for adding subtracting multiplying dividing you're always going to get the same to me no matter what because we started with that we don't add anything new huh because look at that the square root hasn't changed so so that's still the same domain that we had the X is gone it looks like that's gone the the problem for an x equals zero is gone but because of where it came from we have to retain it but our denominator cannot equal zero anymore that right there can't equal zero so we don't gain any new problems you still have the same old ones I hope that makes sense hope you understand this point that your domain does not change from your original functions when you're doing these operators you can only kind of make it worse if you divide so practice that and I hope that makes sense next time we're going to talk about some graphs of functions determining if if one inlet has to give you one output what that looks like graphically so we'll do something called the vertical line test you've probably heard about it but we'll explore why it makes sense we'll talk about even versus odd functions and in small things like that so have a great great day and I hope to see you in the next video you