### Domain and Range of Relations and Functions / Set Builder Notation

so we will be looking at warm actually resistant okay so I’m Jolene rig review we did talk about this during a warm-up the domain is a set of influence might hit the bag of numbers and those are things that are allowed to input into your organization the range is when included in that input return some here but right through the rank something locked out that’s an object so for this particular relation being a set of ordered pairs what is the domain one okay and one my name is will squiggles here the curly brackets represents what yeah so the curly braces have a set of collection of these numbers now what’s the range for each these are the outputs please be careful when I write these down I could write them in any ordinary one it does not be quiet any sort of order of their communities students are not napping this is the mapping of the table this is is this particular relation of function the points are what’s the domain of this particular relationship Oh it’s okay so tell me how you found that do that so we give up things that you what it’s according to this point to it it’s negative to something right for a predominant I’m going to worry about too wide so much this one it’s maybe that one something and this one is fixed something right now to find the range I care about that something right yeah so what if you look to right now you’re this guy’s the y-value right down what’s the wipeout in here what so what’s my range right the guys right begin this order we got supplying them back in here I could have written this as negative what happened just so happens if I put this sticks here okay throw do I have any range value now no so the domain and range are not affected it with a function I’ve heard it over here not a function why he put six at that point adding up one doesn’t necessarily change your domain or range right now I’m looking at an empty diagram this is example Tina what’s the two main on this day right now these are my inputs over here and then

what are my outputs right you can lose the arrows as long as there’s only one power coming out find out I do this function it doesn’t achieve my rate that we just added the added negative to negative 2 then you two in my codeine already linked to my reins over to get energy to change all right let’s say we got here yeah what do you mean Oh yes sir awesome she’s a knot right in there okay little extra this was not in your notes here so just look up here I’m giving you today and I’m giving you a rule did you find the range then y is negative Wow what magic is this how’d you come up with these numbers in his head he said if you plug in negative running here you make me think it is if he wants a private at once 1 + 10 11 very impressive if you put a zero in you get 0 + 10 if you put a 2 in negative 2 plus 10 a beautiful job all right now so domain and range so far you can only reveal discrete domain meaning countable points I can see what all the domain values are now what if you’re not comfortable what if it has these are called continuous relation we’re going to be using inequality symbols to talk about donate so this example pierre at the bottom I hope everybody’s really think coolest attention I feel like I’m not everybody is so please Nick tension this I’m not in the bottom page one for example a so look at this thing now I have a points here if you are receiving the whole point of choice the Barrows points me was that mere eyes alive snaps right very nice what are my inputs into this routine well first of all is a function let’s answer that question first budget every so what’s my domain what do you think negative too much are you losing or equal to I want to say that or equal to write negative 4 & 4 does that describe it next because it give me command we’re used to thinking about it in that way well number of X values behind domain so

infinite number what am i right every one of these possible values sighs I can’t I how long would have to be there right Wasel right I couldn’t the purpose of doing this so what we do now is we have a mathematical way of doing exactly what matters like we said if anything between negative four and four including that so here’s how that works all right you want to stay between two numbers they say you have an x value and you say what’s the so you put this smile it’s like the number line right there number one has to smell over here and the big number here so you do that and then these are number lines right aren’t just wrong numbers here so negative 4 will be less than whatever you have here which will be less than 4 as well always going to be less beds knowledge number on the left bigger number on the price with your ex in the mill this set X is between negative 44 by putting these right so is negative for its four possibilities so the domain is any number between negative forty four including it now what about the range now Rach or Y values right so let’s say if I put two and then i inflate what’s my output well you’re right because no one can you see how you find your outputs okay so what about negative two what’s the output so I’m basically looking at what money order to give the tour what’s my Appa so that mean my only outlets are negative one or is it everything including a woman right I want all the numbers between negative 2 & 2 including negative 2 or 2 yeah and either wide right range of Y so this is all the numbers between negative 4 & 4 including them that’s you’re jumping and all the values and included to mix that you know makes that alright let’s try another example you know this one has an open circle when you think that when we look at inequality in the last room what did this mean it’s not an equality like this would that mean this one right here yes right now it’s a hole right if I filled in the hole what happens regular order and they don’t care this is what we call a hole in the back it means that for this input to our original output so we make a hole in the graph so from looking at right zero and I’m not doing anything right I want to be in between how do I describe that anything yeah always less than four why what it was told me is just like winning had an equality like this why do I make this and it’s bigger than one we did this last year right why right why why there is no value here before input – I

don’t have an output it’s not part of the way – does not work you don’t so how do I say don’t include it I say it’s two to two years so June is not belong in this description this description of of numbers it’s saying all the numbers between negative 2 & 2 including negative 2 but not including 2 everything in between we live in to but not to me I don’t see why I gotta for it yeah it’s always gonna be a listen there’s small numbers here so it’s gonna be small in the video okay now let’s think about range range or your y-values that’s looking up and down so what is the the values that your eyes and a yes now why were these equals I love how you thought about who are the equals or maybe not equals why did you make them people zero hi my linen here right I see it the transpired value there if you go to for sure so four is in my range for the salad for negative so brunette one for works so wow that was a lot of work that’s awesome right into things if I’m sitting exit through between you two and at the same time and why it’s better than or equal to zero at the same time it’s less it’s between in this case it also includes like that any magic question dysfunction no right every employed look it up every year but I’ve only seen one out awesome okay tough stuff here now this was a little different to the arrows so think about what are valid input cell function so what about introducing itself when you think that one that one one trip well for works what is this arrow meet so one over here bargain right how about anyone see anything no is negative one part of your domain now what’s the first thing that’s going to be part of your domain you know they look at zero and then write the domain you’re looking this way and it was zero I do have an output so it looks to me now in words this domain is X to 0 or bigger that were doing we just think about this over 2 million you go up eventually or bigger you could do it like that well that’s every infinity is not a number so this is X is a number there are big noes 0 or bigger than 0 all the way up to infinity what’s an easier way to see that this is 19 correct that is correct that’s another way so you can sign for it if you think that we got something too now think about our range now what are valid output and this is sometimes will trick your neck through your head a little bit now that what it is anymore

so you see the outputs between 0 & 1 whenever I get to want to see something right 100 so what do you think the range why regular work with and why awesome this makes sense or wake it up what does one say you could say X is between an include zero but it’s smaller than infinity is that the same as saying X is bigger than you could well let’s do this people never hear this keep it simple right over me we support people remember it okay like what are you doing our needs goes on forever and ever so it was you see in the hip in the past year on this side why do you put here right what are they gonna be zero that’d be like point one head of the way up here at five million Earth’s is there an upper limit on what to tell me there’s a lower third right that’s what this says X bigger than zero now the range I’m looking up this way I’m looking for where do I see the line when I travel this again here so one is in my range and then look there’s all kinds of stuff and this arrow says if I go all the way up to 1 million and I look over I eventually see something yeah so my rate rise bigger than 1 or equal to what if this had been an open circle a whole domain for me never did and why was not included what the weather business a cola or a filled-in whole matters let’s see another example Oh first of all is this thing a function let’s answer that question it is a function put any words down what can we say about the jumping valid input noticing I’ve got arrows here what do you think it’s great I can’t be greater than infinity but you know something yeah it’s a borders of infinity in that direction positive infinity is going to infinity in that direction so you’re saying is between whoa now is there another way to see in English what is that every number every number and we use the real number system aren’t you main are all real numbers is the number we think that they work I like the way she said it near the register yeah right yeah is there a we located what to mention we’re going if you want to go to dimension a number about opening down that’s a great question right on the y-axis we’re doing all the way up here well what is my range ready y-value now you’re looking up and down up down what do you think Ryan always less than you remember where do I go – how can I go less than or equal to which Kennedy one created to be – Wow range no

good range there’s an easier way to see that you can just say Y is less than or equal to smooth hi this is that imply that if it’s less than or equal to two I’m going down to negative infinity you say so any number you want works for the domain now the range on the other end you’re looking this way and you’re basically saying where do I sleep stuff nothing nothing until until now when I hit the lovely there’s no stopping away so why is because I can’t down so it’s a way the world so you’re making these little guys so you make them just this you’re separated this condition we’ve been doing that is listed and so once you figure out the inequality number of hecklers here this weren’t here by the Declaration and where you say I’m talking about is what is this what number system does have to be monitored this says in each box is an element of the world numbers such that any X is a real number if they didn’t see him free it’s in myself so what did the numbers this is building what else is gonna be on this set X right so business builder notation so the the things that have to be careful but if I want all the real numbers like for this one all the real numbers what’s the condition for that you say well that’s just a real number such that think about that when do I include the real numbers here all the time what numbers do we include only the ones so let’s kind of revisit this now let’s go back the domain you’re you already told me this domain you said it’s between and including negative 4 & 4 let’s see where we used to put this executive irrigation go watch it alright break down the back right let’s down on the back here first of all make your brackets make your brackets I’m starting with a little room here okay put it in brackets elements of the real numbers this such that X is a real number such that X is between a nucleus and the range on this utopia was between negative to you and to and including those values right so how do I make it set builder notation you had two curly brackets and you’re saying that working with this variable right one for the numbers life why is a real number that’s back it is between and includes negative to experience that’s all you

have to do to make sector of the notation that’s all you have to do you makes watching you make two little brackets tell me what kind of number why it’s a real number not for us when you go over that what if I said why is an element of you know just in degrees please note this amantadine beads in here don’t be white as an integer that describe two different that’s going to be negative did your darling is that casino mr. horn what’s in this step all this was everything in between so this is coming in later years so we need to understand the notation now why is a real number such that y is between an includes negative 2 & 2 that includes the fractions right those are decimal numbers all that stuff now the further I think the simplified that you might think it’s complicated this right here if you leave it out if you just say this it assumes you mean real number so this this describes the same set of numbers elements of the real why is the number one of the real special this says awesome surprising but just assume that you don’t say it’s an integer let’s do an example here again we’ve done this right we’ve done the hard part you don’t go hard court is bigger you’re the inequality that’s what you told me you said X it’s between and includes negative two but it’s less than two right and then somehow I I could I could someday this or I cannot when I said you can leave it up now this one our rage so now any number between zero and four including zero before so that was this I want to read into second organization add some brackets or – if you can say this or you can say this they are both correctly – that’s it I have to do this one are doing zero on this way this one our range over the range why bigger than or equal communication right nearly just put the glasses yeah the hard part is this coming up with these qualities right the other two are excluded in brackets okay this one that’s all they did you figured this out already I’m just putting it in the notation now this one with a little trickier I’m gonna start at the range by less than two I need to see if that y is the real number such that right now we have a little trouble it’s all real numbers so I know it’s is a real number right what’s the condition that I want it to be intersect any X works right so any X you pick from the real number what’s the condition I there’s no condition take it it’s in the stuff maybe it’s that you take a real number it’s good to be in the set that’s it that means all real numbers okay tell me that’s going this way

what is the domain doing in words first don’t worry about what equality is what’s what’s the domain here whatever value like so we’re valid inputs in between what I do great and then three now why is that because this is up close whole this is the multiple mega three numbers only now how do you write that as an inequality anyway there’s an equality please you serve the small number three X is a real number such that this condition has a history that’s doubly now what’s the range now Y values are going this way going up and down so one of the Y values here where the valid alcohol okay tell me it working right in industry and then stuff yourself yourself to where right please never nautical building all right building home we include that thing all your foliage and going to do that one and I’m gonna provide the ride looks like it’s less than or equal to negative 3 but only less than three they’re gonna want fall in this one hard part then I say okay you just put it in secular organization yeah that was gonna be equality because I guess how many say it’s between them why is it your than or equal to but it’s go forever right I have to stop at three that says it’s between those numbers now I put a greater or less than here because I can’t actually because looking a little follow the breath hold the ball right through my qp3 that’s why it’s just a beautiful year but if I go down here I can equity negative that’s X let’s look at X now 1 3 is not an input because I fall in the hole but three visited so four domains go this way for me to go that way let’s try another one oh my function because look at that look at that you them – they are existent input with more than one output the output submenu not that I can still domain so close in this way where are the inputs into this relation you see anything happening here we think any yeah I go kitchen all right so my X’s are bigger than negative 2 or maybe negative yeah that’s a native Texas so again that little hard part turning to set builder notation now what about my range look at this range up and down here this goes forever this goes forever what’s my rage okay all lovers think about look at your

range this error you got a dictionary going forever so no matter what why’d you stick control you’re going to hit something now all real numbers Emma Turner have stayed at in set-builder notation so why is the real numbers such that this means try to figure it out Oh Renee doesn’t function look how fast he did that he said look I’m laying this way maybe the date has a value I don’t anything twice that stuff snow put nothing past six pull over here I’m going to make it less than how you turn this into set-builder notation I hacked your bracket to enter and Escobar is set builder notation somebody else written range range somebody else we did right you’re looking at this aggravate right now this way look that way last example this function yes yes one remember don’t let you have your reason every input I want you only one output think about that jumping whatever value because there’s arrows here there’s arrows here remember what am i valid input all your numbers excellent right yes any accessory what is my dream now going up and down nothing nothing nothing whoops and where are all my range values that’s awesome right because nothing until I get some pre-work and then in smaller than four look at it the good negative are you gonna find stuff go to negative ten units go to- thousand with my stuff now let’s turn it into step 1 in your brackets why do you know this like a little bit what we find the top of this up tonight this little video won’t be just yet

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