well welcome back to another lesson in our flipped classroom series this is titled domain and range so let’s go ahead and get these definitions down one the set of all inputs or X values are known as the domain and to the set of all outputs or Y values are known as the range maybe that sounds familiar to you and that’s awesome you probably talked about it in algebra 1 but we’ll review it again here in algebra two so what do we need to get down is that the x values are referred to as the domain and the y values or the outputs I referred to as the range so there’s several ways to do this and the first we’re going to tackle is finding domain and range graphically so again to find the domain here’s what we want to write down we’re going to identify the X min and the xmax and then in the range we want to identify the why men and the wimax so let’s go ahead and look at this example now you probably don’t have graph paper and that’s okay if you could just make a quick sketch in your notebook like a quick x and y axes and then just sketch that in take note that there’s an open hole and there’s a closed hole okay so our goal is to first start off by identifying the X min so what does that mean exactly well I’m going to start by going to the x axis okay for x-men and I want to find the smallest value on the x axis that this graph goes to so if I follow the graph and I go to the x-men if I go out this way and i follow this graph the farthest to the left that it goes i would say is negative 3 on the x do you notice that graph doesn’t hit negative 4 out here anywhere it doesn’t it negative 5 the smallest so the X min I would say is x equals negative 3 now I want to do the same thing on the x axis but I want to go in the Y in the mix direction so if I go out to the right for a mess and again i’m only looking at numbers on the x-axis the graph has a height at one has something at two it’s got a hole there but does it ever hit three no so my xmax the farthest this graph goes to the right would be too and once you identify the xmin and xmax you have your domain how far to the left it it go and how far to the right did your grass go now there’s two ways to identify domain you can use something called interval notation or you can use something called set builder notation now doesn’t matter us which one you go with but you should be aware of both of them in case it’s written as a multiple choice question so I’ll review interval notation interval notation has you looking at these endpoints to determine if they’re going to be open or closed brackets so if I look at this x-men we set at negative 3 it’s a closed circle so that gets a bracket for a closed circle and then i’ll just put negative 3 in there comma I’ll put my max number in there and then you’ll take note if i look at this open circles that gets a parenthesis okay so maybe you need to make a side note in your notebook for yourself a bracket is very close dot and a parenthesis for an open dot now in set builder notation it’s similar but we use inequality symbols so i start with negative 3 I put an X in the middle because I’m domain and I put a two on the end here okay and it always goes an order from min to max min to max and then I just say negative 3 is less than and again because it’s a closed dot i’m going to say less than or equal to x which is less than and it’s an open dot so i’m just going to leave it as less than 2 ok but i can’t stress enough the min comes first the max comes second now for range we’re going to do the same thing with the Y values so this time I’m going to go to my y-axis this time and I’m only focusing on those numbers so I’m looking at the y axis and these are my only options for numbers and I want to see what’s the y min what’s these Louis this graph goes well if I get to this point can you go any lower than this point no the graph stops there so I look to the right on the y-axis I would say my Y min is y equals negative 5 my white max I’m just going to go up to the highest i can see on the graph and the highest looks like right here and i would say that’s y equals 4 and again once you’ve identified that you’re pretty much done with domain and range you’re just looking for the min and the max so back in interval notation I’m going to go negative 5 comma 4 and now I just need to determine if it’s a parenthesis so if i look at the negative 5 on the why i see that had the closed

dot so remember that’s the closed bracket and if i look up at four you’ll notice that there’s not a big hole okay it’s not open it’s closed if you don’t see big hole it’s it’s closed so this gets a bracket as well that in set builder i’m going to go negative 5 notice I’m going to put Y in the middle because i’m talking about range and four and then because they were both close they’re both going to get the less than or equal to symbol and again I can’t stress enough mins come first Maxim second so again I think would be helpful if you just made a quick axes and sketch quick axes and sketch and we’re going to do two more here now what I want you to do is I want you to pause it and try this one right here on your own and basically I just want to identify the xmin xmax anyway you want to write the domain y min wimax and the range and then we’ll work on this one together so pause it give it a whirl okay I tried to squeeze squeeze all my answers on here my ex min so i looked at my graph and I said the farthest to the left or the smallest number this reaches on the x-axis I’m only looking on the X is 1 and the xmax the farthest it reaches to the right would be five so i have an x-men of one x max of five both closed dots take note so i said brackets 125 or in set builder you could use the inequality symbol then for y min I said the lowest the graph goes this to a height of 1 and again i’m only using the y numbers and the highest would be five again both close so close brackets and if you set builder make sure you put the y in the middle now I wanted to do the other one together alright so X min I want you to go to this x-axis and I want you to tell me this grass ever stops notice this graph has arrows on the end so that means it keeps going and going and going and going so can you tell me where the x-men is does this graph ever stop or does it just keep slightly going out and out now now okay since it never stops the x-men we’re going to say is negative infinity it goes on forever the xmax does this graph stop moving to the right or does it slowly keep creeping out as this gets bigger so I’m going to say this is positive infinity therefore my domain is negative infinity to positive infinity and this gets parenthesis because you can actually reach infinity now if you wanted to write that in set builder notation here’s the goofy symbol you could write out the words all real numbers so go ahead and do so or the quick notation is a capital R with two vertical lines let me show you when I make two vertical lines and draw an R and that stands for all real numbers so get that notation written down for yourself okay as far as y min go so now again I’m going to the y axis does this graph have a low point well yeah it doesn’t go past this point on the y axis so if I look over to the y I would say that’s a y equals two now how about a wimax does this graph stop going up or does it go up forever okay again those arrows imply that it’s going up forever so my Y max is going to be infinity so again two ways to write the range I can say from 2 to infinity now you’ll notice to has a closed bracket because it’s a solid dot and you can never reach infinity so that’s a parenthesis or you can say the Y values are always greater than or equal to 2 okay now you’ll notice this left last graph that we tackle here is a little different its kind of split up into two sections there’s a piece here and then it breaks apart and there’s a piece here so how would I describe this graph is I would say that it is discontinuous okay so it’s not one graph all connected it has a break in it therefore it’s discontinuous if you were to try to drive on it so to speak you would drive up this way and then you’d have to jump off and basically drive onto this section it’s not one continuous flowing pattern so since it’s discontinuous you need several pieces to write the domain and range basically one for each section so I’m going to start with just this corner here because like I said we need several pieces so in that section I need an X min and in x max ok so again I’m looking strictly at the x-axis does it ever stop going to the mins on the x-axis or does it go forever again that arrow implies that it’s going forever so I’m going to say that’s towards negative infinity now the wine arms are the xmax as I go this way in just this section okay you’ll notice that the graph doesn’t go keep going forever this way it goes up forever but it doesn’t hit

this line here and there’s something called an asymptote there and I’m going to slightly draw it in there’s this magic line called an asymptote and we’ll talk more about that later in the year you definitely talked about it in algebra 1 when you did exponential functions but that might have been a long time ago now you’ll notice the scale is counting by twos right two four six eight so this is getting close to two but it’s not touching it so I’m going to say my xmax is too even though it doesn’t touch it ok now I’m going to do the same thing for this section here i’m going to write a separate X min and a separate xmax ok so if I look at this graph again be on the x-axis only it looks like it gets close my bad that should have been a negative two it looks like it gets close to negative 2 but it doesn’t touch it because of that asymptote and if I look for my xmax that arrows implying it goes forever so I’m going to say my ex min is negative 2 in my xmax is positive infinity so since I had two sections when I go to write the domain I have to break it up into two pieces and i’ll show you how I’m going to say from negative infinity to negative 2 again I can’t reach infinity so I got a parenthesis and I didn’t actually hit negative 2 because of the asymptotes that gets a parenthesis and I’m going to union Unite that with the interval from negative 2 to positive infinity so when there’s two sections don’t forget you need two pieces to your domain likewise for range when there’s two sections you’re going to need two sections to your range so let’s go ahead it’s the same graph and let’s just look at the range now ok so I’m going to look at both sections I’m going to start with this section that’s to the left first I need my Y min my wimax ok so now I’m looking at the y axis how high and how low does this graph go well again you’re going to see like this asymptotic line in here that it looks like they’re both getting close to something to draw it in with my red pen and I’m going to say it looks like they’re getting close right in the middle there but they’re not touching it so I would say the lowest this graph goes is close to a positive 3 again they’re counting by twos and if I follow it up it goes up forever to infinity when I look at the other section I need again a while oops min and a wine max I look at this section again on the wise it goes down forever so my Y min is negative infinity and it doesn’t go up forever it only gets close to this asymptotic line which was at three so when I write my range i’m going to say from 3 to infinity Union negative infinity to 3 so moving on from graphing they can just give you different ways to look at functions and you can just identify the domain and range in a roster so remember the X comes first wykeham second domain is your X and ranges your why so if they ask for the domain of this mapping they’re really just asking you to list out the X values so I would say my domain are my X values so 2 3 4 and 5 and i would say my range are just my Y values so 8 9 10 and 11 okay so I’m just identifying the X and the y again in question 2 same idea I want to identify the domain so I’m just going through and saying okay domain is my input or my X values so I am just listing out the X values so 25 8 and 9 and then my range of course is my Y values and i’m just going to list them out in roster form 576 and 2 now if you want to put them a numerical order that’s fine as well our last piece is to identify the domain and range given in equation so let’s go ahead and title the entire notebook so it’s easy to find and how do you do that well the first thing I do is I ask myself if there are any restrictions once I come up with an answer to that I’ll write my domain and range and then of course I want to make sure I verify graphically you can’t get it wrong with the graphing calculator will just type it in and compare so look at three quick examples number one y equals 5x plus 3 hopefully you’re familiar with this equation this is just the equation of a line okay has a slope of five and a y-intercept of three so I asked myself that question are there any restrictions is there any number you are not allowed to plug into x I would say no I can plug

in a zero if I want a to if I want a hundred thousand if I want a negative five if i want i can plug anything I want into x so what that tells me is the domain because I can plug in anything I want the domain is from negative infinity to infinity I’m using interval notation or if I you set builder I can say all real numbers or you could write that out all real numbers okay now my second step like we said was to verify with a graph so let’s grab our calculator and get a picture of this graph so I hit my y equals on my calculator and typed in the 5x plus 3 and this is what I got okay and again it’s a line like we predicted and if i look at my ex min okay if i go to the x axis this graph just keeps going forever and ever and ever so I would say negative infinity and this graph goes forever and ever and ever so again positive infinity so my domain is all real numbers okay let’s take a look at one word they’re probably is a restriction number 2 y equals the square root of 3 minus X so I get I’m asking myself is there any number that I cannot plug in so thank square roots in your head do you know the square root of every number in the world well even though some of them aren’t noise you may know a lot of them for example if i plug in x equals 10 I would get y equals the square root of 30-9 so the square root of 21 now even though I don’t know what that is off the top of my head I can still do it on my calculator i get some random decimal but there are numbers you can’t take the square root of can you think of them can you take the square root of negative 4 know if you try it on your calculator it’s going to give you an error you cannot take the square root of negative numbers Chrysler is a restriction when you have radicals so what does that tell you to do well basically that means whatever is under here and let’s make a note must be positive because I cannot take the square root of a negative number so if it has to be positive basically i’m going to say whatever’s under there has to be something to 0 if you’re a positive number what are you compared to 0 why would say your greater than or equal to you can equal 0 because the square root of 0 is 0 but you can’t be negative and now all you have to do is nicely solve this equation so I’m going to add 9 to both sides so 3 X has to be greater than or equal to 9 and then i’ll divide by 3 so X has to be greater than or equal to 3 and that’s my domain X is greater than or equal to 3 now notice that’s in set builder notation if I wanted that in interval notation i would say three is the min because the X numbers bigger so I would go from 3 to infinity it’s equal to 3 because that equal sign and I can never touch infinity so it gets the parenthesis okay so anytime you see a radical you have a restriction I just wanted to take a quick moment to verify it graphically you can always just type it in the calculator and you’ll see that the smallest x value right here is 3 and that’s exactly what we said and then the X values are all bigger than because that goes on forever so again if it’s multiple choice i would type it in my calculator and just look at it all right I’ve got one last one for you and it’s probably the ugliest one so again i’ve got y equals x minus 6 over x plus 3 and i’m going to ask myself are there any restrictions is there any number i cannot plug in well when you have a fraction there is one rule there is one number in the world you cannot divide by let’s get this down right now you cannot divide by you’ve probably set it by now zero okay again if you try to divide by zero if i type in 5/0 on my calculator it says error divided by 0 you’re not allowed to it doesn’t happen it’s impossible okay and what that does is it makes that fraction undefined when you try to divide by zero so when you have a fraction you don’t really care what the numerator says so to speak at the moment all we care about is the bottom I cannot let this number on the bottom equal zero off the topic red could you tell me what x value makes that denominator equal to zero good negative three X cannot equal negative 3 now if you couldn’t come up with that it’s simple we’ll just say X plus 3 cannot equal 0 this denominator can equal zero that’s the only rule so i’m going to subtract my 3 over and X cannot equal negative 3 ok so my domain I’m allowed to plug every number in the world I want except negative 3 how do you write that out well there’s two ways you could say all real numbers except x equals negative 3 so that’s one way to

say it or you could say from negative infinity to negative 3 Union negative 3 to positive infinity now notice even though I use negative 3 twice I don’t have a bracket around it so it’s never equal to so two ways to say the same thing and again my last step is going to be to check graphically so here’s a quick graph from my calculator you’ll notice it’s in two pieces that’s why it makes sense that I have these two intervals one for this section and one for this section and you can kind of see that it looks like they’re getting close to the same point but not touching so it has what we call that asymptote in there and it happens at negative 3 so this can be all real numbers except negative 3 and lastly let’s go ahead and make a table all i did is after i graphed it I hit the table future and you’ll notice and you can circle this in your notebook there at negative 3 what do you see an error that tells me all the other numbers work except negative 3 and again that’s where that asymptote is well we look forward to seeing you tomorrow have a great night