today's video is on section one two domain and range the topics for this lesson are interval notation and set builder notation this is going to be the way in which we're able to describe the domain in any particular problem and in order to be able to solve these problems we'll need to be able to handle linear inequality so that's one of the things we'll need to do and then we're going to define the domain and range in more detail recall from section one one that the domain were the valid inputs to a function and the range were the kinds of things that came out the possible y values associated with a function and how we see domain and range when we're given a graph and then lastly how to read piecewise function notation which is a particular type of function we'll see so first off inequalities and interval notation and inequality notation would be something like you see here that 2 is less than x is less than or equal to five this is actually called a compound inequality because it tells you something about what happens above and below X if we use set builder notation uh you'll see these braces we actually read this as the set of all X such that 2 is less than x is less than or equal to five so that's kind of how you read the braces and then integral notation which is what you'll see most commonly in all the homework assignments you put a parenthesis for things that are less than or greater than so they don't include that number and you put a bracket when you actually include that number one you have a list that are equal to or greater than or equal to and then there's a picture of what it looks like on a number line so those are all the different ways we express an inequality so what kinds of skills do we want to have we want to be able to know how to make a number line graph so if I'm going to do something like this and when you do the online homework the graph will be there for you and you'll be kind of clicking on it so let's do negative two negative one zero one two three four five six seven eight nine and it says go ahead and include negative one so we're going to put a closed dot there but don't include eight so we'll put an open Dot there and then let's take everything in between so that would be how we would graph that um and now if we want to do the next thing so the u means the union meaning you're gonna have both those things in the picture so that would be something like I'm going to use blue for this one we're going to go from negative to to and then we're gonna include both of those and include all the stuff between negative two and two so that's what that part is and then we are going to go from three but not include three so we're going to open circle at three and then open circle at eight and then we'll include everything between three and eight so that would be that whole blue thing so both pieces put together would be how you would graph that sorry and again if we're practicing with interval notation something like this then what we're going to do is have a parenthesis because it's less than all the way up to four so we put a bracket and here where we have two pieces put together we're going to say include two and don't implant five or we're going to have this whole other set anything bigger than eight so anything from eight to Infinity so that would be interval notation in that cell now we do have some integrals you just saw it in the example where it was unbounded maybe it went on forever and ever and ever so we use infinity negative Infinity describes all the things less than that number going on forever and positive Infinity includes all the numbers bigger than that number um beyond that area so you can see the example here at the Top If I'm dealing with this function square root of x then square roots are defined as long as the things you plug in are zero or greater so they have to be X greater than or equal to zero we're not going to take square roots of negative numbers in this context and so we would write that as zero include zero all the way up to Infinity so we have two examples here what would the answer be if I want X greater than or equal to 3. so because it has to include three I'm going to pick something that has a bracket on three and all this stuff greater than or equal to anything but negative one that's what this one says anything but negative one oops highlight that you can cross it out anything except negative one actually we're going to describe that like this all the things less than negative one or greater than negative one and that way we're just left out negative so what I want you to do is pause the video and try problems one and two on your worksheet they should follow pretty closely from what we just did so linear inequalities these are what we're going to have to solve basically in order to find the domain and our goal so our process is to isolate the variables so we're trying to get like the X by itself y by itself T by itself usually an X so whenever we say isolate something we want to get it alone on one side and you do have to be careful about the math you do you can add or subtract whatever you want from both sides and you can multiply or divide both sides by a positive number and then you can combine like terms on the side that would be simplifying the side or like distribute around parentheses none of those things will change the inequality however if you multiply or divide by a negative number the inequality actual switches so if I have X is less than three then Negative X is greater than negative 3. that's what and when I multiply by a negative it actually changes the direction and you can't just switch left and right sides because that would change the inequality as well so here are a couple that we can walk through and do and remember isolate the variable combine my terms and be careful if you want to swap or multiply or add and multiply or divide by Midwest so for this first one I'm going let's start by Distributing so I'm going to get Negative 66 minus 2W equals negative 55 minus 7w equals less than or equal to and so I'm distributed and now I'm going to have to try and add subtract stuff from both sides with my goal ultimately of isolating my variable so I'm going to start by adding 7w to both sides foreign so I get 5w is less than or equal to 11. and the Finish would be dividing both sides by 5. so w is less than or equal to 11 fifths and we're going to write that as anything up to 11 fifths but we're going to need a bracket there because it's less than or equal to next one this one I'm going to do on a blank paper so negative 6y plus 18 is greater than 48. okay this is 5 plus 18. so I'm going to start off by subtracting 18 from both sides adding and subtracting doesn't change our inequality now I want to divide by that negative 6 but now I have to be careful that's going to actually change the inequality and I get y less than negative 5. so this is everything up to negative five did not included and our last one right here 12 plus 4z and 9 plus 13 Z two nine that's 13z subtract 4z from both sides and 12 is greater than or equal to nine plus nine z subtract nine from both sides and now we want to divide by nine thankfully it's positive so that's not going to change our Direction and here we want all the Z's that are less than or equal to one-third so we want to include one third and we want to include all the stuff less than or equal to so those are examples of what we call linear inequalities now you might want to pause the video and try worksheet three worksheet problem three what is then the domain and range just as a reminder the set of all legitimate inputs is your domain and then the outputs that it can occur remember we already talked about it at times when we don't have a certain output those would be the range we talked about this pet grooming problem before CMP represents the cost of grooming a p pound dog what is a reasonable domain name rate so first of all um in terms of a reasonable domain I'm gonna probably say it starts off at zero I'm not gonna have a zero pound dog but there's probably some really small dogs and I don't know exactly how small they go so leaving an open circle at zero says anything bigger than zero Could Happen teeny tiny dogs but I definitely know there's no negative pound dogs and I don't know what's the heaviest dogs can be so it's somewhere in this range of something less than infinity probably you could Google what's the biggest dog there's ever been and you could make that the top of your way the top of your double now what is the range the range is the cost my guess is that we are never seeing costs that are zero dollars like come in and we're doing this for free um and so the range is never going to be negative it's not going to be less pain to being a groom your dog um so for zero and again we're going to use like zero timpani but there's probably a cat maybe the place says Max a hundred dollars to groom any kind of dog and so that would be the top of the range here's another example the average height of an American male in age a years what is the reasonable domain again the age shouldn't be negative and you have to be born and then Ella and then the average height um it probably could go on forever but let's just say you know it didn't say what may measurement let's do like in inches and so let's make worst is a 10 foot tall but that's probably Way Too Tall could be your uh range show your range is anywhere from zero to 120 and then your domain is anywhere from zero to I don't know if there's been like 150 year old guy but those are the possible ages so something in that range and again the more information we have if we googled something we could get Tighter and Tighter fits usually we'll have a graph or no specific function and that'll help us out when you are trying to think about what the domain is you're looking at where does it start and finish with respect to the x-axis so in this case we would say the domain it doesn't include negative two because of that open circle and it does include three so that would be the domain the graph is between that zone and when you're looking for the range you're going to look at the bottom and the Top Line and so in this case the range thank you appears to be going from negative one it does actually hit negative one at one point all the way up to eight and it does actually hit eight so we're going to put brackets on there so what are some common functions and things we need to know about the debate if I have a nice polynomial function like you see here I can plug in any X I want so the domain is all real numbers let me write that as from negative Infinity to Infinity if I have a rational function the one thing I'm concerned about is could the denominator be zero so when you have a rational function you're just looking at a need to avoid anything that's going to make the denominator zero lastly when you're looking at a square root you can't take square roots of negative numbers so you're trying to look at whatever is inside has to be greater than or equal to zero so these are your first I would have a catalog in my notes of functions and key things to know and so these types of functions is what you need to know about the domain so we have an example here we have this function which has a root in it so our rule is that what's inside has to be greater than or equal to zero so X has to be greater than or equal to 3 and that would be that one there and our rule for rational functions is that our denominator can't be zero so X can't be negative ones and so that will be RW here are some other functions following the same rules the denominator can't be zero and so this is all numbers up to zero and all numbers after zero and the only thing we do is not includes here square root we want what's inside to be greater than or equal to zero and this is a linear function it's also a polynomial doesn't have a need denominator or square roots or anything else right so this is remember all real numbers which we're going to use minus infinity to Infinity foreign how do I find the domain take the bottom and I set it equal to zero so two X plus one equals zero when X is minus one half so I want any number but minus one half square roots I want when that inside is greater than or equal to zero so X is greater than or equal to two-thirds now it gives another strategy here so I just saw this as an inequality the suggestion here is to figure out what the solution is to that equation and then to ask yourself which side is it is it bigger or less than zero that meets the criteria so it says plug in something so we're going to do a test value like zero and we're going to say the square root of 3 times 0 minus 2 and that equals the square root of negative two ah doesn't exist so the solution must be the other side of the function so that's another way you could approach it I kind of like just solving the inequalities since we practiced those earlier in the lesson but if you're kind of visual you might want to try that method uh now you might want to pause the video and try a worksheet problem four on your own if I have a piecewise function you'll see a definition that looks something like this what this means is we're actually going to use this only on one piece of the function and we're actually going to use this only on another piece of the function you don't use them both at the same time so here's how we approach this if I'm asked F at negative 2 your job is to figure out which P supplies is negative 2 less than zero or greater than it's less than zero so I'm going to do negative 2 minus 4 which is negative 6. so depending upon which piece applies to it that's how you decide where to plug it in do not plug it into both pieces plug it into the appropriate piece so let's look at one one is greater than or equal to zero so I'm going to plug one into that second function 1 squared is one and lastly when I go to devaluate F of 4 4 is on this piece of the function so I do 4 squared which is 16. so P slice functions you're going to follow different rules x minus 4 x squared depending upon the value of the function so here's one for us to practice and then we'll see how to graph it once we do some of the evaluation so negative 1 up negative one is not on that piece negative one is not on that piece so this is not in the domain that means they're not going to be graphing even on that part of the graph zero goes with this first part so I'm going to do 0 squared which is zero so at zero make sure we have the point zero zero okay one one again Falls in this first piece so I'm going to do one square which is one two actually is this break point and I'm actually going to fill in the circle using the second piece so what I'm going to use is I'm going to use the first piece to figure out where to draw my open circle so F of 2 is using the second piece so I'm going to do one half 2 minus one which is one minus one which is zero so two one is where that closed Circle goes now it's not F of 2 but if I plug 2 into the top function I get four and so that actually represents the open circle where that piece ends but it's not the function value it's not one of the closed circles three is greater than or equal to two so we're going to use this part of the function this is three halves minus one which is one half so at three oops yes positive one half straight down and so that piece of the function actually looks like that so that's what that piecewise function is piecewise functions here's an example with Uber ride any distance up to two miles might cost you six bucks you can tell I've never taken an Uber before but then there's an additional mile cost two dollars per mile after that so we get this different sort of function um so if you travel one hour that's in the flat six Buck range because that's in that first part if I travel three hours then I'm in this range of the function so this would be a twelve dollar Uber ride and if I traveled five miles this would again be in this part of the function so that would be a sixteen dollar Uber ride if for example I did C of two I'd still pay the six dollars so for that to us absolute value which we'll cover more into five is a very specific piecewise function it is sometimes talked about as it turns everything positive really absolute value is measuring like your distance from zero giving that as a positive value so the at negative one you're one step away from zero so the absolute value of negative one is one absolute value of negative two is two Etc and but the absolute value of one is one and the absolute value of two is two and so we get this V shape graph so that is a really common piecewise function that you'll see more of later that's it for one two go ahead and try the rest of the worksheet