so we've just talked about lines we're going to be a little bit more General today we're going to talk about section 0.2 we're going to talk about functions now here's a deal anytime you have such as uh one variable depending on another one such as like y depends on X for for most of our functions where Y is dictated by what you you put in for x uh some independent variable X and some dependent variable y we can call that a function the only thing that you really need to have for a function is that every input has one output not two outputs other otherwise you don't have a function because you plug in a number and you don't would know where to go so when we say a function we mean some expression where each input determines exactly one unique output usually for inputs we we say those are like X's so typically that's uh an X so some expression where each input X has exactly one unique unique means doesn't happen again one unique output and we typically call those Y's for or f ofx um for for us you know we can represent functions a lot of ways though we can represent with tables graphs formulas uh one one time I was fishing and I caught four fish and because I'm a math dork I I made a table table of it because you don't do that is that not [Music] WR anyway uh here's one example of of a function very easy function here's my fishes I think that's that's fish uh fish caught and the number of pounds that they were so my first fish second fish third fish and I caught four that day first one was 3.2 lb not not bad next was 1.4 then 2.8 and then I caught a massive bass for 7 .3 that's a good day good day good day threw them all back them all back firstly let's define what the the inputs and outputs are for this particular well this would we're going to see if it's a function in just a second firstly my my inputs are well the the fish that I caught in this instance the outputs would be well their weights when I weighed them because the scale would be like my function right I put my first fish on there it' give me a weight put my second fish on there it' give me a weight uh now the question question is is it a function is it a function does each input does each fish have one specific weight yep yeah okay uh you know what I need to erase something right here one exactly one not unique but one output uh unique output would be a one to one function we're not there yet so when we say it has one output I mean that that I don't have fish number one weighed 3.2 lb and 4.7 lb cuz you'd say how much did your first fish weigh I'd say 3.8 or 4 3.2 or 4.7 would that make sense to you well no seriously how much your fish weigh uh 3.2 or 4.7 and you like that doesn't make any sense you're giving me two weights for the same fish does that make sense to you would that make sense if if you asked me that question how much did your first fish weigh and I said oh well either 3.2 or 4.7 he said aren't we talking about your first fish yes we are 3.2 4.7 you're like well if I'm having dinner with you tonight I need to know if I'm going to be be hungry of 3.2 lb or if I'm going to be satisfied 4.7 I go 3.2 4.7 it doesn't make any sense right you have to get I have to give you a specific weight for that one fish that's what a function does it says if you say fish number one you're talking about one specific weight okay uh maybe I should say the word specific output not not unique as one unique output we will be talking about one to one in just a little bit how about number two does number two give me give me out just one weight yeah it doesn't say 1.4 and then something else over here that would not be a function so this thing is a function any any fish that I have it has one specific weight that we're talking about if I did this would it still be a function yeah V yeah yeah it would be a function this we this fish four weighed 3.2 lbs not something else it doesn't matter that these things are the same that can happen uh let me give you for instance as far as a graph goes that right there has the same output at several different spots right that that would be some sort of an X squ it says if I'm at this point or this point I still have the same exact output that that's okay this is still a function we're going to talk about vertical line test too it's a function um it wouldn't be a one to one function it wouldn't pass a horizontal line test but it would be a function we can also talk about functions really we don't see fish caught normally we see some sort of function like this especially if you're doing any mathematical modeling you might see this where you have set of inputs our X's you got a set of output outputs that thing would be a function every one of our inputs or X's has one output how you would say that it wouldn't be a function if you came back and did something like this would that be a function no no if your inputs are repeated with different outputs well then you don't have a function there so that would be a no bu no sometimes we actually have formulas too that are are functions uh this is a function what is that by the way sure and it's a function because if you give me a radius it's going to give me out one area isn't it so our area depends on the radius you have for your circle it's not like I say you have a radius of three what's your area and you give me two different answers that wouldn't make any sense it wouldn't be a function the formula would fail also we can have some graphs graphically if we did something like that we can have we can represent them lots of different ways basic tables here formulas which we spend most of our time in formulas and graphs that's another way we represent functions um just one one note functions have to have only one output for each input that's that's the key thing I I hopefully you got that from from this now one more thing about this one let's say say we change it just a bit and we say y = FX could you find F of0 what would that mean to do I mean if Y is f ofx it says Y is a function of X and I'm asking for f of Z can you tell me what is f of0 in this case Okay is how much two because it says you go over to the input of zero you look up the output for that particular input so here it says find your input remember this is f ofx right so go to your X go to zero tangle with output is oh it's two how about F of three what's F of three everybody that wasn't everybody I'll take it fine whatever yeah it just says you go to your input of three you find out what that output was typically we'll use this in this type of situation where you have some sort of equation Y = 3x^ 2 - 4x + 2 the Y equals thing isn't always the best for us to represent a function the reason is it's because in this class we're going to look at a lot of different functions at once and you want to be able to distinguish between them if I have just y equals a function then y equals another function y equals another function I say look at the function y you're going to be like there there's three of them which one you talking about we often use this type of notation to distinguish between them so if I said instead of yals f sorry y equals that function I want F ofx or G of X or H ofx that way we can dis distinguish between those graphs and those formulas um and these equations also What it lets us do is if I ask you to plug in a number it will tell you inherently what number you plugged in for instance if I say uh for you here can you find me F of zero well what does f of zero do what's that supposed to do for you I'm supposed to plug in okay you're supposed to plug in zero and find out what the output is can you plug in zero here yeah two okay so you said two okay so you plug in zero zero yeah you get the two what's nice about this is if you plug in two from this one well you're going to get yals 2 but does this tell you what you plugged in to get the two no no does this this will tell you what you plugged in get the two yeah yeah this actually will give you a a coordinate point it will say you plugged in zero you got out two and that's kind of nice this is one other reason why we use that function notation let's go back to those graphs too uh can you tell what is a function just by looking at the graph so for instance for instance can you tell me whether these things are functions or not just by looking at them we know we can tell with tables right because if we have an input repeated with a different output that says it right there we're going to be able to tell formulaically in just a little bit but right here just by graphically what what's that thing called where you test a line to see whether or not it is a a function or not veral line yeah we have a vertical line test imagine bless you wow that was a powerful one it was like a sneeze grenade going off um sneeze grenade that would be so gross so disgusting that's where my mind is right now okay so if you imagine every vertical line it's supposed to touch your graph at only one spot if it touches it all so is this thing a function yeah for sure every vertical line hits this diagonal only one spot so yeah this is a function how about that one is that a function yeah parab are basically are basic functions every vertical line hits this is it a one: one function do you know that no one to one function would be the horizontal line test saying that every input has one unique output it says it doesn't happen again this is not one to one but it is certainly a function how about this one is this a function yeah sure now what about this is interesting case what about this one is this a function no does every vertical line hit the graph at at most once yeah yeah now a couple people get hung up because wait a second don't you have to have something at this point zero and the answer is no not every input has to have an output but if it does it only happens once do you see the difference there this doesn't have an output it's undefined zero this would be like 1 /x uh but if it is defined then that definition has to be one exact point to be a function so yeah this is still a function how about this one no this fails it because if you plug in this point you actually get one two three points out we can't deal with that so this is not a function so vertical line test verbally I'm not going to write it down because I know youall you all know it says that you imagine every possible vertical line uh that vertical lines have to touch every point of that graph at at most one spot so touch the graph at at most one spot it can't ever cross over more than one spot through how many people feel okay with our very quick introduction to function so far you having fun yet enjoy enjoy Joy well let's consider one more thing what is that Circle say louder circle circle did you all know it was a circle did you read the section on circles I I told you I said read circles right did you read circles hopefully you read Circle that's a circle what's it centered at center of the origin very good z0 the way you shi circles around hopefully remember this from your intermediate algebra days is you have some parentheses in here like x - h and y + K that would that would shift that around okay uh what's your radius good because we know this is the the radius squared so our radius would be five is it a function is it a function why not well yeah I mean visually we know it's a circle right that's circular reasoning isn't it get it thanks where's my drums um it's a circle so it's not going to pass the vertical line test that's one way we Define that if we actually graph this it's a center 0 0 radius of five it looks like this it's certainly not going to pass that vertical line test however can you see it formulaically as well specifically can you solve this for y and see that this is not a function let's try that how would you solve this for y your first okay so probably isolate the Y get the x^2 over there somehow you know you're going to get y^2 = 25 - X true now Y is not completely isolated what would we have to do to get y all by itself let's do that so if we take a square root of course that means both sides that's legal to do on the left hand side we get y on the right hand side tell me what I'm forgetting right here or oh yeah every time you take a square root of something you got to have a plus and minus so if the square Root's on your paper no big deal but if if it's not there and you put it on your paper like we did up here right we didn't start with that square root we in uh we introduced it to the problem when you do that you absolutely must have a plus or minus do you see the situation now I want you to try to plug in a number and tell me how many answers you get out how many outputs you get out how many you're going to get yeah because if you plug in something like I don't know four you're going to get 25 - 16 right right you're going to get N9 square of n n three but then you're going to take plus three and minus three that's giving you those two those two answers as soon as you have that out of a formula it's not a function now the question is could you work with it to look at parts of this function and that answer is yeah absolutely if we Define this a little bit differently if we say well let's call F ofx theun of 25 - x^2 and G of X the ne < TK of 25 - x^2 now let me say a question is this a function yeah sure is this a function yeah together they're not a function but separately well we could talk about each piece that'd be fine what would this be the top half or the bottom half of a circle top top half bottom half then we could talk about them uh but Al together when we look at that thing certainly we don't have a function there uh the other types of functions we need to talk about one of them is called peace wise functions before we go on to that are there any questions on for of the line test or what we're doing over here kind of just showing that all formulas aren't functions I mean we we don't have that necessity and when we solve them though we can talk about pieces of them as functions are you guys all with me on this so far you ready to talk about peace wise functions sure you sure sure all right here's what a peace wise function basically have you guys seen a peace wise function before okay I I know you have you supposed to have seen it is to be in this class you've seen it before uh one very basic PE wise function we're going to deal with for just a second uh the idea is though with PE wise in general that the formula depends on the value of x so the formula for the function depends on what value you're trying to plug in so PE wise functions work where the function changes depending on the value of x your input move over here some room the most simple one I can think of and this is really a really simple one that people introduce peace wise functions with it's one you deal with man you've been dealing with this probably since like seventh grade six seventh grade it's the absolute value function what's the the symbol for absolute value what do you do with that bars yeah those vertical lines okay so if I say the absolute value of x what is absolute value do the distance away from zero yeah that's right and and some other people you said what's it do like more applicably what do you do with that if I put a number in there I say uh absolute value of five what's absolute value of five okay and I say absolute value of -12 and you tell me it's 12 why it's a distance from zero we're counting over what does it do what does it do okay so make everything be more specific you actually have to say this in two parts right because one of them's already positive so what what does it do if the if the number is positive does it change it no okay if the number is negative does it change it so you tell me that this function does two different things depending on what value X is that's really what you're telling me right if x is positive I leave the number alone if x is negative well I change that sign somehow does that make sense so really we can Define this as a peie wise function if we say okay uh f ofx is absolute value of X2 it's really hard to think about that as as far as a graph goes I mean you might know what the graph looks like but how would you say you might have memorized that do you know what the the graph of absolute value of x looks like looks like a V yeah show me with your hands how it looks don't throw up gang signs that's I'm just kidding no it's yeah it's it's a v now why how can you get that from this you can't you have to either plug in numbers and figure it out or you have to Define it piecewise here's how a piecewise definition looks how that funny bracket that says all this stuff goes together and then we have to Define it on a piece by piece basis here's what the absolute value does it says you're going to do something if x is less than or sorry greater than or equal to zero you're going to do something else if x is less than Z some people Define it as strictly greater than strictly less than and then when x equals z itself we're going to Define it like this what do you do if X is bigger than zero do you have to change it at all now like this five right you didn't have to change the five you just pretty much dropped the absolute value so we leave the X alone if it's bigger than zero what do you do if x is less than zero you say what now X okay so we change a sign he said Negative X that's the when we could change it right if I did this and said how do I get from -12 to 12 what math did you do magic I I did I'm Harry Potter this my my wand I did Magic math negative gone right there all right Voldemort I'll show you who boss too much what this really says is you have to find some math way to change a sign the only math way we have to change a sign is either multiply by a negative or divide by a negative so absolute value of -2 really says follow this formula and says you take the negative of -12 does that give you back pos2 that's the peace wise definition right there that's it and works it works for anything you can follow these directions for your P wise function it will tell you which part to use now of course we know this one from a long time ago but can you see that this is the definition of that this would do it every time what's kind of cool is that any piece-wise function can be graphed by using their pieces so we're going to do that next you can graph any pie wise function by graphing each piece individually the only thing you have to be concerned about is that you use the appropriate range that's really it so we're going to tack on just a little bit you can graph each uh you can graph piecewise functions by graphing each piece individually but you you have to do it for the given range so we're not going to graph the whole line of f xal x we're going to just going to do it for for a little bit of it I'm going use that word domain let's give that a try so here's what we do with graphing pwi functions I'll write this out in a little bit for you when we get to a more advanced example uh but for right now basically what you do you ignore one of these functions one of these pieces you graph this whole thing and you erase it for the parts that it doesn't actually exist so what right now I want you to think of the the line FX = X how's that look what does f ofx x look like sure you can do it with slope uh intercept form what's the intercept this is in MX plus b form yes okay the plus b well that's zero so we know it's Crossing at at the origin what's the slope of that so it means it's going up one over one so if I were to graph this whole thing this right here is FX = X agreed the problem is is this right right now the whole thing where does it actually exist and the directions will tell you that where does it exist to the right of the y or to the left of the Y right don't all speak at once to the right of the y or left of the Y come on you got to be participating yeah that's because we're looking for the X's that are positive these X's are negative it's saying it doesn't exist over here so we'd erase that part that doesn't even make sense so right now we know this is the X where X is bigger than Z bigger than or equal to zero that's why we have this closed Circle there because of the equality that includes that little piece since we've graphed that piece already let's go down to the next piece negx just takes that makes a slope the different way same intercept since we know this already has the piece of a graph we can't put anything else otherwise it won't be a function we're going to leave it just like that that's where you get your your V from this is the f of x = x are you ready to try something just a little bit more advanced can we do that okay you guys have any questions on the absolute value you've all seen that before yes good let's see if we can can sketch something a little bit funner is funner a word I'm a math teacher it's this is funner I guess I should know words and stuff that's better are you ready we're going to graph this piece by piece now here's a little hint for you what you want to do break up the the the domain first your x-axis first and the appropriate ranges graph the pieces that'll work out for you so what I'm looking at first is where this starts and stops I'm looking at the key intervals here the key intervals are what's what's one point that I'm going to have on my graph where's the X start and stop Z zero is not up there don't care about the zero I care about the negative one because that's where we're going to trade off between one piece and and the next piece do you get what I'm saying I want to I want you to find the places where you're switching between fun one place is at NE one where's the other place sure here's what our directions say PE wise functions have directions it says for a certain range that's less than or equal to Nega 1 so everything over here I'm going to be doing something between these two numbers I'm going to be doing something else after this number I'm going to be doing something else that's what how PE wise functions work nod your head if you're okay with that now we just grab the pieces making sure we don't overlap these functions or these these inters what happens when the X's are less than or equal to -1 what are we doing for this range so we've already broken it up we're going to go piece by piece we're looking at this piece right now which of the directions has to do with this piece of information do is this this piece no this is bigger than one that's that's that's the wrong way is that this piece this is between negative 1 and one so we've got to be talking about this piece and if you look at it says X is less than or equal 1 so we know we're we're this way what do we have to graph for this this piece right here Z zero wow what's zero mean what's zero mean horizontal line good where y0 this which one this do you know have a lost you yis have lost you gu have a lost you hor like that it would have to be horizontal listen if if the the whole grouping of this kind of confuses you just write them out differently say that you have y = 0 for a certain bit say you have y = < TK 1 - x^2 for a certain bit say you have Y = X or a certain bit it's the same thing you're just grouping all these together in function and graphing them piece by piece do you see that okay so this is the piece that's working where your X is less 1 or equal to it this is the piece that's working when you're between1 and 1 and this is the piece that's working when you're greater than or equal to one split it up if you have to but you need to be able to graph each of those functions so can you all tell me now what does yal 0 look like the XIs that is the xaxis yeah that's a horizontal line at yal 0 y equals a constant right we talked about that last time it's a horizontal line so we're talking about this right there uh one question I have for you should I have an open circle or a closed Circle here and why CL Circle and Y good good very good okay check got let's go to the next piece the next piece Works between -1 and one now do do you also understand why why we might have to have no equal sign here and why if I did this it would not be a function be overlapping if these were were different yeah that's right so you're you're never going to see equals equals you're always for the same value you're not going to see that so let's go and see what what is that oh my gosh what is that do you recognize it we had it before on the board except the numbers were a little bit different what is that a circle if you squared both sides you get y^ squ over there wouldn't you if you added the X squ you get x^2 + y^2 = 1 that's a circle it's centered where do you think origin origin somebody else tell me what's the radius one mhm that's one guys if you have questions now and that would be a good time are you okay that this is a circle how many people feel okay that that's a circle you can see it if you can't see it don't reach your hand can't see it don't raise your hand can you see it not so much if you can't see it if you're like oh my gosh what in the world is that square both sides you get y^2 = 1 - x^2 if you add the X2 you get x^2 + y^2 = 1 now this isn't we cheated a little bit we cheated because this is not an entire circle an entire circle would be be that you clear that would give the top half and the bottom half what are we talking about the top half or the bottom half which one top we're just the top half right now so we want the top half of a circle the top half of a circle centered at 0 0 with a radius of one well that looks to me like it's going from here to there it's going to have an open circle right here but it's been closed in by the previous function that's kind of cool right we don't have an over leing point is it going to have an open circle or closed Circle here open yeah again why why is it open it's not equal mhm doesn't have that little that little equals so we've got our first piece we got y equals 0 we got our second piece the top half of the Circle Center 0 0 radius one the last piece oh the last piece y = x how we going do that yal X what's yals X look like it's a it's a line that goes through the origin okay okay so so normally we would just have a diagonal line Yeah we actually already to graph that it's right here so if we were to graph that we'd have this thing it' be through the origin it would it would go through the point one one wouldn't it because when you pluged in one you get out one so I know that it's going to go through that point and in fact it's going to be solid because there's an equals there so if I were to graph it it would be that diagonal however I can't have it exist I can't have it exist over here because then this whole thing wouldn't be a function we're talking about just the piece that's that way so I'm going to extend that line erase this piece because I can't have it there I've already got my piece of the function for that part that r that interval this is a the interval we're looking for that's the piece of the function is that kind of cool looking I think are you able to understand it can you follow help me feel okay about that one good so piecewise uh delineate your I use that word again uh delineate that x-axis by the appropriate intervals graph each piece and you got down oh cool the last thing we're going to talk about for this section we're going to talk about domain and range more about domain than range hey when I say domain to you what do I mean if I talking about the domain X more more more General than just x what if I'm using different variables what is do someone over here what does domain mean input yeah the inputs yeah you're absolutely right so when I'm talking about domain what would mean is all the values you can input into a function and yeah you're right there usually X's typically unless you're dealing with like a position function then you're talking about time has time if domain means all the inputs what does the range mean these are usually your y values or your F ofx or whatever function you're dealing with now most cases we have some sort of constraint in real life especially we have some sort of constraint on the domain let me give you a couple examples on this uh let's say that that I give you an example here the area of a square the area of a square now the domain is all the lengths of the sides of the square that you could plug into this this function and get out an appropriate area can you tell me if there's any problems with numbers I plug in here for instance does s have to be is s restricted in any way if we're talking about an actual Square here oh sides do have to be the same but I'm talking about can't be negative why can't it be negative because in real life there is no negative sure in the formula you could plug in3 couldn't you it's going to give you n but in real life can you draw me a square right now with the side length of3 can you do it draw me neg3 oh you can't do it right you're not going to make a square that has when you measure on takee measure it gives you ne3 that doesn't happen because we don't measure uh actual distances and units of length and negatives so we'd say here yeah the area of a area of square is S2 but we have a restriction the side length has to be greater than or equal to zero you could have a tri trial a trivial area trivial Square here it is that's it has zero area okay no no length of the sides you get have a zero length but you can't have a negative that's impossible we also could have some formulaic restraints like that one like that one y = 1X is there any number I can't plug into that sure why not yeah because your teacher the first time they showed you a fraction say now what number came you divide by and you're like zero I'm like oh good job Little Johnny you got it right never told you why right but you can't divide by zero any other number is going to work positive negative you're fine but not zero how about this one FX equal the square root of x here we couldn't have xal 0 can I have x equals 0 there can you plug in zero to a square root yeah well let me ask you what's the square root of zero then you can do it what can't you plug into square roots yeah not inside so we'd say sure this has a restraint where X it could be equal to zero but it can't be less than zero it can't be negative otherwise we well in the real numbers at least now let me make a little little statement there if you're talking about complex numbers can you do itag sure yeah you have an I but if we're talking about complex number I'm sorry real numbers and graphing them on a a real number system well we can't we can't do that now what we what we know is that we're going to redefine domain just a little bit I don't know if your book does this or not but how we Define do we're going to define domain a special way in this class called the natural domain the natural domain is basically everything that works in the in the formula including the maybe natural restraints of the of the problem like the side length of the square or the formulaic restraints of like a square root or dividing by zero so natural domain basically means everything that works in your formula or your function all values that work in the formula would you guys like to do some examples of finding natural domain or domain in general I was hoping you'd say yes natural domain ask this question it says are there any problems with plugging in numbers basically so let's look at a real simple one we got f ofx = x Cub it asks is there anything that you can't plug in that you can't input into that can you think of anything can you plug in positives can you plug in zero yeah can you plug in negatives are there any problems you can think of can you plug in fractions yeah sure you can plug anything you want if you can plug in anything you want and there's no problems what we do is we say domain is simply all real numbers all real numbers for those of you who like to be very symbolic you can do it differently uh you can say X is an element of the real numbers that's another way you can say it so there's no problems there that says that you can plug in any number within the real numbers and you'll be just fine how about that one can I plug in any number I want to this problem and get something out of it that's reasonable that's defined okay uh can I plug in zero yeah yeah even though you know we can't have zero in the denominator if I plug it in well I'm not getting zero in the denominator what numbers can't I plug in sure now you're doing this in your head but I want to show you here's how you find domain in general you look for situations where you could have problems those situations are Roots square roots specifically maybe fourth roots and denominators to find out what your domain is really here's what you're doing in your head you're saying I want to make sure that I know that if this denominator is equal to zero I've got a problem here now the zero product property is going to come up with your answers it says I know x -1 could possibly equal Z and give me a problem I know that x - 3 could possibly equal Z and give me a problem if I solve those I'm going to get your answers of one and three mathematically that's what you're doing right here do you guys see see what you're doing you're you're saying I can't plug in one I can't plug in three because if I do bam it's going to give me zero and even if I multiply by something it's still going to give me zero and I know I can't have zero on the denominator of any fraction so by setting a denominator equal to zero or not equal to zero is how I used to teach in in some other classes you can do this too I know it can't be equal to zero therefore that can't be zero that can't be zero and these two things I cannot have x equal to zero that's another way you can do it by setting that equal to zero solving it down you find out your problems within your domain for denominators so here we'd say x is all real numbers except or but X does not equal 1 and X does not equal 3 please don't ignore that does not equal okay because if you do this I know I I just use the equal sign that's what a lot of you have been practicing doing for your math careers but if you do this except xal 1 and 3 that says one does work do you see that we have to have the not equal to so once you find those problem numbers put the not equal to we're going to find out later that what these things are if you cannot simplify them out of your function those are going to be asmp tootes vertical asmp tootes so this is not going to be defined at x 1 it's not going to be defined at x 3 those are either going to go up like that looks like I'm doing my dance moves but that's so they're doing they're going to go up at an Infinity down Infinity something like that do you guys feel okay with this particular domain let's try a couple more I want to get through a few more problems here um okay how about this one how about Tan x t x is that defined everywhere s goes like this right and cosine goes like this does tangent go like this how's tangent go it makes those those kind of weird s's of the snakes all the way down your paper why does it do that why does it do that wasn't wasn't uh rhetorical why does it do that sure it's undefined why why is it undefined at certain points skipping the X you remember how I just told you how functions aren't defined if they have denominators where those denominators equal zero we'll check it out if we if we looked at this this is tangent right if we set this denominator I I'm going to use the not equal to zero because we know we should not have that equal to zero if cosine X can't be equal to zero which is what I'm saying here what vales of cosine make it equal to zero where do in terms of radians pi two is one of them so X can't equal that's why I'm using this right here x can't equal Pi 2 x can't equal if you keep on going you're going to get let's see for you guys here's Pi / 2 cosine Z true you're going to keep going to 3un / 2 if you keep going around the unit circle you're going to get a lot of different values it's going to keep repeating both backwards and forwards every PI from that so Pi / 2 plus or minus U Pi K Pi where where K can be positive or negative integer it's going to give every part where tangent is not defined because cosine will be zero at those points do you see why this this works the way it does you see why the tangent is not defined at those simply because cosine is not defined at those points you guys see that kind of neat right doesn't matter if the numerator is equal to zero S zero over something that's defined that's okay but something over zero that's not okay okay last one we're going to talk about and then we'll call it a day do I have any denominators here any denominators anything over anything no I'm good as far as that goes but man I got a square root so we're going to be having some trouble doing this square root what do you know about things the radican that's what's inside the radical what do you know about the radican of square roots they have to be they got to be positive so when you're dealing with these roots denominators I showed you how to do that what you're going to do is set the denominator equal to zero or not equal to zero solve that down that will give you your problems in natural domain uh as far as the radicals go you know for a fact that this thing is going to have to be bigger than or equal to zero you know we're going to start this next time I'll show you how to do that all right so we're going to continue finding some domains and some ranges of some functions now now basically when you're finding what I said was the natural domain that's just really what works in a function what we got to do is look for our problem areas if we can Define our problem areas that really defines what we can plug in and specifically what we can't plug into a function really what that comes down to for us is you're always looking for denominators and Roots if you have denominators you know that at some points you might be undefined if you have some Roots there might be some ranges of numbers that you can't even plug into your function so with us up here we're going to have an issue inside that root what do you know about square roots I think we might have talked about this last time what do you know about those what numbers can't you have in there now can you plug in a negative here yeah you might be able to provided that when you work it through the the radican the inside of your radical it's positive that'd be fine really what we want with any root with any square root we want the inside part the radicand to be greater than it could it be equal to zero is that okay to have a root with a zero inside of it sure so so really we want this if you remember from last time we had some denominators what we did was we set our whole denominator equal to zero or I even use the not equal to zero because it can't be zero and you can solve it that way with the equal to zero finding out your problems with our Roots what we're going to do is make the inside greater than or equal to zero we're going to find out those ranges that work oh my gosh that's a quadratic inequality that's math C that's Intermediate Algebra days for you how do you do that what would you do in order to solve that problem Factor we would definitely have to factor at least you probably can see that because it's a quadratic right go ahead and factor that see what you get let's see I think so- 3 -2 do you all get that too okay we pass Factory scar if we didn't my gosh all right how do we do the rest of it though set it equal Z if we set each one equal zero you know what it's going to tell us it's going to say that X is greater than or equal to 3 and it's going to say that uh X is greater than or equal to two do you guys see that which actually is not going to be the right thing for us I know you want to right because you're so ingrained in saying make this equal zero make this equal Zer by the zero product property however we don't have the zero product property because that's not an equation that's an inequality if you know how to deal with your inequality here's what you do you do find the places where X would equal Zer you kind of temporarily set it equal to zero in your head you find those points hopefully in your head right now are the points xal 2 and xal 3 do you guys see those points here's what you do with this this is called a basic version of a sign analysis test so we know that x = 3 and x = 2 are some important points for us here's what you do with those points I want you to make up a number line put those points in order on your number line so which one's going to come first Forest going from left or right sure this is basically a graphic representation of of your interval what's going to work in your your function here so notice what we've done we've said okay we know the inside it's got to be bigger than or equal to zero we factored it we have some key points here we've got x = 3 x = 2 we put them on a number line now here's how to determine when you're going to be okay in your function when you're not if you test a point for each of these intervals how many intervals do we have yeah it's like you're slicing bread right if you have a loaf of bread and you cut it once you get two intervals if you cut it twice you get three intervals so you get three pieces of rad here we have our three intervals if you test each of those intervals with a point in your expression here it's going to tell you whether it's positive or negative that's going to tell you what you can plug into your function and what you can so let's try this can you and you'll see what I mean after we plugg in our first point can you tell me what is a point uh to the left of two zero zero that's the easiest one to plug in I want you right now to test zero so we're going to test zero test zero on the inside of my function if you plug in zero how much are you going to get six wait positive 6 or negative 6 so would you say that we came up with a positive answer yeah so this is going to be a plus what that signifies is that every number try it if you want every number to the left of two is going to give you a positive answer are you with me on that every if you tried one go for it you can try you tried a Nega any negative number that's going to give you a positive POS there do you believe me if you don't well just sprend the rest of your life try out numbers and then you'll come back to me I believe you when you're like 8 years old and I'm dead put on my EP letter or something so we've tested every number over here we know that every one of them is going to be a positive the reason why we know this if you think about it it's a quadratic right and you've just found the roots right it's actually upward facing quadratic so it goes like this positive positive positive dips down at two negative negative negative comes back up at three positive positive positive that's we're finding out right now we want the sections that are positive because we know we can only plug in positives or sorry get out positives for a square root that's the idea that's what a sign analysis test will do for you uh how about a point that's greater than three what are you going to try five five or or four try try five or four the only points you really can't try are two and three why what are two and three going to give you yeah that's not going to tell you anything right that's not positive or negative try try four right now on your own plug that in see what you get you can do it in your head if you want I don't really care about the actual value all I'm really concerned about is whether you're getting a positive or negative what' you get yeah you really you should this is a quadratic it should alternate so you tested four and you got a plus can you give me a point between two and three cuz we got to test a point in there too just to make sure we have this right 2 and two and a half great try two and A2 use your calculator if you want bet your million dollars is going to be negative you want to take the bet no it's quadratic right it's positive here it's positive here it's got to be negative there if it crosses the x-axis so it's going to be a negative how many people feel okay with what we've done so far so can you just set this equal to Z and this equal to Z and get it right and the answer is no no you can't because if you did this watch if you did x - 3 is greater than or equal to Zer and you did x - 2 is greater than or equal to Z you're going to get X is greater than equal to 3 that's true but you're going to get X is greater than sorry greater than equal 2 two that's wrong it's not greater than or equal to two it's actually less than do you see on our on our graph this tells you the areas that work right it says I can plug in any point over here and it's going to be okay because it's going to give me a positive and positives are okay inside of square root some of you're zoning out you can't zone out right now this says every number I plug in over here that's okay because it's a positive it's going to get this is the inside's going to give me out a positive and square roots of positives are okay these numbers aren't so good why aren't these numbers good when I plug them in the inside part becomes negative you follow if we have a square root of a negative well that's not so good for real numbers we can't have that defined so what this does it says your domain is all the intervals that actually work and this is how you can do all of your roots if you have a a function with a root and you're asked to find domain well you do this you set it greater than or equal to zero and you work it out sometimes you might have to do the quadratic with a a basic sign analysis but this tells you those intervals that actually do work so take those those ones that you know of and write them in interval notation can you tell me where this interval starts good and where does it end so Infinity all the way up to two now I know that's got to be a parenthesis because Infinities always have parenthesis is this a parenthesis or is this a bracket why is it a bracket because it's equal to good it's equal to zero because if I plugged in the two it would give me out zero right and that's that's okay for us and then to show any other interval what you do is you put this U standing for a union I know the three is going to work all the way up to positive Infinity that's a different way and your book likes to do this a different way to define your domain instead of all real numbers except you use the intervals that you can actually show so on your book sometimes instead of saying all real numbers like I showed you last time they say things like negative Infinity to positive Infinity you have you seen that yet have you look through your book that just means everything everything from the far left to the far right you guys ready to try another one did this make sense to you okay cool okay let's go ahead and try to find the natural domain for this what you ask yourself for domain is am I going to have any issues or basically do I have one of these situations do I have denominators that could possibly equal zero or do I have any Roots do we have any Roots here no we don't have any Roots that's kind of nice we don't have to do this whole mess mess of crap right we do have what though what's going to be a problem for usat we got a denominator how we how we deal with denominators domain issues with denominators is we set them equal to if you want to set equal or not equal to we know that the x - 2 can't equal zero if we do we're going to get something that's undefined so right here the way it is I know that x - 2 can't equal zero which means X can't equal which which number yeah that's problem so our domain would be all real numbers except X cannot equal two you'd have all real numbers but X can equal two that's that's the deal now I want to show you something this is going to come come back later in our class when we deal with continuity check this out true yeah difference of squares true yes you can do that they're factors true can you plug anything into that if there's no roots and there's no denominators then yes you can can you plug anything into that but wait a second this came from a function that we knew had a problem didn't it we knew we could not plug in two there so what's it mean that I can cancel out the problem can you actually cancel out a domain problem no well that doesn't seem right and the answer is no you can't so even if you can listen carefully even if you can simplify your your function which you can here you still have to keep the original domain otherwise you'd be eliminating some of those problems that you know exist did you follow that I say one more time if you can simplify your function do it but you have to keep the original domain because by manipulating your function on this particular case you've actually eliminated one of your domain problems which that that's not right you have to keep that original domain this says you can't plug in two right just by manipulating it you can't all like say hey no I can Harry Potter here's my math one done you can't do that so if you're going to do this this simplification just make a little note that with simplification you've got to keep your original domain so our our our our domain stays the same X still cannot equal two so for simplification keep your original domain with simplification keep the original domain I'll say it to you another way and this is how I always like to think of it you can't ever make your domain better by combining or simplifying functions you can't ever eliminate problems all you can do is make more problems so how your functions start those are the problems you're dealt that's a hand you're dealt okay if you add or subtract function together compose them uh the only thing you can do is create more problems you can't eliminate problems what you start with is what you start with so if you have a domain with right here with this function that's the domain unless you mess it up even more okay you can't ever make it better by allowing more points does that make sense to you now would you like to see what this actually is you want to see what this does the answer's always yeah yeah okay good I hope so can you graph this can you graph this in your head actually the answer is yeah these are the same graph check it out this graph right here is well that's a y intercept of two yeah what's the slope that means you go up one over one this is x+2 this is the graph of x plus2 with no domain restrictions it would say if I was just given this function and I didn't know about this that's how I graph it now what this function is what this one is with your domain issue it says that you shouldn't be able to plug in the two right if you were able to plug in two how much would you get out of it plug in two how much do you get out of it okay so what happens what this says is that when I plug in two I should be getting out four however I know for a fact from my original function I can't plug in because that would create an undefined situation in My Graph you with me don't zone out you with me still so it says you can't plug in two what that means here is if you can't plug in two you can't get out four that means you have a hole and that's what those are that what that is called in the future is called a removable discontinuity it's not continuous because you have to pick your pencil off the paper so it's not continuous however it's a removable discontinuity which is defined as if you allow one single point to fill that hole if you can fill the hole with one single point it's removable uh so that's that's what we consider that as so what are situations in your denominator there they're two categories I'll say them verbally but I'll go over them more in depth later if you can cancel out you know you love the use those words cancel out it's really not a mathy word but if you can cancel out your domain problem it's a hole you with me it's a hole if you can't cancel out the domain problem it's an ASM toote those are your two categories that makes it kind of simple to if you can cancel out it's a ho if you can't cancel out it's an ASM toote that's it you me write that down for you yes hoping I get away let me give you another example before you you write that down I'll give you the same one over here and I'll give you this one what can't x equal here yeah that's pretty I gave you a very easy one just so you can see of course X can't equal 4 here's the difference can you manipulate this way to manipulate this to get rid of the x- 2 okay if you cross it out that means you have a z over zero if you were to plug in the two check it out you're going to get 0 over Z right you see that when that happens that means you can cancel it out you can simplify it out of the problem that's a whole for you so when you get zero Z this is a hole like that when you can remove the discontinuity AKA when when you can cancel it out of your problem you always get Z over zero if it's a whole yes okay so as a test inad add whatever number you're not supposed to use and it comes to zero with pols absolutely okay radicals it'd be very hard to factor that out maybe I don't know I to do some work on that I'll get back to you on that one with polinomial is absolutely though because if you plug in a number and it is a zero that's automatically a root which means you can factor xus that number out of your equation and that's math that's that's proven so when you can remove the discontinuity we even talked about continuity before but when you can remove the domain this means domain problem that's a ho that's a hole zero over zero for pols in general when you can just when you can simplify it out of your problem and it's no longer a problem for your domain that's what we call a hole it's just a little spot where you don't have a point now your Happ you're okay with the idea of a hole okay now now the other the other thing we got to consider is well what what happens here can you simplify the x - 4 out of your problem here is there any way to factor it and simplify it if you plug in the four do you get 0 over Z no no you get well you get 12 over zero and that's that's an issue right that means that you're you're not going to be able to factor that in any way to be able to simplify that what this is is a vertical ASM toote vertical ASM toote this happens when you have a number over zero typically and it means you can't get rid of the domain issue I also say one more thing but I forgot it those are going to come much later I'll show you an easy way to do this we going talk about something called limits all right does this outline it better for you so you're going to have two classifications main issues if you can simplify them out they're called holes if you can't they're going to be vertical ASM tootes um we're going to find out a little bit later on how to determine what happens with those we'll do another sign analysis test with four on our number line we'll figure out that these ASM tootes can either go upwards or downwards or a combination of those two things so they'll they'll either be like this going towards four in this case or like this going towards four or like this takes a lot of practice to do that by the way it's kind of like uh rubbing your belly and patting your head it's hard to do can you do that let's see no I'm just kid camera's not on you guys I don't care okay now let's do one more example we'll go on uh two more examples we go on to a word problem kind of figure out how we can talk about domain with word problems and then we'll continue on to some uh trig stuff so I want us to find the domain Main and the range of this problem find domain and range to that problem hey first thing are we going to potentially have any problems in this any problem what what I'm asking you is basically are there denominators or are there roots are there any of those two things Ro uh there's definitely Roots so we potentially could have some problems we don't have any denominator so we're not dealing with holes and vertical ASM tootes what we're dealing with is areas of our graph that we can't even plug in a number because it's undefined in the real number system do you see the difference there it's not even defined in those those areas now what do you know about the roots how do we how do we go ahead and and find those areas set them they could be equal to zero right inside or less than zero or greater than zero have to be greater than zero so what we know is that the two is this okay they doing anything the only problem we potentially could have is the inside of this route because we know if this thing is negative that's not so good you okay with that not so good so we know that whatever we do whatever we plug in this has to be greater than or equal to zero that's a must otherwise we come into a a serious problem where we're not in the real number system anymore can you solve that how would you solve that folks what would you do add what one add one this isn't even one of those quadratic inequalities this is actually pretty straightforward one this says you're going to be okay provided X is bigger than or equal to 1 so for our domain you can say one of two ways you can say okay the domain is X is bigger than or equal to one or if you want to use the interval notation which again you're going to see a lot of answers like that in the back of the book um if you use interval notation it would say you're going from where to where infity negative Infinity 1 now that would be X would be less than or equal to one we want to be bigger than one to Infinity sure I know parenthesis here parentheses or bracket here sure yeah very good do you have any preference on which way write the answer I don't care uh you're going to see this a lot right now so you may as well do that uh when could you not have it equal to zero inside a square root I'm going to change the problem just just slightly so you see it if I erasee that and put it over one something or or four any number on on the top uh this would be the only case for for your roots when when you wouldn't have this equal to zero you see if if you actually plug that in if you plug in your one sure you're going to get zero but the square root of zero is z and it's on the denominator so this would say oh wait yeah even though that's a square root I can't have it equal to zero because if I did the square root of Z is still zero and that's on the denominator of my Frac see the the problem there FR that's you now how do you find the range what's the how do you get range how do you get something in the range it's it's an output it is an output so what wow that was that was a sneeze grenade oh my that was is a sneeze rocket launcher I'm allergic to this stuff allergic to calculus me too I start getting all amped up and excited uh anyway what are my inputs up here what can you input give me an example that you can input you can input one give me some something else three three someone else give me something any real number bigger than or equal to one yes that's everything I can plug in now now this says what you can get out right says plug in something you'll get out something so start with the first number you can plug in and plug it in if you plug what's the first number you can plug in plug in one what are you going to get out you're going to get out what two two so this is starting with two now take another number any number plug it in and see which way you're going higher or lower so pick a number in this domain again like three or four or five see which way you're going where where you going pick out pick two what's two going to give you or pick something easier than two Pick 10 I don't care what you pick pick five pick something like that pick five okay what's five going to give you four ah there you go are we getting bigger or smaller than this so we know that this is going to as I plug in bigger and bigger numbers towards Infinity this is getting bigger and bigger towards Infinity so my range goes from two to Infinity so how do you find range well you plug in your domain with limits we're going to find out how to do range a little bit differently but right now you plug in your domain you still feel okay so far y yeah yes yes yes indeed indeed all right so let's try one more we'll go on to our super fun word problem are you guys familiar with odd and even functions oh boy okay we'll talk about two more things then refreshment yeah refresh we want domain and we want range first things first let's talk about your domain are there going to be any issues with our domain here anything things that we can't plug in one what about negative one is that okay we'll get 0 over -2 is that all right to have sure zero over a number that's fine that's zero but something over zero that's not okay so for our domain we go okay I know that x -1 can't equal Z so X cannot equal one well that that's that's pretty clear so the remain is all real numbers except X can equal 1 now the question I have for you is is that thing a hole or is that thing an ASM toote what do you think if you don't know here's how you check you plug in the number that you're not supposed to be able to plug in it's like one if it gives you 0 over zero that means you can Factor it out if they're polinomial and simplify it out of the the problem if it doesn't give you that then you can't do that you can't Factor out so plug it in you get 2 over Z is that a whole or an ASM toote definitely an ASM toote so we know we're going to have a vertical asmt at that point you can't really abbreviate ASM toote more than that you don't want to vert ass so that' be just weird anyway uh how about the range how about the range of this thing that's kind of tough right you you don't know what the range is well there is one thing you can do uh on certain occasions this isn't always possible on certain certain things if you can solve it for your X variable then look at those domain problems for your y that's actually the range it's kind of like you're flipping the script in the problem so you can find the range by solving for the independent variable that's going to and then looking at those problems so if we were to do that JZ I don't even know if I want to do that I don't really want to I guess I will you can do it I can't well I know I can such confidence been doing this a while you were in Math League College I made fun of a keep Math League in college actually it's a horrible person Math League is cool do math League it's it's fun I swear it really is I was just I was too busy with other things to do that um anyway do you have problems here answer's yes same oh yeah it is the same that's just it's kind of weird that that worked out exactly the same I did this right trust me you you can follow it down this is correct but now that we solve for x if you find the domain for your y's that's actually the range of your function do you see the it's kind of cool right you say well if we have any problems on our y's that means we have problems in our range now well we're going to get out of the function what's the problem what can't you get out one one what's the what's the problem here what what can't you get out of this you can't get out one because y couldn't equal one do you see how y can't equal one here yes or no okay so if you do this whole idea of back or and you you you solve this for your independent variable and you make it so you're kind of like you're finding the domain of your dependent variable your y here that's going to give you your range so our range would say well now y I know y cannot equal one can you simplify out this domain issue which is actually a range issue can you simplify out that then this is a Asm toote as well only it's not a vertical ASM toote what is it it's a hor Asm now we're going to have a better way to find horizontal ASM tootes in the future because if you haven't noticed this process for here this was really easy but for every function can you always solve it for your your independent variable not even close no no no that'd be ridiculous uh so we're going to have a better way to do that in the future but for right now that's how you can find your domain and your range how people feel okay with what we talked about so far all right you ready for word problem no answer is always yes yes you ready for a super fun word problem then sure why not not come on you know you're not getting out of it anyway so you may as well enjoy it wait do that exist on the word problem yes it exists I'll prove it to you right now uh it exists just brought it there it is look at that here's what we're in the business of we're in the business of making cardboard boxes what we're going to do is we're going to take a cardboard box and make it by doing this taking your flat piece of cardboard and we're going to cut out a square here here here and here and fold up the sides will that work to make a box we'll use some tape around the sides and be good- looking box so very cheap way to make a box so our our box idea is we're going to take a piece of cardboard that is 16 in by 30 in and we're going to make a box by cutting out squares in the corners and folding up the sides tell me what you know about the squares same can the squares be different sizes you have a really stupid looking box I me you're like yeah that's not going to that's not going to work so well so if these aren't the same lengths all the way around you're not going to be able to make your actual fold flat like shirt box right that that people like to use for presents so I know if I call this x to in order to make my nice Corner that's also got to be X right yeah but as soon as we do that every other corner has to be exactly the same otherwise our box is going to really not be that great and you're going to get fired from box making how bad do you have to be to get fired from box making I mean come on no I'm just kidding uh I I'm guessing you you have to be pretty bad so we got this piece of cardboard is 16 by 30 in we're going to cut out those Corners so we get this machine that's going to come down and maybe it it it uh indents it here here here and here and we're just going to fold those sides up and and crease them I think they actually do is they probably just make one cut and then fold the sides over but you know we're a little more advanced than that so our box when we're all said and done let's see if I can draw this should look something like this you took a 3D drop in class it's amazing oh no that should be never mind y I was you guys are ging way too much credit uh so if I'm folding this up it's still sitting the same way it's going to be on on this Edge is this still 16 in long cuz what I want to do is I want to find a formula for the volume dependent on X I want to find a formula for the volume depending on the size of the cut we're making in in this uh this cardboard so find the volume as a fun function of X how far is that is it still 16 in how much is it 16 - 16 - x yes 2 why 2x you have to do Each corner yeah if we cut both corners and fold that up we're missing not only one X but the other X as well so if I were to find this yes I know the maximum length is 16 right and if I subtract both those those cuts I'm going to get 16 - 2x absolutely what that means is how about this length what's that length sure we still have the same X right because they're squares so we're going to have 30 minus 2x what about the depth of our box how much is the depth of our box the depth is X okay how do you find the volume of a rectangular pris like like this is say what now makes sense so as long as we multiply those three dimensions we'll have the volume so our volume should be okay well we know that one side is 16 - 2x the 2x again comes from the fact that we're cutting out two boxes from each side or a box from each side and folding it up then we're going to get well the the length of this is 30 - 2x and the depth of this is X do you guys feel okay on on how to do that by the way if you graphed that could you find out an approximate maximum volume if you put that on your graphing calculator yeah if if you did if if you worked all that out and plug that into your gra or just plug in just like that you're going to get some sort of graph right it happens to be a cubic graph so if you found the maximum height for the for our domain that we're about to find out you could find an approximate maximum volume for that that's kind of neat right I think it's neat is awesome let's talk a little bit though about the domain are there any issues that we are going to run into with the values of X that we're supposed to plug in so firstly we got to check for any denominators are there any denominators okay that that's okay that that doesn't fail that part how about uh Roots do we have any Roots so we have no issues with that how about some realistic constraints though is there anything that I'm not supposed to be able to plug in for X CU I tell you when you put that a graphic calculator it's going to go negative measur what now negative measurement explain why why not a negative measurement you can't measure netive cim yeah I can't tell you would you make a box up please and uh put -2 in cut in it bigger it's imaginary can you do that yes it's not going to hold very much right be flat like here well yeah you can't even do it you can't say make a negative cut out of my piece of paper here that doesn't make sense so we know for a fact that X is got to be greater than zero for sure could x equal zero but could it equal zero could you make no cut yeah here's your box okay that try to put a package in that yeah it could it could equal zero for sure is there a maximum cut that we can make for our box so I know I can't make a negative but I could make a cut of 1 in and 2 in is there a maximum to that nothing bigger than 16 16 okay 16 could I make a cut of 16 no why not because I cut off your side would that make a difference yes okay let's let's also I want you to think about that that number for a second so here's your box right and what you're telling me right now is if you have a maximum cut of 16 check this out this side length is 16 right you're cutting out two squares one from the top and one from the bottom can I cut out a square of 16 and still cut out a square of 16 remember this would be like a square of two four and four would go there if I take a square of 16 it's the whole thing that doesn't leave any room for the other Square to be cut out of it do you get that that's a problem so let's rethink that idea it's not 16 here we can't do that what's the maximum length I could cut it would be eight it'd be half that length because if I made a cut of eight here I made a cut of eight here that would be the most I could go without overlapping if you overlap you're cutting uh stuff that's not there anymore yes but you still wouldn't be able to make a box out of that no you wouldn't but we weren't able to make a box out of this either so if if we cut eight and eight can you take off that whole piece of material yeah your box and you did it from the other side your box is going to look like oh you cut this side off and cut that side off equal amount that's now your box it's a flat piece of paper that's much smaller you just wasted all your material but you could do it right again you'd be fired from box making but you could you could do it and that's our domain you got to think about these things don't just go with the formula I know look for two things in the formula we look for denominators we also look for roots but even if those things don't exist this is a realistic constraint you got to take into account if you're dealing with realistic stuff you got to really think about don't let yourself get tripped up by the 16 really think about what you can and you can't do with your your product I hope if you're all right with that okay now I wasn't going to do this but I'll give you a little refresher on some odd and even functions let's talk about odd and even function for just a little bit even functions are functions that have twos fours sixes and eights in them well that's not completely true but the powers should actually look like that odds are usually the ones 3es FIV sevens uh what we say mostly though is that even functions are going to be symmetric across the Y AIS odd functions are symmetric about the origin which means if you if you took them and you rotated them 180° about the origin it's going to make a mirror image with what you have already so for for even functions here's what even means like algebraically or formulaically when when you plug something in even function says if you plug in a negative number or if you plug in U negative x per se it's going to give you back out theun function just as if you had plugged in the positive version of that number so if I plug in -2 or two it says it doesn't matter that's even it is symmetric about the Y AIS now odd functions say this it says if you plug in a negative number it's like taking the function plugging in the positive of that number and making it negative that's what it happens with the odd function it says if I was to plug in -2 it'd be like I plugged in positive two got the answer and then made it negative do you see the difference between these two things all right this is this is odd and it's going to be symmetric about the origin like can see an example of some functions that are even and odd let's let's find one out each one each why you guys are asking for the moon today that work here's how you test whether something is even odod what you do is you plug in thex and you see what happens so I want to find F ofx what that means is that for every place I have an X I'm now going to put a negative X inside some parentheses it's it's like you're uh you're composing a couple functions okay you're just replacing x with negx so that means means for this you'd say I want to see what happens with x 4 -^2 + 1 so far so good don't use an actual number you don't need natural number just use the the negative X it's still going to work out for you how much ISX to the 4th x to the 4th because the the negative also gets to the fourth power right it is going to go away posi so this is going to become x 4 how much is what's this going to become so Min - x^2 and then + one did we get the same thing back again exactly the same thing back again yeah this actually equals FX if you plug in the negative X and it gives you back out your original function your F ofx does that mean it's odd or does that mean it's even this is definitely an even function yeah that's how you tell this is going to be symmetric across the y- AIS Mirror Image gee I bet you don't know what this one's going to be well I could trick it uh let's see what do I want to do yeah again to check odd or even you plug in the negative X see what happens so with with our problem we'll have G ofx but that says you're just going to take that Negative X and plug it in everywhere you see X Sox Cub minus X just like that notice X Cub okay we got it and the negative X takes that place that minus is still there regardless of what you plug in so that has to be there what happens with this problem sign switch well yeah negative negative X cubed is that back to our positive X or do you get a Negative X cubed out of that and this one yeah that's for sure plus here's how you tell the the odd if your every term in your function is opposite of what you started with that means what you could do is you could actually Factor this out if you factored ative 1 you would actually get look at thatx Cub - x do you see how if I factor out that negative I'm getting back my X Cub - x yes no this is negative of your original function this is negative FX negative that's my original so I now oh sorry I used F instead of the G shame on me oh my go I Mark you off two points for that on a test I really do actually so be careful with that um so if G of X when I plug in the negative back negative G of X that means it's even or odd odd what is it definitely odd be symmetric about the origin usually S curves are like this you have a good understanding of the even and odd concept now do you feel okay with what we talked about today it all make sense