hey there we're going to go ahead and get started uh my name is Tyler I'm an instructor with Manhattan prep and this is a free GRE prep hour and today we're going to be talking about something called function shapes so if you've never been to one of these before the way this works is I am hosting this session live in Zoom right now uh it's going to be uploaded onto YouTube later so whether you're in the room with me now or whether you're checking this out on the internet sometime in the future big welcome to you good to have you here uh and if you're in the zoom room right now why don't you open the chat box and tell me where you are connecting from all right we got some New Yorkers I used to live in New York during the pandemic actually it was a wild time to be in the city for all that um got some Texans I used to live in San Antonio when I was a kid so gohost Spurs wanyama taking over um got some Californians as well that is where I live currently I'm in the San Francisco Bay Area uh but yeah as I mentioned we're gonna be just going deep on a math topic for the next hour so without saying too much more I just want to show you all a problem to get started so if you're in the zoom room take a look at this one and chat me your answer when you think you've got it if you're watching on YouTube feel free to pause the video here is your first question e e e e e e so one way to approach something like this would be to pick some values for x and take a number picking approach which there's nothing necessarily wrong with that way but I want to show you a technique that I find is usually a little bit faster and it's something that revolves around knowing the shapes of different functions and kind of visualizing what these things look like the most basic shape is is a linear shape right if Y is just equal to X that's just a straight line and then for a more advanced function if you have X squar or any even ordered power on X you're going to get this curved u-shaped parabolic function and then if I had any odd powers like X cubed x to the 5th Etc you're going to get this kind of like curved where basically this half of the parabola has been flipped upside down so if I'm really familiar with those shapes that's going to allow me to tackle something like this with a good amount of speed because I know it must be true that X cubed is less than x so in my head I'm going to think okay X itself that's just a straight line and then X cubed I'm G to draw that in it's going to start off higher than x itself until I cross over one at one it's going to dip down for fractional values for positive fraction values of X then it's going to curve around until I hit negative one and then come out like that so I'm just over laying the linear in cubic shapes now where is X cubed smaller than x and other words if I just took like a single vertical slice of this graph where is my green line underneath the red line well it's going to happen in a couple spots the green line is below the red line right here and right here in other words having this visual allows me to determine that the domain of X that is the values that X could possibly be it's either going to be positive fractions or negatives that are more negative than negative 1 so now that I'm clear on what x could potentially be I can look at my two quantities and compare okay what would X as a positive fraction be by just pick like one2 then 1 half squared would be a quarter so that's a case where quantity a is bigger or I need to test something that's in the negatives down here so if x were -2 x SAR would be positive4 and that's a case where quantity B is bigger so I've gotten two different relationships sometimes a is bigger sometimes B is bigger that means we've proven that the answer is D so let me show you a similar problem and I want you to see if you can take this kind of function visualization approach to figure out what the potential values for the variable would be try this one e e e e e so the first thing I'm going to do here is do what I can to simplify this upfront information that n 5us n Cub is less than zero I'm going to get that into a form that's a little more digestible so first thing I'm going to do is I'm going to add n Cub to each side so I think that makes it a little easier to see versus comparing to zero I'm just comparing my two n functions now so this tells me the fifth needs to be smaller than the cube but I can actually take it a step further because these are both bases of n i can divide out an n s from each side now why am I dividing n s rather than just a single n good I see some people getting this in the chat n could be negative and if I were to divide by a negative that's going to flip that inequality symbol around and I don't want to do that but N squared squares are guaranteed to be positive here so if I divide that square Square out then I get this form that n Cub is smaller than n and now you might be getting Deja Vu because this is pretty much the exact same information we had in the previous problem so again I'm going to grid this out so a straight line that would be n the 1 just a linear line like that and then my n cubed is going to be this kind of zigzag curvy shape looking something like that so once again where's the cube smaller than the first the cube is smaller there and there that means my domain is either positive fractions or negatives that are smaller than negative one and I'm going to do a similar move to simplify my two quantities kind of like what I did to The Upfront info I don't like having this kind of longer expression in quantity a I'm just going to add an N to each side so that what I'm actually comparing is just n squ versus n itself and it's the same thing as the last problem if I had a fraction if n were 1 12 n^2 would be a quarter so that's a case where quantity B is bigger but if I had a negative value like -2 n^2 would be positive4 that's a case where quantity a is bigger so I have two different relationships and I've proven my answer is d so I like to use this technique it's often on quantitative comparisons where there's just one variable involved and I'm looking at functions of that variable like often squares cubes fourth powers and we're going to go a little heavier on this uh domain finding here so if you've got some questions in the chat I see a few people asked about this I want to take a look at some other scenarios so this first one this is similar to what we've already been looking at if I'm comparing two different odd powers like a first power versus a cube power the first power is just a straight line and then I'm going to draw in my hashes for one and negative one one and the cube crosses over at one and then goes under and then crosses zero and goes over and then crosses negative one and it's back under so the areas where the first power is underneath the cube right where the red line is under the green line that's my positives bigger than one and it's also my negative fractions and for the opposite if I wanted my first power to be larger than the cube that happens in two spots first Power the red line is on top for my positive fractions and the red line is on bottom when I have negatives smaller than negative one so let's keep going I want you all to try and chat me answers for this next one if I want a square to be smaller than the fourth power what's my domain of X where is that possible so a square is going to have this kind of parabolic shape something like that so if I call that the square how is the fourth power going to look in comparison to a square well you know it's going to be equal at three points the square and the fourth are equal when my input is one then they're going to both output one when my input is zero they're both going to Output zero and when my input is negative 1 they're both going to Output positive one right negative 1^ sarga 1 to 4th those are both positive one but what's different is because I have a higher power here for my large positives my positives that are bigger than one it's going to be a bit steeper so my blue line is going to going to be on top of the purple line until I hit one and then for my fractional positives the fourth is going to be less because when you multiply fractions they get closer to zero same thing for my negative fractions the fourth is going to stay under that square until I hit negative one and then the fourth power ramps up again so where is the square smaller than the fourth so those two areas on the outside and if I want to look at the opposite if I want my Square bigger than the fourth that happens anytime my input is a fraction Okay so we've seen what happens when I'm comparing two different odd powers and then two different even Powers now I want to look at the case where I'm mixing an even and an odd so the first variant here would be I have a square versus a cube in other words my odd power is the larger magnitude of the two so I'll do my Square in purple something like that and then my Cube because it has more magnitude it's going to be faster for those big positives then it connects at one and drops under and then compared to a square right even ordered Powers always stay positive when I have a cube these ones are going to Output negatives when the input is negative so it'll look something like that so where is the purple line underneath the green line it only happens when I have positives bigger than one and the inverse where is my Square bigger than the cube where is the purple line on top of the green line it's for those other three three zones basically anything smaller than positive one with the exception that they're going to be equal at zero and in fact that's kind of a similar thing up here that you're going to get equality at negative one and zeros for a lot of these so in my head I'm kind of always thinking about seven areas there's three numbers negative 1 0 and one and then there's four ranges I've got my large negatives my fractional negatives my fractional positives and my large positives so when I'm Number testing or I'm just trying to figure out like where these functions might overlap those are the big seven that I'm trying to visualize when determining the domain okay we got one last scenario here it's when we have a mix of powers but this time I'm making my even ordered power a larger magnitude so for this case we can imagine what if I have a linear just x to the 1st I'll draw that in red and then my Square put that one in purple so square is going to be on top until it hits one drops down below but my Square stays positive over there in fact I do that a little bit sloppy let me I can do that a little better something like that looks a little cleaner so where is the first Power the red line under the purple line it's in three different spots it's for my big positives for my negative fractions and my big negatives the opposite is going to be true only for positive fractions now what I've got up here this isn't necessarily something to memorize but ideally I want you to be able to be familiar enough with these shapes so that if you saw any one of these pieces of information as The Upfront information in a quantitative comparison you can kind of visualize what those shapes would look like and then determine the domain of x from there so let me give you all another problem to try take a look at this one e e e e e e so first thing I'd like to do here is take my upfront info and add X squ to each side so now I'm just comparing my two functions and I want the fourth power to be smaller than the square and that's actually the same diagram we just drew where that purple line is the square and that blue line is the fourth power so where's that purple line on top it's inside that range it's only when my input is a negative or a positive fraction so knowing that whether you want to do this mentally or actually pick some numbers right I could either have a positive fraction like one2 or a negative fraction like negative one2 no matter what the absolute value of that is going to be smaller than quantity B so B is is always going to be greater and I had someone in the chat mention this that if you wanted to you could try and keep simplifying that upfront information and and at that point you'd get x^2 is less than one with one caveat that year you know assuming that X is not equal to zero because you don't want to divide by zero uh but it's the same thing where is the output of X squ smaller than one it's only when that input is a fraction so even do that way you should still get b as the answer okay so at this point I'd say we're pretty familiar with the basic shapes linears parabolic and cubic functions and now I want to show you some ways that the GRE might start to modify some of those functions and the first of those is going to be absolute value and what introducing absolute value to the entire function will do is make it so that function Cannot drop below the x axis so on the left I just have y equals x on the right if I wrap that X in absolute values it can no longer go negative right here it's got to flip up to stay positive or similarly if I wrapped a cubic in absolute value right this chunk that was going negative before now it's flipping up so everything is staying positive in quadrant one and two so doing this to a cubic effectively makes it look like a parabolic function right it's that u-shape that we typically see with even ordered Powers another common modifier is if you introduce a negative to the entire function and what that is going to do is flip everything vertically across the x-axis and I want to make sure you're conceptualizing this it's a vertical flip I know you could conceive of it as being a horizontal flip across the y- axis but that's just kind of an illusion right you'll have a much easier time with this if you think of it as a vertical flip so there it is linearly or if I did it to a cubic same thing we're flipping it vertically across the x-axis so that that moves up and that moves down so using that new information I've got another quantitative comparison for you all to try give it a shot e e e e e e so in my upfront info here the absolute value of x cubed is less than 64 so normally X cubed is that kind of cubic shape but the fact that it's wrapped in absolute values is going to turn it into that more parabolic U shape so I get something that looks like this uh and then the fact that it needs to be smaller than 64 means I'm kind of limited on the left and right by 4 and four but in terms of my number types I haven't limited things much I've still got some larger negatives some fractional negatives fractional positives larger positives negtive one01 all that stuff is still on the table at this point so really I'll need to take a closer look at quantity a and quantity B so quantity a Negative X that's just my linear shape flipped vertically and technically you know I'm only looking at values between ne4 and positive4 but that's the basic shape of it now if I go to quantity B the opposite of the absolute value of x so regularly absolute value is going to be that kind of vshape but if I have that negative sign it's going to invert that we're flipping it vertically and again we're only between -4 and positive4 but you can see when I'm in Quadrant 4 negx and the opposite of the absolute value of x those are going to be equal but things change around when I'm in quadrant three so I can either have these two quantities being equal or not equal depending on what numbers I select that means the answer has to be D let's do one more of these involving absolute value here you go e e e e e for this problem we're not giving any upfront information so there's nothing limiting the domain of X right all of these seven areas are still on the table here and the graph of quantity B we've got that coefficient two in front of X so it's basically taking that linear shape and just speeding it up making it a little more aggressive so how is quantity a going to look at by comparison so we have the absolute value of x plus the absolute value of the opposite of X and remember when it comes to absolute value those two things are equivalent no matter what x is the absolute value of x and the absolute value of the opposite of X it's the same number so I can reconceptualize this as either being the absolute value of 2x or the absolute value of the opposite of 2x either of those it's going to be similar to that graph of quantity B except it cannot go negative it must stay within quadrants one and three so kind of like what we saw in the last problem when I have positive values of X these two are going to be equal to one another but when my input is negative that's where I get some differences so the relationship is not consistent and the answer will be D so earlier when we were talking about modifiers we saw how wrapping a cubic inside absolute values makes it look more like a parabolic even ordered function so now I want to ask you how could I go the opposite direction how would I have to modify this x squ function to get it to look more like a cubic anyone have any ideas well it's again going to come down to absolute value but instead of putting the entire higher function and absolute value symbols what I would have to do is just put one of those x's in absolute values so I'm breaking that square into x x x if I put one of them in absolute values then my parabolic shape is going to look more like a cubic shape so on the test if you see a single X in absolute values times a regular X in your head recognize that's going to be a cubic shape and that's going to allow you to tackle some problems like this with speed give it a shot chat me your answer e e e e e so when I have the absolute value of x times x that's going to look like that cubic shape but I also have a negative sign here so that mean things are going to get mirrored across that x axis and I flip the whole shape vertically it's going to look something like this and we want to take that shape and have it so the output is always bigger than four where is that going to happen well if I say that is four it's gonna be all those points Beyond you know whatever that ends up being there so it's only my large negatives that are further left than negative one on the number line so no matter what I do quantity a is going to be something negative so quantity B A positive value will always be bigger so B is your answer let me show you one more of these and then we're going to start wrapping up try this one e e e e e so first thing I'm going to do here is take that up front info where we have X over the absolute value of x is less than x I'm going to multiply each side by the absolute value of x so now I see I've got that x times absolute value of x that cubic shape has to be bigger than the linear X so graphically something like this so where is that red line under underneath the green line it's going to happen for my big positives and it's going to happen with my negative fractions so that means if I had a big positive get the absolute value of two that is bigger than one but if I had a fractional negative like the absolute value of - one2 actually I wrote that the wrong way so in the original in blue quantity a is bigger when I've got fractional negative one2 that's going to make quantity B bigger so I've got my two different relationships we've proven the answer is D all right and that's how you can use function shapes to visualize the domain of X so if you don't want to pick numbers if you instead want to think visually this is how you do it so to kind of recap what we talked about you've got your three core shapes right linear parabolic and cubic when you have upfront info use that to help you V visualize the domain of x if your quantities look like that um it's less the domain of X and just like what x looks like but still useful and then we've got a few modifiers we talked about how you have that negative sign it's going to flip everything vertically over that x- axis and if you have absolute value symbols it's basically preventing the function from dropping below the x-axis so that if I have the absolute value of x normally it looks like that if it's the absolute value it can't go negative so it's just going to get flipped right back up all right so that's going to do it for me if you like this you can go to this link the bottom of your screen there I'm also going to put it in the chat box for those of you here in the zoom room uh you can check out more free prep hours at that link there we also open up session one of our nwe course open opened up free to the public so if you want to check out what that class is like you can check out one session for free so I encourage you to do that um and yeah like I said if you want to check out more of these in the future head to the link as well uh other than that I'll say it was fun hanging out with y'all and talking about math today and I wish you the best of luck with your studies take care