Yeah, is the screen visible now so last class I had introduced a special function, which is the modulus function and we had learned how to Redefine those functions. That's actually very important because you will be asked questions Based on inequalities also from that not only equations, but any equations can also be asked on those redefinition of functions so in today's class we are going to dive more into modulus and and Last class we ended up discussing that if you have a function y is equal to f of x and I have to draw the graph of y is equal to mod of f of x There was a simple rule that we used to follow whatever whatever was the part above the x-axis We never disturbed that part, but whatever was below the We reflected it about the x-axis right So in light of that just a few equations that you will do now Okay, so let's say I ask you this question plot the graph of plot the graph of y minus three is equal to x minus Two more. Yeah. Yeah y minus three equals x minus two more. Okay, so First okay, so first you take the graph of y equals mod x very good So that comes Okay, that will come I The the the tip I mean and the other what extra thing will come at the at the origin for that Zero And then it's the whole thing is shifted tuned to the right and Three units up. So it is vertex comes at two comma three Absolutely very good. So now it has got moved like this. Okay, so this point Right, which is the corner point now has come at two comma three. Okay. Yeah Okay Now let's look into those cases where you are just modding x. Let's say I give you a function f of x and The graph of it is like this. I'm just giving you a hypothetical situation Okay, just let's say I have a function like this This is a curve like this Okay, let's say let's say some hypothetical function is this and this is the graph of that function Okay now what effect will this function undergo if I just Mod the x part of it not the entire thing. So there's a difference between Modding the whole function and modding just the x part So what what will happen if in this very function? I just want to plot the graph of y is equal to f of mod x Just think carefully and tell me what effect will it have and if possible draw the graph for f of mod x So mod x meaning that it will That if there's an Value about the eyes. So the output of that will be an absolute will be a positive value So if it is going to be a positive Value Okay, one second take your time so, okay, so The curve part because it is on the negative And and then on the y-axis Okay, okay, let me help you in this see when you say mod of x means you are making This entire thing has a positive input, right? This entire thing should be positive. Yeah, correct Now that doesn't mean x cannot be negative x can be negative But the output of mod will always be a positive quantity, right? So, yeah, let's say I tried to put minus one as x. So when I put minus one as x The function will perceive as if it has been given one as the input Okay, yeah, okay, if I put a minus three as the Value of x the function will perceive as if it has been given three as the input Okay, so actually actually I'm drawing the graph at minus one and actually I'm drawing the graph at minus three But function will pull out those values Which it is which it knows is f of one and f of three respectively for these two points Okay, let's say f of one is this value. Let's say f of one is this value So for f of minus one also the same value will get replicated. Are you getting this one? But what will happen to the positive values of x Those will not remain those those will not change at all because even if you do f of mod one it is f of one So when you're already in the positive values of x the function will exactly trace the same graph as it was So this will not disturb this part Okay, so this part of the function would be undisturbed The one which is on the positive x-axis. That would be undisturbed And what it'll do is for the negative values It'll start copying the same graph which it was which it had on the positive x-axis. That means the same graph here would start getting copied Oh, but okay, right and this part it will ignore this part would be ignored because They will never be able to plot this curve because they came for negative inputs And no matter what f can never realize a negative input even though x is negative mod x will make it as a positive quantity Yeah Okay, it's ignored it's ignored Therefore the rule here is let me write down the rule for you the rule is If you want to plot the graph of y is equal to f of mod x Okay, then first step is plot the graph of y is equal to f of x second step is Don't disturb Don't disturb the part on the positive x-axis. Yeah, don't disturb The part of the graph on the positive x-axis Okay, the graph On the positive x-axis, okay, and third step is Reflect the graph on the positive x-axis about the y-axis Okay, ignoring the part which was already in the negative x-axis Okay, so if that's about the positive y-axis and then you ignore that was already on Negative x-axis absolutely Ignored it because it was because x because absolute value of x is never negative You ignore the part on the negative x-axis Yes The negative x-axis graph came because of the fact that the function was given a negative input Correct. Yeah, but this time because there's a mod of x involved you will never be able to give the function a negative input Yeah, okay, so that part will never be able to the function will never be able to trace that part Okay, fine functions is like machine So when you are putting some value into it It gives you an output and that's what is what we trace Correct. So since it'll never get a negative input It'll never be able to trace those those parts which were which were in the negative x-axis direction. Okay. I miss it So let's say I want you to plot the graph of y is equal to mod x-square minus 3 mod x plus 2 3 mod x Plus 2.  Absolutely. So step number one. Yes, you draw simply the curve ignoring the mod first. Yeah So that's what me y equals x y minus 3x plus 2 It's roots at x equals 1 and x equals 2. Right, and and it's And it's point. It's gonna be a minimum point, right? Yeah, that would be a minimum point between 1 and 2 so it'll be coming like this for you Yeah, it'll be coming like Okay Minus 1.5 Alternately you can do one thing you can actually complete the square here as well So you can do x minus 3 by 2 whole square. Yeah, okay, and plus 1 by 4 1 by 4 is half square Okay So if you do this also Sorry minus 1 by 4 So this will give you 9 by 4 right? Yeah 9 by 4. I need an 8 by 4 so 8 by 4 will be minus over here Okay, okay So in this way you can actually Shift the graph of x square 3 by 2 units to the right and 1 fourth down So this this point is at 3 by 2 comma minus 1 by 4 Okay, that's another way to draw it and of course your way is also correct No, you already know the roots of the graph which is at 1 and 2 and it's an upward opening parabola How do I know it's an upper opening parabola because Yeah, the coefficient of x square is positive so you can also trace it like this Now when it comes to modding just the x value remember we had discussed we will not disturb the part Which is already on the positive x-axis so One on the negative side we don't we know that part And then what we do is we simply Yeah, so this will be the graph of mod of x square minus 3 mod x plus 2 Actually, it's a stupidity to write mod of x square because mod of x square and x square will be the same So even if you try to plot this Your result would not change Right. Yeah, this is because Mod x square and x square are the same things Okay, yeah, let's try to see on GeoGibra. How does it turn out on GeoGibra? So y is equal to x square Minus 3 absolute value of x Plus 2 see this graph Yeah Yeah, so that's how the graph is being done Let's take few few more questions Let's say we have to draw the graph of Y minus 1 is equal to mod of x minus 3 plus 4 Okay, okay, so y minus 1 equals 1 by 1 of x minus 3 plus 4 How should we plot this graph? Okay, one second. Okay, so this is So of course, you'll start with this step y is equal to 1 by x, right? Yeah, y equals 1 by x so and then Okay, one by x and then it's a shifted to the to the same graph is shifted to the Right by three units and then shifted to the right by three units. That means you are doing this now Yeah Okay, are you sure about this step? Okay, fine. Let's go ahead. Let's see whether if an error comes, we'll come back Okay So and then What is it? What is plan? Do you shift it down? No, yeah, you know it should up by one unit. That means you're planning to do this Yeah, okay. Go ahead. Yes, then Okay, then I have to do a plus or we went to that comment. That's the problem. That's that's what I was worried about Yeah, how do you bring the plus four in?  Okay. And then the fifth step would be shifting the graph three units to the right That means you are now replacing your x with x minus three. So that's how we end up getting to this guy Okay, okay. So sequence of step is first we go to the graph of the rectangular hyperbola that we studied the last class Y equal to one by x Okay, then we shift the graph four units to the left Four units to the left. Yeah, this is shifting four units to the left four units to the left Mm-hmm. This is the shifting of the graph one unit down Okay Okay, okay. This is like modding the x means whatever is on the right side You reflect it to the left side and whatever was on the left side you ignore it. Yeah And then again, this is shifting the graph three units to the right Okay, so this is okay. I understand. Yeah, if you don't follow this sequence of steps In fact, few of them can be up and down, but you have to mod first and then introduce x minus three Okay So what I do is I will I'll Replicate this on GeoG graph and there you can understand how the graph comes out. Okay. So first step was I'm getting the graph of y is equal to y is equal to one upon x. Okay, that's figure like this Okay, next step is I'll change my x with x plus four So what I'll do is in the denominator. I'll add a four Let's see the reflect. Okay The graph moves four minutes to the left Correct. Yeah, then I will change my y with y minus one Graph will go one is down. Okay. Sorry one way up. Yeah One mistake. We wrote a one. Yeah. Yeah, so let me just make a correction over here We wrote by mistake one and down so it will one in it up because they're changing y with y minus one, right? How do you change change your y with y plus one? It would have gone one units down. This is one Yeah, yeah, okay. Yeah Now next correction that we are making in the graph is we are making x as mod x correct So this this x I'm replacing with absolute value of x Correct Okay, you see what has happened to the graph? Yeah, it's become like a small Yeah So why let me go back a step Let me just release this for more because then you'll appreciate much better. What has happened earlier This was the graph and you see this part only the one which I'm showing with my cursor That part only gets reflected over it Just all these parts are ignored. Okay, I understood understood. So y minus one is equal to one upon Abs of x Plus four Okay, so I'll just remove the previous one now. Okay. I'm sick. Okay. I'm sick Now next is you're changing your x with x minus three So the same graph will appear to move three units to the right I'm making a change over here itself. So minus three as you can see this part has moved slightly to the right Okay, I'm This change in curvature is so so gentle that we don't actually observe it carefully Okay. Yeah. Yeah, so if I like try to zoom in a bit How it is Yeah, I basically just that part which got reflected and Decided yes. Yes So now we'll move on to one more example Yeah, let's say I ask you to draw the graph of y is equal to Three plus e to the power mod x plus one minus three X plus one minus three mod of x plus one minus three. Yeah Okay Spine so First you have to draw them. I think you can draw the graph of y equals x Well, why would eat about it? Okay? Yeah, why do you do power x and then after that? Okay, you replace Why you eat about x with e to the power x minus three? Basically what you're doing is you're shooting it three units to the to the right right and then You are also you're shooting the right the units to the right and you're also shifting it Three three units up because now you've taken y minus three into consideration By my sequence even power x minus three, so you're shooting the three units Can we like can we write down the sequence of events as you're doing it? So first step is what first step you told you will be doing e to the power x perfect. Hello You Sorry and the voice broke actually in between. Yeah, what is the next step you're doing planning to do Arushi? Yeah, I y equals e to the power x minus three Okay, so you're doing planning to do y equal to x minus three perfect nothing wrong in that Then why minus three equals e to the power x minus? Okay, absolutely fine then Then after that Now you mod the X now we mod the X Absolutely correct Okay, so when you model it basically shift, I mean Reflects about the y-axis like whatever is in the passive x part, right? Then You Yeah, you replace your x with x that means you shift the graph one units to the left Right. Yeah, so so this is exponential graph that we can draw. This is just an exponential graph This is shifting the graph three units to the right Right. This is shifting the graph three units up Mm-hmm. This is performing the modulus operation on X mod just on X So we know what goes in that operation and then this is like shifting the graph one unit to the left Correct. Oh, yeah, so let's let's try to plot the sequence of events on our Geo Gebra Okay So Let's move everything We have y is equal to e to the power x first So y is equal to e to the power of x first Okay Next is we are going to change X with x minus three So here itself I can make that change Minus one so shifting the graph three units to the right Okay, then I'm going to change y with y minus three shifting the graph three units up Okay And you put the modulus Next is you put the modulus on X. So it is y minus three is equal to Mod absolute value of X minus three This is the effect that will have on the graph. So whatever whatever was See whatever was on the positive X axis Only that part got reflected to the negative side and this line as you can see I am showing it with my cursor This line vanished Okay. Yeah, so it became like this Now what are we are going to do we are going to replace our X with X plus one meaning I am shifting the graph one unit to the left One unit to the left so plus one so you can see this will move one unit to the left Okay, nice Understood the sequence of events. Yeah, I understood it wonderful Now we'll move on to the fact where We are again making certain changes in the graph Let's say I Tell you that We know the graph of y is equal to f of X Okay. Yeah, and I Need to know the graph of mod y equal to f of X now try to understand this is different from doing y equal to mod of X Okay, yeah, right this and this thing are quite different from each other How can you tell me how they are different these two are different from each other? Yeah, so if you have mod y equals f of X then Then you would have Okay, then and y would be equal to f of X Y would be equal to negative f of X But then when you have y equal X mod of f of X then f of X equals negative y So, okay, so basically Yeah So just in the plain and simple words just answer this question in this expression can my y be negative No, no, why can't be negative. No, I should hear why can be negative. Oh Oh Okay, wait, how the output can't be like the output of it cannot be negative f of X can be can be f of X cannot be negative Okay, fine. Yeah f of X Has to be positive Okay, so why can't be negative And f of X has to be positive. Okay. Yes. Yes, because you're comparing the modulus to something So this will always be a positive quantity. So f of X can be always be positive. It has to be positive But why can be negative because there's no restriction on why because Mod is making it positive. So why can be negative as well? Okay Why can be negative but f X that's the output has to be positive. Yes Then the other one it's opposite. It's y has to be positive Yeah Y has to be positive in this case Whereas f of X can be negative Okay, so why are we positive but f of X can be negative Understand the point. Okay. Yeah, of course, since we have realized both are different cases. Now. How do I plot the graph of? Mod y equal to f of X. What should I do? Let's say I hypothetically give you a graph of y equal to f of X like this Let's say my function graph is like this Mm-hmm Okay, let's say I have some sine waves over here Okay, and here. Let's say I have some Triangles being made Okay, just a hypothetical graph and I've made up this graph on my own. Okay. Now. I just want to know What can I do to get the graph of mod y equal to f of X? Mod y equals f of X. Yes, how would the graph of this be? Okay, fine, so Here so here y can be negative but f of X has to be positive So if f of X has to be positive then would you keep the one on the positive side of the x-axis? No, there's no y-axis Positive side of y-axis. Yes. Yeah the same and simply reflect the ones in the negative side Okay, so you're trying to say that I keep only these two Yeah Okay, as per my given graph and just reflect it down like this side. Yeah, that's absolutely brilliant Wonderful, that's correct. Okay. See again since formally I explained it to you Just just for the purpose of recording it see when you say mod y equal to f of X This can never be a negative quantity. This can never be a negative quantity So you will never never get these values that means this part of the graph and this part of the graph can never be obtained Okay, yeah, this cannot be obtained. Yeah Okay Now where f of X is positive Let's say I have these two graphs and let's say here at X equal to one I get a value of one Let's say okay. So this means you're saying mod y equal to one Meaning why can take two values for the same X which is plus and minus one? Yeah, okay, right? So for the same value, we will have a graph value down as well So for all these values, we will start getting a value down as well. Are you getting this one? Yeah, that's how this actually gets reflected This actually gets reflected about the x-axis like this and that's how we get this structure So the rule here is the rule here is whenever you are plotting the graph of mod y equal to f of X So first thing is plot y equal to f of X without any restrictions on it. Okay, secondly Reflect the graph Let the one on the positive side be the same and simply reflect the other side about yeah You can you can you can introduce that step before writing this don't do not disturb the part Which is already on the positive x-axis positive y-axis Do not disturb the part of the graph the part of the graph which is already on the positive Y axis direction Mm-hmm and first step reflect the graph Which is on the positive y-axis? About the x-axis Ignoring the part which Was in the negative y-axis direction Okay, I'm innocent. Okay. Great. Yeah So let's do some questions based on this plot the graph of mod y plus 1 minus 2 is equal to More mod x square I Think I can write it without Yeah, just write it x square because no book will write it as mod x square. So just x square Ah Plus 3 mod x That's 3 mod x. Okay. Thank you So first okay, so first what you do is that Okay Okay, so first you can plot the plot the plot the the normal quadratic equation You can plot x square plus 3x plus 2 Without without without the modulus sign Can can I just bring plus to that side and simply plot like that? Can I the sequence of tips? What can I write the sequence of stuff that you're telling me? Yeah, okay? Okay, fine. So First step is what you're saying I was thinking that we could we could bring plus 2 to to the other side and making like a proper corner equation with x Square and plot x square Okay, so first step is you're trying to Yeah, restructure this equation and write it as mod y plus 1 is equal to x square plus 3 mod x plus 2 Yeah, and simply plot x square plus 3x plus 2 without the mod Okay, so y is equal to x square plus 3x plus 2 without the mod. Okay, then okay, so plot that first Then After that So once you plot it And then x also has a modulus sign So because x has a modulus sign what happens is that? It is The One of the So next you want to make a mod on x, right? That's what you're saying. Yeah. Yeah, okay So I'll give you x square plus 3 mod x plus 2 then Okay, so you factor e to the reflection and then okay, then you do the mod of y Paper paper Yes, sorry Yeah, you do the mod of y Are you so you would basically just reflect one of us on the positive side of the y-axis? To the you just reflected about the x-axis. So you just planning to do a mod on y This is what is it modern why then what do you do? Just tell me what you're going to do How to do it. We know it right Then we shifted yeah Shifted one unit to the left. No, no one into down right because you're changing your y with y plus one Yeah, so this is how you get it very good So this is fine. This is you know what to do this, you know what to do this, you know what to do So let's do this on our Geo zebra So that will be much faster So first you said we'll plot y is equal to x square But see x plus two right see x yeah Okay, so this is going to be a parabola like this And you said you're going to mod your x Yeah, so Here we'll mod our x Okay, okay So the part of the graph which was on the positive direction that only got reflected the whole entire Parabola over here got vanished.  So yeah, okay now you decided to take a mod on y p So let's say abs of y Okay, so as you can see the part of the graph which was about above the x-axis that graph got reflected below Okay, yeah, okay Because of some graphic issue you'll see a breakage over here, but that's not a break actually. It's because of the graphic issue Okay, some pixels go missing actually It's a heavy software. That's why it happens Okay, yeah, no, it's proper. Yeah. Now next we plan to do y with y plus one Yeah, so we'll change this to y plus one So what will happen to the graph it'll shift one is to the down Okay, as you can see both the beaks have gone one one in a down each. Yeah Now is this a function is This a function if you take the No, actually, it's not a function. Yeah Mod y equal to f of x is not a function. Yeah is not a function Because for any input of x, right, I do let's say let's say this answer comes out to be five Then you are saying mod y equal to five implying y can be five also y can be minus five also This will not be a function Okay, fine. Okay Shallow now we'll move on to another special function which is called The greatest integer function The greatest integer function one of the most important functions that you would deal with Mm-hmm. Also called as the floor function Also called as the step function Floor or step function. Yeah, okay. I would abbreviate it as gif always Okay, gif means greatest integer function First of all, what is this function all about? Let's understand the definition of this function. So let's say Let's say When you say f of x is the greatest integer function of x It is first of all represented by a square bracket like this Okay, now that doesn't mean any square bracket that you will come across is a representation for gif There in the question you would be given specifically where this represents The greatest integer function, they will not probably use the gif in a short form. They'll like the whole thing Okay Well, many people will not understand what is gif they will confuse it with the gif that we put on Facebook and WhatsApp Okay, so this what does it mean? This means this function will return to you the This f of x will return to you the greatest integer The greatest integer lesser than or equal to Whatever input you have put inside so if I put inside x then it will return you the greatest interior lesser than or equal to x So if I if I ask you what is the gif of 5.34? The greatest integer lesser than or equal to 5.34 What is the greatest integer lesser than or equal to 5.34 5 5 it cannot be equal to of course Because no integer can be equal to a fraction like this. Yeah, so it's 5 Yeah, five. So if I ask you what's the gif of 4.37 4.37 will be 4 absolutely What's the gif of minus 6.24 Okay, the greatest integer It's the greatest Than that right so it would be minus seven Absolutely correct. This is a mistake which many people do they think it's minus six. It's not minus six It's minus seven as you rightly said What's the gif of minus 0.00024 0 minus 0.0004 Yes, so that would be Basically going to be So minus 0.00024 so that is the final the greatest integer lesser than that So the value lesser than that would be Hmm It should be an integer. Okay, so then Wait It's in minus zero point. It is. No, is it minus zero minus one minus one. Absolutely correct Okay, now you have known the definition of this. Can you plot the graph for this? Let's see. I give you a free hand now try plotting the graph of this and Tell me what do you see in the graph? Okay The graph of f of x equals Gif of x Yes, okay So one second So let's say I just ask you what is the graph for all values between zero and one not including one Yeah, but all values between zero and one not including the graph for all values between zero and one Not including one. Okay. So if it's between zero and one not including one then At one it'll be at one at 0.5 it would be at one Okay, I want you to tell me what would you see when you are dealing with such values Which are between zero and one not including one. Can you see you'll always get a zero for that? Okay Can I say the graph will be just a straight line like this, and there would be a hole over here. Okay Okay Yeah, okay, so the whole signifies that you have not reached one yet. You are just before one Let's say point nine nine nine nine nine nine nine nine nine. You have stopped there. Okay. Yeah Not tell me what happens if you take the interval one two two not including two 1 to 2 not including 2 so that will be that whole graph will be at y equals 1 yes so can I say it will be a line like this till you reach 2 value and at 2 there again will be a whole yeah all right yeah and if this 10 continues you will see this kind of graph coming about yeah okay this this down also will continue like this what does this give you a feeling about it gives you a feeling of a step isn't it and that's precisely why this function is also calls okay now I also tell you why it is called for function with some other example is that fine is that fine Arushi this is graph is clear to you yeah okay so can you plot the graph for y is equal to gif of x plus 2 minus 4 okay x plus 2 minus 4 fine so if you simply have y equals x or I call that square bracket of x then what would you have yeah but you basically simply get the graph here and x plus 2 meaning you want to shift it to units to the left so it gets between the left and then y plus 4 so gets a full units down so yeah so you're ready with that framework in your mind yeah okay so can we draw it on G o G graph yeah all right so first of all we'll draw the graph of y is equal to gif of x and in the app also it is called as floor only okay okay but remember in the app it will not show you those holes because it is just a pixel missing for the graph okay okay yeah so what I'll do in x is I'll replace my x with what was my function x plus 2 right yeah so I'll replace my x with x plus 2 here x plus 2 as you can see the graph will shift to units to the left yeah see that it is shifting to use to the left yeah okay next the graph was requiring me to do what are the other operation that we're supposed to do on this minus 4 that means why plus 4 yeah so it will shift four units down four units down so I'm changing it here only y plus 4 shifted for example okay as you can see it is happening like this okay it shifted four units down yes and what is that like it's that it's again a fault of the graphics so much but it's not like that it's not like that this graphic sometimes starts showing you different versions okay anyways so the model of the story is the all the transformation that we have studied so far would apply on this yeah so we'll start with y is equal to gif of x okay second step is why is the y is equal to gif of x plus 2 means shifting the graph to units to the left and then then we are doing this that means we are shifting the graph in step 2 by 4 units down yeah okay yeah now the problem comes if we start plotting the gif of a non-linear function for example if I say plot y is equal to gif of x square how would you do this dot gif of x square okay y equals that so okay it'll be it'll be so if you put in the value of one so it'll be I find out the greatest integer value that is lesser than or equal to one so it will be like it'll be it'll be a line along the x axis then if I put in the value of two it'll be values of that of x that is like lesser than four so less so it'll be like it'll be an interval between three so okay now let's let's discuss this first in this case what is going to happen first of all let's draw the graph of x square without any restrictions on it so this is the graph of x square okay let's say yeah okay now what is happening this value is zero yeah where do you think will attain a value of one so you'll say it's very simple it'll attain a value of one at one correct okay so when you are looking at this part of the graph when you are between zero to one you know your x square is also lying somewhere between zero to one yeah correct so what I'm going to write it here itself so when you're looking at the interval zero to one your x square is also lying in the interval zero to one correct yeah so when you take a gif of such functions which are between zero to one what will gif do to it it will straight away bring it down like this yeah and there'll be a hole over here yeah okay because for all these values your gif of x square is going to be zero understood this part understood Rishi this is slightly tricky see between zero to one your x square is also lying between zero to one yeah so any value which is lying between zero to one not including one will always have its gif as zero the same thing with the previous one right yeah now yeah same thing as that now ask yourself when will it attain a value of two what will be this value of x be here yeah so for what value of a case will be why will become okay when x is plus or minus root two let's say positive on the attic negative side I will also deal separately okay okay now see it in this part of the graph when you are between zero between one and root two your x square actually happens to lie between one and two right oh yeah so when you're in this zone your y is between one to two so what will be the gif of this the function which is lying between one to two won't it be just one any value take between one and two not so can I say this part over here it will be at one it will be at one like this again they would be a whole at two over here understood yeah can you guide me with the next so when is it three ask yourself yeah for three equals x so x should be minus root three so that'll be there so basically when x is okay when x is between root two and root three x square is between two and three yeah okay the gif of x square should be two yeah so what will happen to this part of the graph it'll fall on like are you getting this okay do you see this trend happening okay now if you continue on the other side also it will try to replicate the same thing because ultimately it's going to be a kind of even function the same trend will be continued like this okay so if I show you the graph on geojibra this is how it happens it's a very cool graph so let's say y is equal to x square I'll show you okay okay now see what is going to happen I'm just going to draw it here itself just to show what is happening nice okay see what is going to happen so between zero to one again broaden it a bit so that one is shown yeah yeah now between zero and one between zero on this part of the graph will fall on the x-axis like this yeah yeah between one two root two root two maybe somewhere over here okay one point four yeah that part will fall like this I'll just type in root two for you here so or I'll do one thing I'll plot y equal to one y equal to two y equal to three y equal to four like this okay now see what is going to happen this makes my life pretty easy so this part of the graph which I'm showing you with the red parts this part this will fall on the orange part similarly just how this part actually fall like this yeah okay similarly this part will fall like this that means vertically down there will be a arrow like this yeah correct this part again I'm showing you this part yeah will fall like this right and the same thing will happen on this side as well so this part will be like this this this red part has all this this red part will fall like this okay and this red part will fall like this and it continues on and on okay so do you really want to see the graph now yeah let's show you the graph of this okay so I'll just remove this line I'll bring it back one time is there so y is equal to abs of thank you for correcting that's floor floor of x square you see that if I remove this this is what you're going to see okay now there's a trick over here if you see these lines will hide those ladders or steps okay so what we normally suggest is a simple rule which I'm going to discuss with you the rule is whenever you are asked to draw the gif of any function let's say I let's say the graph of this is known to us this is known to us okay and you are asked to plot the graph of this function okay where again I should as a customary thing I should write this is gif because any square bracket doesn't mean gif unless until stated don't treat any bracket as gif okay okay what do we do is we first draw the graph of that function let me just take a hypothetical case so let's say this was the graph of your function okay correct let's say f of x graph is like this okay now what do you do is you make lines at a unit distance like this y equal to 1 y equal to 2 y equal to 3 y equal to 4 if the graph is going down also let's say I take a graph like this down okay then you have to make it at y equal to 0 is already made y equal to minus 1 you have to make y equal to minus 2 you have to make let's say suppose this many intervals then what do you do you start bringing this part let's say this part will fall like this yeah so it is falling on the floor think as if it is falling on the floor and hence it is getting the name of floor function because all these graphs start falling on the floor think as if this is the next floor okay this graph will fall like this this will fall like this understood okay so this is the graph you're the lines unit distance and use that for reference to absolutely absolutely so just start plotting the lines which are at a distance of 1 1 each and start making the part which is sandwiched between the two lines fall on the floor below it okay yeah understood how it works so I'll give you some questions okay plot the graph of y is equal to x minus 2 whole cube floor of this plus 1 and let's say mod on y try to include everything that we have studied so far okay should I do modern eggs also you just draw this in fact you just try without mod we will try to see what happens after we have done this yeah first of all this let's learn step by step so so first okay so first what I'm going to do is so why equals x of x minus 2 whole cube plus 1 so first we draw plus cube then shifted shift x cube two units to the right and one unit up so okay so the x x minus 2 whole cube so two units units out of the right then why you replace y minus 1 equals x minus 2 whole cube so what's that going to be you have to then you're going to move it to one unit one unit up so 2 is right one unit up then after that I have to take the but you're replacing your y with y minus 1 without taking the GIF on the function but after that if you take GIF then it will be including one also within the GIF bracket so that way that may become wrong yes that's why I think yes you don't take you don't shift across y axis you simply first do you shift the x by x minus x cube by two units to the right that's fine then you take the GIF absolutely the absolute then you have to take GIF correct correct so if so when x so first you take when x is when x is between zero and one no you just have to follow the method which I discussed with you okay so the rule is discussed follow the rule don't have to do the do it from the basics okay so first is this step right let's let's write it Arushi what you are planning to do first is this step correct if I'm not wrong then is this step if I'm not wrong correct and then the GIF step then is the GIF step if I'm not wrong yeah and then you're planning to do y with y minus 1 correct correct that's how we get to the our graph okay yeah absolutely correct let's plot it on the geojabra and see what is the chain of events okay this cluttered I will go to and open a new geojabra tab have you started using it in here I started using it okay so y is equal to x cube this is the graph okay undoubtedly now next is y is equal to x minus two x minus two whole cube whole cube so this gets shifted by one units to the two units to the right two units to the right correct then we have to do the GIF of this correct so GIF is floor of x minus two the whole cube you see that okay and again this can be very easily be drawn if you just start making lines like this if you just make lines y equal to 1 y equal to 2 etc so I'll just minimize this yeah so if you make a if you start making lines like this lines like this this this this and start making them fall on the floor next to it for example this guy will fall like this yeah this fellow will fall like this this fellow will fall like this the ladder length would become smaller and smaller yeah okay it'll start falling like this okay so that is how you get the graph for this okay fine understood okay next what you do is you replace your y with y minus one one minus one yeah okay so y minus one means you're shifting the whole thing one unit up yeah one unit up so minus one see how it'll move up okay right this torsion is because of the graphic issue okay okay so basically moves up yeah is that fine so the vertical lines that you see they will not be there it is just because of the graphic issue okay we'll try one more question or what if I take a mod on y what will happen just tell me what will happen if I take a mod on y okay if you take a mod on y then um so again so it will whatever was the one on the positive side of the y axis will remain the same but then it will remain the same but then it'll also get reflected about the x-axis down x-axis down very good very good let's just try to plot this again okay so we have abs y minus one is equal to floor x minus two the whole cube see the graph yeah you see this okay okay yeah so this is the ground that you'll see okay okay this simply got reflected simply got reflected about the x-axis and the negative part was removed okay try this one okay uh y is equal to plot y is equal to uh gif of one by mod x mod x minus one uh plus three okay minus two minus two okay well let me let me just put this entire thing under mod whole thing whole thing under mod yeah okay so first uh you would just uh okay so first the basic uh graph is going to be y equals one by x absolutely then then if it's y equals one by x then after you you shift it you shift x by three units so y so x is replaced by x plus three so you shift it three units to the left yeah three units left okay then after that x is then you mod x beautiful you find the actual x plus three okay so when that is done what happens you know what happens okay that's fine we'll leave it to the graph yeah now what is the next step then after that you replace x with x minus one again absolutely you shift it one use to the right yeah so once you have that um wait wait wait one second so you do that okay um yeah now you take the um the value the gif of this thing absolutely okay gif of that okay and you know what to do in that you just have to make lines parallel lines okay at one one unit distance and start making it fall on the floor yeah then okay then after that uh this is um okay hmm so this this thing is shifted it shifted it should have two units down down that means you are now doing this yeah very good two units down which is equivalent to saying that which is equivalent to saying that you are sending this minus two on the other side oh sorry there's a plus two so minus two will come on the other side okay okay now you do that you should have it like two units uh down and then after you take the modulus value of the whole thing that means whatever was below the x axis you send it above right and whatever was already above you don't disturb it okay yeah this is f modulus of fx yeah absolutely that's perfectly you know roadmap is perfect so let's try it out on geogebra first i will just plot it on the geogebra so first step is we are going to plot y is equal to just follow the changes that happen to the graph okay yeah so y is equal to one upon x this is fine yeah then y is equal to one upon x plus three so i'm just going to make a change over here x plus three okay shifted three units to the left yeah then you're just modding x so this y is equal to one upon mod x plus three yeah okay as you can see only this part of the line which was below the oh why did i write two plus two this plus three plus three yeah so this part of the line only gets reflected over here and the blue one disappears okay then x replace your x with x minus one means you're shifting this one way to the right as you can see one you're to the right yeah okay then what do we do we take the gif of the whole thing correct yeah so we can say uh y is equal to gif which is floor of q okay everything falls on the floor because it is it is below the one line okay yeah and now you're doing and now you're doing what you are taking minus two on this right yeah minus two so you will be doing y is equal to r function minus two as you can see let's come down by two correct yeah and then you're doing y is equal to abs of whatever function you have got which is s in this case it'll go to unit sub so it'll be a straight line y equal to two so much we did just to get a straight line so this is your straight line y this is the straight line y equal to two is that fine okay let's now let's now introduce you to a next uh important special function which we call as the fractional part function fractional function just a question and before we go to the fractional part what if i ask you to plot the graph of uh x is equal to gif of y okay so first and foremost uh it will be just use our inverse function concept right absolutely very good so first we start with this yeah y is absolute in the yeah gif of x and then simply uh reflected about y equals x axis reflect about y equal to x line absolutely correct so can we see how it looks it look up floor in the vertical direction yeah so since this is already cluttered i'll open a new page so x equal to x equal to floor of y yeah okay let's now go into the vertical way ignore the else yeah yeah simply this one okay okay yeah let's say if i want to draw um x equal to gif of y square minus three y plus two how will i do this yeah first you would simply plot y equals x square minus three x plus two then you would get the gif of that minus three x plus two you get gif of that then you reflected about y equals x line absolutely absolutely good i suggest a final question what if i do the gif on both okay so if you do the gif on both hmm so okay gif of y equals gif of x so basically if i put in uh like a value of x is as one no as um i don't know maybe like 1.5 then the gif of that would be um would be at two right so two would be two that give you a gif of x would give you two but then uh i have to find out the two equals gif of y wait wait two equals gif of y so um if two equals gif oh it'll be it'll be like a line um two equals gif y we align between uh the values of two and three really like we align between two and three uh think again carefully i mean uh can i say if x is between zero to one y should also be between zero to one yeah okay yeah okay so okay let's focus on this box zero to one and zero to one okay in this zone if i take any point let's say i take this point okay x will be somewhere between zero to one why y will be also between somewhere between zero to one so can i say their gif would be equal okay the gif would be equal so can i say it will be valid for any point which lies in this box excluding the tip so their gifs are equal okay fine for example if i take a point on the tip over here here gif of x will be zero but gif of y will be one so excluding the tip all the point would be included excluding the tip means excluding the corner i'm shading it with white okay so this white is excluded from this box okay okay and can i say the same thing would be true even if i make a box like this excluding the box itself excluding the edges of the box itself if i take any point like this let's say this point is uh 1.7 comma 1.6 okay can i say gif of x and gif of y will also be gif of x and gif of y will also be equal okay yeah okay can i say for can i say for every point within this box again excluding the edges of the box this condition would be satisfied okay and actually this continues on and on and you'll start getting not a graph per se but an area which will satisfy this okay well can we see the geojibra once geojibra actually doesn't show this but let's try it out okay let's say floor x is equal to floor y it doesn't show the complete region actually it shows you the line it's actually a shaded area it's actually a shaded area okay it's actually boxes like this floor y okay yeah this entire box will be your answer this entire box will be your answer except the edges of the box i cannot choose a point on the edge for example if i choose this point what will happen here x is 1 but y is let's say 0.5 correct so gif of 1 and gif of 0.5 they're not equal right yeah so these points will not be in my list similarly if i choose a point here let's say 0.2 comma 1 0.2 comma 1 so gif of 0.2 is not equal to gif of 1 correct so this point will also be not included so on the edges of this box the points lying on the edges will not be included just one within the box will be included okay okay so there was a question in itge where they asked to calculate the area bounded by this curve okay from some value of x to some value of x for example 0.24 so what are the area area bounded by this curve from 0.24 so as you can see in 0 to 4 unit square you'll have four squares yes yeah four unit square so four unit square would be the answer okay yeah so now we'll move on to the fractional part function okay so basically once i can give you back i just need to do that part once which is just the previous this one yeah okay so basically the the whole box underneath bounded by those two lines will be equal like the whole okay this this would be your area but not including the surface periphery of the box okay that is because of the the gif of x should be in that area the gif of x is equal to the gif of y yeah but not on the edges not on the edges not on the edges okay maybe in the maybe in the this edge it will be maybe on this edge it will be because if i take a value here y is always 0 right yeah so 0.2 comma 0 the gif of 0.2 and gif of 0 would be same but it would it would not be on these two edges these two edges will not be included okay but these two edges you can include these two edges you can include okay yeah so fractional part of x so fractional part function fractional part function is a function which is represented by curly brackets so whatever who's whoever fractional part you require you feed it as an input to it so let's say i take a very simple of simplest of all cases where i find the fraction part of x so what does it return by definition of this function this function returns the function is a machine you put some input and return you something correct yeah so it returns x minus gif of x okay okay so you can treat fractional part of x to be just x minus gif of x okay so for example if i ask what is the fractional part of 7.2 fractional part of 7.2 so it'll be 7.2 minus uh okay 0.2 yeah so 0.2 is the fraction part correct what's the fraction what's the fractional part of 8.867 0.867 0.867 what's the fractional part of negative 2.36 negative 2.36 okay no no no i'm just saying it out to myself okay so negative 2.36 it'll be negative 2.36 minus the fractional i mean the gif of this so this uh the smallest integer yeah the the greatest integer of fraction yeah correct okay fine it'll be um it'll be um one second one second i'm sure lesser than or equal to x okay so uh lesser than that would be so be minus 2 minus 2.36 minus of minus 3 so minus of minus 3 so 3 minus 2.36 okay that's 0.64 yeah okay yeah absolutely correct now given that you know this definition can you plot can you plot for me the graph of fractional part of x again any curly brackets please do not treat it as a fractional part unless until stated yes so the question will say this represents the fractional part of the x this represents the fractional part of the function okay so tell me how would the graph look like so at at um at one two three and four it would be at um zero yeah start looking at intervals as you rightly approached start looking at what happens between zero to one what happens between one to two what happens between one to three what happens between zero and minus one if you if you find for these intervals you'll know the trend of the graph so but but at one two three and four it'll be at zero right and then yes between zero and one what would be okay so if you think 0.5 um so let's say i am between zero and one then what will happen to the fractional part of x what will it return to me so 0.5 if you take 0.5 for example so 0.5 minus zero so it will be 0.5 only yeah so return 0.5 right 0.5 right it will basically just return that uh between zero and one basically just return the value of x absolutely so you got the answer so from zero to one can i say it will be just be a part of the line y equal to x yeah okay this is just a part of line y equal to x just a small part okay and exactly at one as you rightly pointed out this should fall down to zero it should come over here yeah correct so there will be a whole left over here very good now try from between one to two between one and two um okay okay one second between one and two okay add two again it comes to zero but if you say 1.5 for example 1.5 minus uh 1.1.5 minus one is 0.5 okay if you take point 1.7 also it'll give you 1.7 minus one if you take 1.9 also it'll give you 1.9 minus one so in general can i say it is giving you x minus one yeah okay yeah x minus so isn't x minus one is basically a line y equal to x minus one mean the same line shifted one is to the right oh nice okay now it will appear like this okay and if you follow the trend it will keep on going like this okay nice okay so this will be the graph of gif what will be the tip be at or what is the value that it will try to approach here at the tip um try to approach one one but it'll never become one it will be 0.99999999 but it'll never be able to get one value oh never touch one never touch one absolutely is it almost like an asymptote can you say that you you you cannot say asymptote this values are not achievable okay achievable okay so never so let's say i want to plot it on geogibra so it will be y is equal to x minus floor x there's no direct function for fractional part actually yeah as you can see this is the line that you see again those holes the graph will never show you yeah okay understood understood okay so now let's ask you some questions uh please plot the graph of of fractional part of x plus two minus three okay x plus two minus three okay fractional part of x plus two minus three okay so first so the first i have i'll have to plot um okay so this uh first i can plot the fractional part of x plus two okay i need to plot the fractional part of x plus two is going to be x plus two minus um gif of x plus two plus two minus first you'll plot this graph right okay okay fine okay so you first plot that then you shift it to the left by two units then you shift it down by three units absolutely done yeah into y plus three equal to this so you shift it down by three units understood so let's check on geogibra again so y is equal to uh y is equal to um y is equal to my f is already a fractional part okay so fractional part minus two you want uh no no no here the function is slightly different so again i'll write it no worries so fractional part of x plus two minus and then minus uh x plus two minus gif of x plus two the function was gi uh function part of x plus two yeah so you write x plus two x plus two minus floor of x plus two x plus two as you can see the graph has got uh shifted units to this right but you don't see an uh you know change in the graph because whatever was yeah whatever was occupied two units to the right the same moved not too used to the left so the graph doesn't make a difference okay now you change your y with oh yeah with um y um y plus three y plus three it'll go up yeah then no no you change your y with y plus three right so yeah for us it has to be from here have to be g minus three after right oh that's why okay so the graph will go three units down okay is that fine okay let's do another one do this very very carefully okay y is equal to under root of okay gif of x okay mm-hmm plus greatest india of x okay plus greatest integer of x yeah i'll be back in one minute okay let's try this first ask yourself whether you know this graph or not root x graph and it'll actually be this i already told you y square is equal to x it has two graphs for it y equal to root x minus root minus root x so this part is y equal to root x and the part which is below like this okay y is equal to negative root x so i just need the upper part so i just need this part of the graph okay that would be helpful helpful for you in plotting this okay part of the graph y equals in a curly bracket of x yes i can hear you um the first okay of x but how do we root this okay um root x y equal to root x yeah okay okay then what okay so after that after you do y equals root x then should i give you a hint okay work with intervals first see what will happen between zero to one then see what will happen between one to two like that that would be more helpful okay you know the trend once you do uh one or two uh intervals you'll know what is the trend of the graph yeah so i'll help you with this first try between zero and one what will happen to this function between zero and one okay so if you take uh zero point five for example you've only figured out that gi uh fraction part was just behaving as x and this will be zero only so it is as good as just plotting root x graph only for this interval okay only for that interval okay now what will happen when you're between one to two uh one to two um uh basically become x plus one oh yeah root of x minus one yeah because this this thing is as good as saying root of x minus floor x yeah yeah so basically basically between one and two x will remain x nothing will happen to it the floor x function will become a one right yeah x minus one root plus one okay fine so what will happen after that let's say uh two to three two to three okay then it'll be x minus um plus two absolutely very good then three to four three to four the x minus okay now NS evidence is there for us to plot the graph now let me start plotting the graph okay so for zero to one your graph will be just the graph of root x graphs only this part I will drop okay and I'll stop just before zero sorry just before one so I'll put a hole yeah so this graph is plotted okay what about this graph how do I plot y equal to root x minus one plus one plus one means you're doing this okay so can I say it is just the shifted version of this graph one unit to the right and one unit up yeah simply shifted okay so this part only will shift one unit to the right and one unit up so won't it appear like this yeah okay remember this the tail of the next one will fill the hole of the previous one so there will not be any discontinuity okay and same thing for between two absolutely absolutely keep on going like this okay yeah so it should be some kind of a you know cloud structure let's let's see okay so our function was y is equal to under root of x minus floor of x to the power of 0.5 plus x you see that yeah it will happen if I ask you plot the graph of I just make a site change in the question yeah plot the graph of y is equal to under root of fractional part of mod x fractional part of mod x gif of fractional part of mod x plus gif of no plus uh plus gif of mod mod x yeah fractional part that's not fractional part right it's the gif of mod x did I say by mistake fraction part sorry about that so it's under root of fractional part of mod x plus gif of mod x okay plus gif of mod x okay so first okay again deal with intervals you're already done 90% of the job you're just replacing x with mod x right so what should I do with this graph which I've already obtained okay all you do is you have to simply reflect it you have to reflect it about yeah you have to just you have to keep the positive side as this and simply reflect it correct so it'll become something like this yeah like batman yeah if I show you here uh so y is equal to uh p abs x correct and if I do a mod of p only what will happen let's say I do y is equal to absolute value of the whole p of x function inside the twist yeah whatever was below will go up yeah I just reflected yeah are you getting it yeah yeah so let's take a few more examples mod y minus one is equal to mod x minus one equals fractional part of it was fractional part of mod x plus one minus three minus three hmm okay so first you have um okay so once again I'll just make it slightly easy because okay let's let's do this one yeah let's do this one you already know how to do fractional part of x plus one correct yeah you already know this yeah okay so now my question is if somebody asks you fractional part of any function will you be able to plot it fractional part of any function yeah for example if I say fractional part of x square minus three x plus two will you be able to plot it yeah okay so you just first you plot x minus three x plus two and then you um find out this fraction part so basically yeah um okay so what would you do yeah so it would be um get into different intervals and plot it that way okay you first do uh between the interval of x and one you would do um x square minus three x plus two and between one and two it would be in the x minus one whole square minus three times x minus one plus two doesn't make sense right out I just want to see how much you can think okay so okay so am I going the right way first what is the step that you're planning to do first okay first do routine uh between the interval of zero and one okay but in that case uh calculating value for all the points would not be easy right okay I'll tell you a simpler way out let's say I want the graph of GI uh fractional part of this entire function so first of all I'll plot without it so I'll just simply plot the graph of a parabola like this okay now see what is happening in this case when you're doing a fractional part of x okay fractional part of the whole thing what are you doing you're trying to assign why some fractional some value which is between zero and one isn't it yeah between zero and one yeah so what is what are you doing actually okay so you're trying to assign it a value between a fractional part you are trying to assign the value between zero and one so if let's say this value was five point two four yeah and then when you're doing fractional part of this then what are you actually trying to do you're actually trying to do five point two four minus five so my take is wherever is the point you are actually bringing it down by the integer value at that position for example okay I'll just show you a simpler case here okay let's say I draw these lines y equal to one y equal to two and let's say y equal to three so any value here any value in this zone hope you can see the color and change the color to blue any value in this zone okay its value should be brought down by how many units let's say it is this value is 2.6 should be brought down to 0.6 yeah can I say for this value of x this graph will start falling over here like this why because you are bringing it down to zero to one gap because this answer can never exceed zero to one fractional part of anything should always lie between zero to one okay fractional part of anything we should lie between zero to one yeah it cannot be greater than one it cannot be equal to one also it has to be somewhere between zero to one right okay what you're doing every part you are bringing it down directly within the zero to one block so this fellow will fall down like this okay fine this fellow will fall down like this okay should start falling like this and you will be very surprised to see the actual graph also so y is equal to let's say I draw the function x square minus c x plus two okay this is the function I just see what I am going to do y is equal to c minus floor of c c of x I have to write I think let me open another one y is equal to x square minus 3x plus 2 minus floor of x square minus minus 3x plus 2 you see that yeah that's what I'm trying to explain over here see what is going to happen is minus 3x plus 2 what is happening is yeah see this part what was happening is this part over here I'll start with the shading of the graph this part this has fallen over here so this is the response for it okay yeah why is it falling over there again again because this value is somewhere between let's say this I talk about this zone it is somewhere between two to three correct so if I take any value like 2.7 yeah it should have given me 0.7 correct so this will fall by two units down over here okay this fellow this point will drop at this point okay it's gonna fall by two units down okay so every point in this zone every point in this zone Arushi is going to drop by two units down okay so this this guy will fall over here this guy will fall over here so like that every point will start falling on this line this okay okay are you getting it yeah wait that was 2.7 and now it's become 0.7 right so gif of sorry fractional part of 2.7 should be 0.7 right yeah so it started falling two units down similarly any any point between three and four will start falling three units down correct yeah okay so this guy yeah this guy yeah this guy will fall like this guy okay getting this point yeah the response on this green graph is your red graph okay that you see okay everything is fine here but what will happen to this part this is a negative already right yeah so remember the gif function if you are already dealing with a negative part so let's say if I say gif function what will be the gif function if you are already between minus one and zero okay so it'll be I'll be x so remember what what we used to do it's x minus this yeah I'll be x yeah x plus one yeah okay that means every point here will get a jump of one up oh it has come over at this point okay understood yeah okay very interesting graph correct interesting back to this so this will start you know making it like this but it'll become more cluttered towards the end because you know this graph is becoming more higher and higher yeah okay awesome so one last question before we close the session okay how would we draw the graph off let's say the same thing now the x has been replaced with fractional part of x okay so here y equals x square minus 3x plus 2 okay so here x has been replaced by fractional part of x yeah okay so if I first deal with x then um so it's very simple it's not as complicated as the previous one okay so if I take a value like um if I take a value like 1.2 I'll get back then I'm getting confused I don't get this see what is going to happen okay without the gif on x the graph was like this correct I'm just going to draw a miniature version of the graph correct yeah now no matter whatever x you put to the function your x will always be restricted between 0 and 1 right yeah so whatever is the part of the graph between 0 to 1 the same graph will start repeating every time okay are you getting this point okay okay because whatever x value put let's say you put x value as 2.3 then gif sorry fraction part of 2.3 will be just 0.3 yeah so at 2.3 let's say I take this as 2.3 it will draw the same graph as what it had drawn for 0.3 okay are you getting this point okay so it will start it will start plotting the same part which is I'll show you over here so now I have to plot y is equal to c of x minus floor x okay it is not showing any response for this I have to type it literally sometimes it irritates so x minus floor x square minus three times x minus floor of x okay yeah okay whatever I said the part of the graph which is between 0 to 1 this part is only repeated everywhere okay so the graph is closing its eye and only replicating this part everywhere because you are making a fractional part on x as your input to the function so whatever it had drawn for 0 to 1 gap whatever it has drawn for this gap it'll start replicating it everywhere okay okay so okay fine okay understood this yeah okay fine okay now before we sign off yeah I'll leave you with that question okay so we already learned how to I'll give you one second but uh one second I'll come back I don't have with me you can just put it out from my bag so we just learned the graph for this function right yeah okay what if I ask the graph for this function that means you're replacing your x with gif of x okay okay fine so this we know this we know how to plot yeah this this was a process we where we started making it fall on the floor remember yeah you make lines like this and we started making it fall on the floor like this okay but what do we do for such case okay so think about this and if you are done just show me on the can take a snapshot on the of the paper or pick up the paper where you've drawn it and send it to me on my whatsapp okay and tell me the answer for this okay okay fine so let's call it a day now okay we'll meet next Sunday or some plans of having classes in between also I don't mind coming up okay I'll coordinate with ma'am for that okay yeah okay I wish you bye bye


Welcome to George H. Smith's Excursions into Libertarian Thought, a production of Libertarianism.org and the Cato Institute, narrated by James Foster. Thomas Hodgskin, Libertarian extraordinaire, Part 1 The Englishman Thomas Hodgskin was born December 12, 1787. His spin-thrift father, despite making decent money as a storekeeper at the Chatham Naval Stockyard, managed to keep his family in dire financial straits, so he sent Thomas, who was barely twelve, to serve as a cadet aboard a British warship. Although he served with distinction during the Copenhagen expedition and rose to the rank of lieutenant, Hodgskin detested his twelve years as a sailor. For one thing, it deprived him of an education. His access to books was limited, so he could do little more than reflect in the midnight watch on the solitary deck on the wide ocean amidst the wildest or the most peaceable scenes of nature before I had acquired a sufficient stock of material. Hodgskin's independent spirit, his intense dislike of unjust authority and his determination to make a powerful resistance to oppression every time I was its victim, were not well-suited to the rigors and harsh discipline of naval life. Thus, when it became clear that he would be passed over for promotion, Hodgskin complained of the injury done to me by a commander-in-chief to himself in the language that I thought it merited. This, of course, only made matters worse. Hodgskin, at age twenty-five, was forced into retirement at half-pay, after which he wrote an essay on naval discipline, 1813, a scathing indictment of conscription and the brutal conditions endured by British sailors. Hodgkin's experience with the horrific punishments inflicted on British sailors for even minor offenses caused him to question both the justice and utility of the supposed right to punish. British sailors, who typically hailed from lower-class backgrounds, were frequently pressed into service against their wills and their officers tended to view them as brutes who could be controlled only by the lash. Hodgkin, who by this time had read John Locke, William Paley, and other moral philosophers, had a different opinion. Humans created by God with similar passions are everywhere made alike. Many individual differences are caused by different social and political environments. If the English tended to be happier and more virtuous than people from other nations, this was largely because they were less governed than other nations. And if English sailors appeared more brutish than other Englishmen, this was owing not to any inherent defects in their natures, but to the barbarous conditions of naval life. In short, if you treat men like brutes, they will behave like brutes. Press gangs and conscription, according to Hodgkin, should be abolished and replaced by voluntary, short-term enlistments. Pay should be increased so that sailors can afford a decent standard of living, and the draconian penal laws of the navy, which were applied arbitrarily without recourse to due process, should be eliminated. If there must be punishment, then the navy should follow the example of the civilian courts in England, which do not punish the innocent, and which are administered according to impartial laws, not by the whims of superiors. As Hodgkin, who was remarkably frank about his motivations, would later note his essay on naval discipline was an emotional reaction to his unjust treatment by superior officers. I was angry at being punished when I thought I was doing the duty of a good man and a good citizen. Hodgkin used his anger to spur further investigations into the right of punishment. This anger made me read books on the subject, and I sought in vain. I sought and still seek in the writings of celebrated authors for any justification of the right to punish, and the result of that was a system of opinion which, as far as I have read, may be called on the whole peculiar. Hodgkin would later elaborate upon these peculiar opinions in the pages of The Economist, after he became senior editor of that journal, the same one published today in 1846, a position he would hold for eleven years. Hodgkin's many articles for The Economist made it one of the most interesting and provocative libertarian periodicals of its time. In addition to supporting laissez-faire, voluntary education and other classical liberal causes, Hodgkin also opposed capital punishment and questioned the traditional wisdom about the efficacy of punishment as a deterrent to crime. According to Hodgkin, most crimes are motivated by a desire to escape intolerable poverty, and such poverty is often the consequence of taxes, economic regulations, and other governmental restrictions of free market activities. If people were allowed to pursue their own interests through voluntary interaction with others and if they were allowed to keep the fruits of their own labor rather than having much of it expropriated by government, then a good deal of poverty, along with a major motive for criminal acts, would be eliminated. Around 1815, Thomas Hodgkin became friends with Francis Place, the famous tailor and working-class radical of Charing Cross Road. This friendship gave Hodgkin access to London libertarians, including Jeremy Bentham, his Scottish protege James Mill, the father of John Stuart Mill, and an elderly William Godwin, who had created a sensation with the first systematic defence of philosophical anarchism and inquiry concerning political justice, 1793. Hodgkin felt that his age and other obstacles would make it difficult to embark on a new career. Medicine would require that he learn Latin, whereas law would require that he smother his love of justice. So, given that peace had returned to Europe in 1815, Hodgkin decided to take the grand tour and write a book based on his experiences. The result was his first major work, Travels in the North of Germany, published in two volumes in 1820. These volumes combined interesting observations and his political commentary on the much-governed countries of Germany. Hodgkin emphasised the inefficiency and waste of government projects. For example, he commented on the inferior quality of state-financed roads in Germany, contrasting them with the privately financed roads in England, and he even suggested that police functions should be placed in private hands. As Hodgkin put it, all business of men, what promotes their prosperity, is always better done by themselves than by any few separate and distinct individuals acting as a government in the name of the whole. When it is remarked that the prosperity of every nation is an inverse proportion to the power and to the interference of its government, we may be almost tempted to believe the common opinion that governments are necessary and beneficial is one of those general prejudices which men have inherited from an ignorant and barbarous age and which more extensive knowledge and greater civilisation will show to be an error full of evil. The hundreds of German principalities were the consequence of artificial political divisions, but the solution was not to unite them under a single conqueror or central government.  Instead, the Germans need only chase away their different masters to make them all sensible that their interest is everywhere the same. Thus did Hodgkin defend the spontaneous social order that emerges from private property, voluntary exchange, and the natural harmony of interests, as explained by Adam Smith and other political economists, while excoriating the square mechanical unity of despotism and economic regulations that transfer wealth from the productive class of labourers to the unproductive class of rulers. Hodgkin did not recommend a parliamentary system such as that found in England as the cure for Germany's problems. The key to England's prosperity lay not in its parliament but in a vigilant public and a free press that acted as a continuous check on the abuse of power. The evils of society cannot be remedied by acts of parliament. Even public assistance laws and proposals for agrarian reform, however well intended, will not accomplish their intended purposes. Such measures merely create the illusion of reform while placing more power in the hands of legislators. The simple means of making the race frugal is to supply the wants of no man and to leave every man the produce of his own labour. The Germans, like many Europeans, had a false conception of political economy, according to which prosperity can be promoted by means of government. In truth, prosperity can come about only through human labour and voluntary exchange, and those social regulations ought to be exposed to censure which have inflicted on us so much poverty and distress. Hodgkin was living in Edinburgh when travels was published, and this city had long been a centre of liberal agitation. While his wife, whom he had met on his travels, was giving German lessons to supplement the family income, Hodgkin was doing his part by writing articles for the Edinburgh Review and other liberal journals. He wrote to Francis Place, undoubtedly the abolition of all restrictions of whatever kind is the great point to be aimed at. We want a destroying legislature whose great business would be to do away with the enactments of their predecessors. Although at this time Hodgkin had a good deal in common with other liberal reformers, his anarchistic tendencies often gave his opinions a radical edge that alienated other liberals. This tension became evident in his reaction to the Peterloo massacre of August 16, 1819. After a coalition of middle and working class reformers organised a mass meeting at St. Peter's Field in Manchester to call for parliamentary reform, the crowd was attacked by mounted soldiers who killed 11 people and wounded around 400 others, including 100 women. Although this incident was universally condemned by liberals, they typically focused on its supposed illegality and condemned it as a violation of the British constitution. Thus, as Hodgkin saw the matter, liberals, while explicitly criticising the government, were implicitly supporting the system of unjust law on which the government depended for its legitimacy. As he explained to Place, the horrid violation of the laws at Manchester seems only now to be a cry and a watchword to support them. Productive as it has been of miseries, it is our miserable constitution we are now told we must defend and support.  I should like to know the single law that is worth an honest man's struggle. I do not know one that of itself is worth supporting, but all men seem to think it is better to be sabered by hussars or shut up in Bastille's according to rule than trust their fellow men. They seem to think it better to be fleeced according to rule than to run the most remote possibility of living according to reason. In an article which he submitted to the Scotsman in January 1820, Hodgkin praised Smith, Malthus, Bentham, and other political economists who had shown the absurdity of almost every regulation and as a necessary consequence had weakened the respect of all men for the authority from which these absurdities had emanated. But the Scotsman declined to publish this article, probably because Hodgkin pushed his critique of law into an anarchistic direction that few liberals could accept. Whereas Jeremy Bentham and his followers had criticized bad laws as the consequence of sinister interests, that is the landed aristocracy, Hodgkin extended this perspective to all governmental laws while contrasting them with the natural laws of social interaction. All governmental laws are designed by ruling oligarchies to further their own private interests and these laws conflict with those natural laws of society that serve the general interests of the people. This recurring theme would later receive an extensive treatment in Hodgkin's most important book, The Natural and Artificial Right of Property Contrasted, 1832.