 So good morning, everyone. So welcome to another session on functions. In this session, we are going to talk about periodic functions, periodic functions, not a new concept for us. We have been already introduced to trigonometry in class 11 where we saw periodic functions, trigonometric functions, which were actually periodic. So all the trigonometric functions, standard functions that we had learned in class 11, they were all periodic. So how do you define a periodic function? So a function, let's say defined from x to y, is said to be periodic. This function is said to be periodic. Is said to be periodic. If there exists a positive real number t, such that f of x plus t is equal to f of x for all x belonging to the domain of the function. For all x belonging to the domain of the function. And this quantity t is called the period of the function. And this quantity t is called the period of the function. It's called the period of the function. Please do not use the word time period. It is not a time period. Had your variable been the variable of time, then yes, then it will become a time period. But in maths, we just call it as a period. A few things to be very carefully observed over here. One is your t should be a positive real number. That means it cannot be zero also. Because if it is zero, then every function in this world will become periodic. Second thing is this relation must hold true for the same t for all value of x in the domain of the function. So these are the two things that we need to watch out for. Now why am I saying for all x belonging to the domain of the function? See, if you look at sine x function. If you look at sine x function. If I just focus on one single value, let's say zero. Let's say I want to see when does sine x become a zero. It becomes a zero at zero. It becomes a zero at pi. It becomes a zero at two pi. It becomes a zero at three pi. So for all n pi, the function sine x becomes zero. Does that mean the period of sine x is pi? Does that mean the period of sine x is pi? The answer is no, because it must satisfy the same relation of repetition of the value must satisfy for all values of x in the domain. For example, if I take a value half, half comes from 30 degree. Or if I take a value, let's say one which comes from 90 degree. Now I do not get a value of one again when I increase the value of x by pi. So if I move to three pi by two, I will not get a value of one. In fact, I will get a value of minus one. So the next value that I get here is when I take it to five pi by two, then only I will get this one back. So in this case, the gap becomes two pi. Now can I say two pi is the period? Yes, if you take any value of x, the sine x value will get repeated if you change your x by a value of two pi, whether in the forward direction or in the backward direction. Okay, now what is period? Period basically is the value for which the function is repeating itself for every value of x in the domain of the function. In this case, you saw two pi could be a period. In fact, four pi could also be a period, six pi could also be a period or two k pi could be a period. Are you getting my point? Now there is something called fundamental period. So what is the fundamental period? So in this case, you can say two pi is the period, I will come to fundamental period. So this is your period. What is the fundamental period? Fundamental period is the smallest or the least value of t for which the function repeats itself. So the least value of t such that f of x plus t is equal to f of x for all x belonging to the domain of the function and of course t should be a real positive number. So it is the least value of t. So yes, two pi, four pi, six pi, eight pi, ten pi, twelve pi, etc. They'll all be period of sin x, but the least of them is two pi. So two pi will be the fundamental period. Okay, many a times the word fundamental may not be used. They will just use the word period for it. Okay, any questions here? So many a times the word fundamental. I'll write it down. Not too many. Fundamental word may not be used. Word may not be used sometimes. Okay. I have a doubt. Yes, sir. Sir, can you drag a little bit? I had to ask something there. Yeah. Yes, sir. So what is that inverted e? Inverted e is there exists. There exists. It is a symbol for there exists. Okay. So I'll give you a list of some periodic functions that you'll be coming across in mathematics. Okay. But these are the basic ones. These are the basic ones. There'll be complicated questions given to you which will involve various combination of different types of functions. Okay. So we'll take those questions as our problem questions. So let me give you a list of periodic functions. List of standard periodic functions. Okay. So now let me begin with. Let me begin with. The trick functions that we had. Done in our. Now when I say period, I mean fundamental period. Okay. If I just use the word period. It means fundamental period. Okay. When you talk about sign to the power nx, what do you think is the period of sign to the power nx? So if n is positive, then it will be part. Okay, let's say n is an actual number. No. So if it's even, then it will be part. Okay. Okay. Very good. If n is odd, then the period here would be. Two pi. But if n is even, your period becomes pi. Okay. I'm sure you have seen. I'll just show you the graph of these functions when you're changing the power. Let's say I talk about sign x. Okay. Yeah. So for sign x, you can see the period is two pi. Okay. So look at the crest difference that is equal to two pi. Even if I make sign cube x, let's say I make a sign cube x. Oh, let me raise it to the power of. Yeah. Now the difference doesn't change. Still this difference is equal to two pi. Of course, there are harmonics introduced in the function, but the difference between the distance of x between two points in the same phase is two pi. But if we have a power of, let's say an even number, and let me just show you that even if you keep it five, it will not change. Yeah. Okay. The period is not getting changed. It is still two pi. Seven. Not getting changed. But I square it. Let's say this. As you can see now the distance between two consecutive crests will become pi. Okay. Now note the difference between the graph of sign square x and mod sign x. Many people make a mistake. Mod sign x graph would be, would be having kink kind of a structure like this. It's better to show on the tool itself. Many people draw the same graph. Sign square graph is differentiable everywhere, but not mod x, mod sign x graph. Okay. See the difference. Okay. In the past, I have seen many students. They draw the same graph for sign square x and mod sign x. They're not the same, by the way. Yeah. So let me hide this first and let me increase this power from two to four. Okay. There'll be no change in the, there'll be no change in the period. It'll still remain a pi. Yeah. Just becoming flatter and flatter here. Okay. So yes, rightly said by Param. When n is even, it is pi. When n is odd, it remains two pi. What about cos to the power nx? Can I say the same trend will be seen for cos as well? So if n is odd, the period will be two pi. And if n is even, the period will be pi. What about, what about tan to the power nx? Or tan x to the power of n? Yes. What's your opinion about tan to the power nx? Yes. Yes. What is the answer for this? Is it pi? It's always pi. Yes. It's always pi. It's always pi. Okay. And let me tell you the same will be the story even for the reciprocals. The same will be the story even for their reciprocals. Okay. So let me write it down here. Same will go for cos to the power nx. Same will go for seek to the power nx. And same will go like tan to the power nx even for cot to the power nx. Okay. Good. What about a constant function? What about a constant function? Is it a periodic function? It's not, right? It's a periodic function, but its period is not defined. Its period is not defined. Don't say zero because zero cannot be a period. Period is always a positive real number. For constant function, any value of period will work because it is constant throughout. Okay. So it's like a degree of a zero polynomial, which is not defined. It's like the argument of the complex number zero, which is not defined in the same way period of a constant function is not defined. But yes, it is a periodic function. It is a periodic function. Periodic function. Okay. All right. Now, what about mod of sin x? We have just seen mod of sin x graph. What is the period? Pi. Okay. Didn't we see that in the GeoG graph? Yeah, this one. So as you can see, the consecutive cases are coming after a gap of pi in the value of x. Okay. Same will go with mod of cos x. Mod of cos x. Yeah. So again, the difference between consecutive cases is pi. So fundamental period for mod of cos x will also be pi. So very interesting questions will be framed on this concept. We'll talk about it later. All right. Now, what do you think? Can you give me a function which is algebraic and which is periodic? I think one is a constant function I gave, but its period is not defined. So tell me a function with a defined period and it is algebraic. No trigonometry involved. The GIF function is not periodic, but the fraction part function is periodic. Yes. Yes. Fraction part function is periodic. Okay. Also called as curly bracket x. This is a periodic function with a period of one, with a period of one. So these are some of the commonly seen functions which are periodic. We'll talk about, we'll talk about the combinations of different types of functions. But before that, we have to look into the important facts about periodic function. Some important properties related to periodic function. Okay. Have you all copied this down? Can I move on to the next page? Yes. Yes, sir. Important facts about periodic function. The first thing is, if you know your function f of x has a period of, let me write period on the top, has a period of t. Okay. Then what will be the period if you constant or subtract a constant from the function? Will the period change if yes by how much? And if no, then of course, why? Okay. The answer is it won't change. Why, why will it not change? The adding a constant is shifting the graph up or down. Yes. So what is, what is basically period? Period is seeing when is the function retaining its previous value again? Or when is, when is it repeating itself again for all values of x in the domain of the function? So if you lift the function up or down, the repetition still occurs in the same interval of t. Okay. That change will not reduce or become more. Okay. What about, what about if you multiply a function with a, will this change the period? No sir, it will be the same. It will be the same because multiplying a function with a just changes its amplitude. It again doesn't change the interval. See it's like, you know, on the x-axis, nothing is happening. If there is a contraction or expansion of the graph along the x-axis, then of course, period is getting changed. But if that gap is not changing, the span along the x-axis is not getting disturbed. Then the function period is going to be the same. What about if I do this? What about if I do this or this? Is it the same? It remains the same. Very good. Okay. So I have a question. Yes. So, so like, is f dash of x plus t also equal to f dash of x for the periodic function? f dash of x plus t also equal to f of x. No, no, no. The periodicity may change. For example, But like, but we can say that f of x minus dx is equal to f of x minus dx plus t and x plus dx is also the same. So then the derivative should also be the same. See, let's talk about the function fractional part from zero to one. Okay. The period there is one. Nice. But when you differentiate it, it becomes a constant function. In that case, the period will be undefined. So it may hold true in some case. For example, if you look at sine x derivative is cost, both period are same. If you take, if you take tan x derivative is, sorry, period is pi secant squared, period is also pi. Okay. So this is the same, but it may not remain the same for all the, the, the trig functions. So I don't think so, whether that comment can be made that the derivative of the function will also have the same period as the given function. So like just for first principles, if I take like f of x plus dx and then I know that my derivative is just going to be f of x plus dx minus fx divided by dx. So if I take that entire thing plus t, it will still be the same f of x plus dx plus t is equal to f of x plus dx. So it should be the same. Now, when you add or subtract functions, the period gets changed. For example, I'll just give you a simple case. Sine squared period. What is sine squared period? All right. All right. Can you add them? What, what does it become? Constant. Constant. Whose period is undefined. Right. So both functions having a period of pi doesn't mean if you subtract or add them, the period will remain a pi. Right. So those variations can happen. Okay. Getting the point. But yes, there may be cases when the period is not disturbed, but I think this rule cannot hold for every function in this world. It's better not to make this as a rule. What's wrong with the logic there? Yes, that's what I said. When you're applying first principles, right? In this case, the function remains the same, but so probably when the function is piecewise, it may make a difference. Just like your GIF was, sorry, a fraction part was piecewise, then it, then it was different. For non-piecewise continuous function, I think so this rule will work. Whatever you're saying. Okay, so. Okay. But it implies that all the higher order derivatives will also be the same. Yes. In that case, if one is true, then for all the higher order derivatives that will remain the same. Yeah. Okay. Okay. Now when you multiply, let's say X with a B, then what will happen to the period? It will become T divided by B. It will become T divided by B. It actually becomes T divided by mod B. Why? Because if B is a negative quantity and if you just divide by B, your period will become negative because T is already a positive real number. Okay. So it's always taken as mod B. Why am I writing? Yeah. Mod B. The moment you write a mod, you feel like you're writing the magnitude of a vector quantity. Yeah. Can somebody prove this? Why it happens? Of course, there are intuitive proofs, but I want you to do it mathematically. No intuitive proofs. You can't give examples like, okay, see sine X and sine 2X. Sine X has a period of 2 pi, sine 2X has a period of pi are not that kind of a proof. Yeah. Mathematical proof for it. Sir, I got a way to do it. Yeah. How do you do it? We write f of X plus T is equal to f of X and then we write f of B into X plus T by B. Okay. Once again. So for this, the proof is. Okay. First of all, T is the fundamental period for f of X. So f of X plus T is f of X. Then what did you do next? So f of B into X plus T by B. F of B into X plus T by B. X plus T divided by B. So only the T by B. Yes. So just the T will have the divided by B. Yes. So and then we open up the same thing in a different way to express it. We express it as f of is equal to f of B X plus T. The same term. Okay. So now let me just write it in a more logical way. See, let's say if I take the function as f of B X plus C. Let's say I take a function like this. Okay. And let's say I claim that this function's period is T dash. Okay. That means f of B X plus T dash plus C should give me f of B X plus C. So it becomes f of B X plus C plus B T dash is equal to f of B X plus C. That means if this function is repeating itself, that is f of B X plus C is repeating itself. That means BT dash should be equal to T. Okay. So let's say this function f X is repeating itself after a period of T. So this guy f of B X plus C is repeating itself after a period of BT dash. So BT dash should be equal to T. That means T dash is equal to T by B, so we put a mod B in order to make it positive. In order to make it positive. Right. So you can replace this with a capital X if you want. If you can replace this with a capital X if you want. Okay. So f capital X or f X will have a period of T. It doesn't matter whether you're changing the name from small X to capital X. So BT dash will be equal to the old period that we had, which is T. So T dash will become T by mod B. So this property, it will be applied at so many places. So please make a good note of it. Any questions here? Next thing is basically a question that I would like you to answer. Let's take a question here. Let's say I have a function. I have a function which is. What is the sum of sine X plus let's say tan X? Okay. What is the period of this function? What is the period of this function? How will you find a period of sine X plus tan X? Sine X, you know, is periodic with 2 pi cos tan X, you know, is periodic with pi. So when you add this function, what is going to be the period? There is a bell which rings after every two hours and there's a bell which ring every one hour. Right? So let's say both the bells have rang now. When will they ring together next? That's what it's trying to ask you. Of course 2 pi or 2 hours. Isn't it? Okay. Let me make a few changes to the question. So the answer here is 2 pi, which is very, very, very easy to solve because you just had to see which is the higher of the two because they are multiples of each other. But what if I say there's a function? There's a function which is let's say sine of let's say 2 pi X by 3 and there is a tan pi X by 2. Yeah. What is the period of G of X function? What is the period of G of X function? What will you say? 6. Okay. So what did you do? You first found out the period of this one, T1, which was actually 2 pi divided by mod of 2 pi by 3. Remember the previous property that we did? That will give you a 3. Similarly, period of tan pi X by 2 will be pi divided by mod of pi by 2, which will actually give you a 2. Then what did you do with the 2 and a 3? Of course, as I can see from your answer, you have taken the LCM of T1 and T2 and the answer came out to be 6. Very good. This answer is correct. Okay. So this is what we normally do. Now when I say normally mean there are some abnormal cases also. Okay. So in general, what do we do is? In general, what do we do is? If you have been given a function, if you have been given a function which is made up of some or difference of various functions or even product or quotient of various functions where those somewhat difference could not be further reduced. Are you getting my point? See, many times as I said, sin square plus cos square, if somebody gives you and if you apply the same logic, your answer should come out to be pi, isn't it? Right? Because sin square period is pi, cos square period is pi. So the LCM of pi and pi will give you a pi. In that logic, the answer should come out to be pi only. But the answer is not pi. The answer is actually undefined because sin square plus cos square is further simplified to 1. Right? Even if you take sin x by cos x, sin x has a period of 2 pi, cos x has a period of 2 pi. Correct? But sin x by cos x becomes tan x whose period is pi. Okay. So when such kind of simplification is happening, then in that case, this rule which I'm giving you will not work. Okay. So let's say if a function f of x has a period of, has a period of let's say p by q and a function g of x has a period of let's say r by s, r by s. Okay. Then, then the period of the combination of these two functions. The period of the combination of these two functions will be LCM of p by q and r by s. Okay. Provided, provided there does not exist a positive number k. That is very important to write here. Provided, provided there does not exist a positive number k. Okay. Less than the given period that you get over here. Let's say I call this as a t. Okay. Less than a t for which or such that such that f of x plus k plus g of x plus k is f of x plus g of x. Now it is this provided condition which makes the game little tricky. Else it would, this chapter would have been a, this concept would have been a very straightforward concept. Because all you would need to do is take an LCM. By the way, how do you take LCM of fractions? How do you take LCM of fractions? I'm sure everybody knows this condition is not, that doesn't make much sense. Right. So because like what if just by chance you had one value which aligned. Sorry. So like, when, when we found the period of sine x, like let's say I took sine of pi by six sine pi by six is equal to sine pi minus pi by six. So in this case, like, if you had something of that sort where just by chance two things just came out to be the same even though it's not in one period then it'll break down there. I could not follow you can just repeat your example once again. The example is for one function. I can't think of one involving two functions, but like, like, let's say I had sine x. Yeah, the period is five. And now I take sine pi by six, and then I find that sine pi minus pi by six is also equal to sine pi by six. So then I can say that that value is like, does it hold true for all values of x in the domain or for some certain values? If it holds for certain values, it's the equation kind of a thing which is which will give which cannot be called as a period. Period should hold for all values of x in the domain of that function that same interval change should bring the result for no matter what value of x you're looking at. Are you getting my point here? Am I able to answer the question that you were trying to ask me? Yes, sir. Yeah, I thought that even if you found one example then you can't do it. An example, two examples, even hundred examples will not work. It should work for all values of x in the domain of the function. That's why I categorically underlined that word for all x because people think that even for few of it it's true then it'll be true for everything. So we will take a lot of questions where such anomalies will be seen. And I'll tell you why those anomalies are seen for those examples that we'll be taking up. By the way, yes, how do you find out the LCM of two fractions? LCM of two fractions are found out by using this formula LCM of their numerators by SCF of their denominators. So when you're finding LCM of two fractions, you take the LCM of the numerators, provided they are in the very simplest forms and divide by the SCF of the denominators. SCF of QNS. Now, let me take another example. So one example was this. Another example was this. Let me take this question. Let me take this question. What do you think is the period of, what do you think is the period of tan x plus fractional part of x? What are the period of tan x plus fractional part of x? What is the response? Please put your response in the chat box. What are the period of tan x plus fractional part of x? Nobody's answering. I'm expecting actually, yeah. Okay. So Rajdeep has given the response. What about others? No solution. Okay. Now, what many people think, okay, this has, this is periodic with pi. This is periodic with one that we have seen the list of periodic functions and there we had discussed that. x minus gia function, which is your fraction part function is periodic with period of one. Now, many people make this wonderful statement that LCM of pi and one is pi. This is absolutely blunder. You know why this particular mistake is being made by students? Because they think LCM is just the product. No. If you see the definition of LCM is the lowest common multiple. The word multiple means integral multiple. That means you're trying to claim that if pi is an LCM of one, that means pi could be obtained by multiplying one with an integer. Can you tell me which integer is such that you can multiply with one to get a pi? Right? Such integer. And Ruchir thinks that pi is 22 by 7. Ruchir, 22 by 7 is just an approximation to pi. Pi cannot be expressed as a fraction. Okay. So pi cannot be expressed as a fraction. It cannot be expressed as a P by Q form because it's an irrational number. And 22 by 7 is an approximation to pi. In fact, there are better approximations to it. If you see the evolution of how we came up to 22 by 7, you realize that in the past, a lot of approximations have been made for pi. Yes. The answer is you cannot find an LCM of pi in one. You cannot find an LCM of, to be very specific, you cannot find an LCM of a rational number and an irrational number. Rational number and irrational number LCM does not exist. Cannot be found out. Okay. So in such cases, what we have to claim this function is non periodic. We have to say this function is non periodic in that case. Non periodic. Got the point. Okay. So let me begin with some questions. Let me begin with some questions. I'm not done with my properties, but I'll come back to questions in some time. Next property is, next property is if I think property number, I have lost track of the property number. Was it property number three or four? If G is periodic, if G is periodic, then what can you comment about FOG? Now F may or may not be periodic. Then what can you comment about FOG? Will it be periodic? First question, will it be periodic? Yes, sir. Okay. So you all agree that FOG will also be periodic? It's both F and O. I'm not saying F is periodic. F may or may not be periodic. If G is periodic with a period of T, then what can we comment about the period of FOG? T only. T only. Okay. Yes, sir. T. T only. Okay. Now the answer is it may not have a period of T. Now I'll give an example. If let's say F itself is a periodic function. Let's say I take cos of sin x. Okay. Now if this is your function FOG, then sin x has a period of 2 pi. We all know that. But FOG has a period of pi. Would you like to test it out? Okay. So just change your x with x plus pi. See what will happen. It becomes cos of minus sin x. I hope you know sin x plus pi is minus sin x. And cos of minus sin x is as good as cos of sin x because cos doesn't care about the negativity of its argument. Right. So you started with a function and you came back to the same value after a jump of pi. So please make a note of this. This is a direct theory base question which Jay can ask you and people will say. Same is same as period of T. No. It need not have the same period as T. But yes, you can find hundreds of examples where it will have the same period as T. For example, e to the power sin x. Right. So if you have e to the power sin x as your composite function, then yes, it will have the same period as a period of sin x, which is 2 pi. It is because this function e to the power x itself doesn't is not periodic. Okay. So it follows what the input function period is because it cannot dictate its own periodicity to the total function. Okay. So e to the power sin x follows the same period as what sin x period is. Got the point. So here we need to be careful. So this is a stage where we need to be careful. Okay. Next is a very simple and trivial property. A continuous periodic function is always bounded a continuous periodic function periodic function is always bounded. Please don't say a periodic function is always bounded. Then tan x will come out as a classic example to violate that. Yes, sir. Will it be periodic with LCM? LCM. Again, cos x has a period of 2 pi sin x has a period of 2 pi. So cos of sin x has a period of pi 2 pi 2 pi period should be 2 pi only. No, but it is not working in this case. Okay. Okay. Okay. All right. So I think we have we have done enough properties time to implement it in solving questions. The first question that I would like you all to solve is this find the period of the following function. Find the period of the following. Of the following. Let's take the first question as mod sin x plus mod cos x. Done. So I'm getting two types of answers. Pi, pi by 2. Okay. Now if you go by the logic that you had learned a little while ago in the property, this has a period of pi. Okay. This has a period of pi. Okay. So the game looks very simple. When you say the LCM would be pi only. But let me tell you this answer is wrong. Actually, the period is actually the period is pi by 2. You know why it happened? It is because see there exists a number lesser than pi for which the function is repeating itself. Now, why there exists a value k lesser than pi for which this function repeat itself is because these functions are interchangeable. So when your functions are interchangeable. Now why we say why we say a period is pi in or why we say the period is the LCM. Okay. Because we wait for this function to come back to itself and we wait for this function to come back to itself. So after pi mod sin x will become mod sin x and after pi mod cos x will become mod cos x. But we forget the fact that for a lesser value change mod sin x can become mod cos x and mod cos x can become mod sin x. So overall the function will remain the same. Are you getting my point? See when you have some of two functions. When we say the period is the LCM of let's say this period is T1 and this period is T2. And when you say period of the combined function is the LCM of T1 and T2. We actually wait for both the functions to come back to this. So after a jump of T in x, g of x will become g. After a jump of T in x, h of x will become h. So your entire function will repeat itself. But what if there was a shorter value let's say T dash for which this became h of x. Let's say T dash and this became g of x. Then also this sum of the two functions will be the same. That means the combination will remain the same. So we have to look out for us value smaller than that value. Now many people say there will be so many values smaller than pi. How can we find those values? See normally it is seen it is half of that value. Okay normally. Again I cannot make a rule out of it. So whenever you are getting a period you just try checking for half of its value. Many times options are also given. So when you see an option lesser than what you are expecting it to be. Check for that lesser of the option. That is the safest way actually. Okay so this is a big eye opener for us. The answer here is pi by 2 not pi. This is your answer. Got the point here? Let's take more questions. Let's take more questions. Find the periods of the following function if periodic. Let's do all of them. Let's start with A. I would like you to give me the answer for A. Okay so we have three functions involved here. E to the power. You can treat it as ln only. So it is just sine x I believe. Sine x tan cube x and minus cosec of 3x minus 5. So here we will f of x not be defined for sine x between pi and 2 pi. Because then log sine x is not defined. So undefined functions when the function is missing at a point. Can't it be periodic? I just wanted to ask if you just like when we take it we don't just take sine x. We take sine x from. It's an identical function. It's not an equal function. That's a very good thing that you have pointed out. There is something called identical function and equal function. Identical function they may not have the same domain. But ultimately on simplification they become the same. A classic example of this is 2 log x and log x square. They are identical functions. I should not write equal to in between. They are identical functions actually. What makes them identical is because see ultimately you say this will give you this but no. This function will work for all real values except 0. But this function will only be defined for x greater than 0. So they have different domain. So these two functions are identical but not equal. They are identical but not equal. But not equal. Equal function should be having the same domain as well. Are you getting my point? For example sine square x plus cos square x. This is equal to 1. These two functions are equal. But sine square x minus tan square x is identical to 1. Because this will remain 1 for all values of 1. All values of x. If I call this as a function f of x and I call this as a function g of x. f of x will remain 1 even if x is any real values. But here we cannot feed multiples or multiples of pi by 2. Are you getting this? They don't have the same domain. So they are identical but they are not equal. Are you getting my point? But for the purpose of evaluating this thing. Period of the function. You can treat identical functions to be the same. Even if you don't treat it the same. You will still conclude that this period is 2 pi. Period of e to the power ln sine x is 2 pi. Tan cube x period is. I hope you understand when I write an arrow like this. I think we are talking about periods. So you can be known that I am referring to their period. Cos x minus 5 period is 2 pi by 3. Now I am sure there is no intra simplification happening between them. And even if it happens it doesn't lead to the change of period. So the period of the combination would be the LCM of these 3 guys. 2 pi by 1, pi by 1, 2 pi by 3. So LCM of fractions is LCM of its numerators. Which is 2 pi pi 2 pi by SCF of 1, 1, 3. So that is actually 2 pi by 1 only. So answer is 2 pi for the first one. So option, sorry, question number A. Answer is 2 pi. Next, B. What is the answer for B? X minus GIF of X minus B. Even though it is not written, oh no, it is written. Treat square brackets as the greatest integer function. Right. The answer for B is going to be 1. Because you can write this as, you can write this as fraction part of X minus B minus B. Sorry, plus B. Yes or no? So adding something to a function or subtracting something from the X doesn't change the period. So it will have the same period as, it will have the same period as the fractional part of X, which is actually 1. Which is actually 1. So for B part, answer is 1. Okay. C. Okay, Vikas. Okay, Chaitanya. Sir, can you scroll up just for a second? Just for a second? Okay. Yeah, this part? Okay. Thank you sir. Thank you sir. Okay. The answer to part C, let's discuss it. See, first of all, let's look at the numerator function. Numerator function, let me write it by different name. Numerator function, NX is made up of mod of sin X plus cos X. Okay. Remember you can write it as root 2 mod sin X plus pi by 4. Okay. Now this will have the same period as mod of sin X because remember multiplying anything to the function or taking a plus or minus anything with the input X doesn't change the period. So it will have the same period as, same period as mod of sin X which is going to be pi. Okay. Denominator function we just now saw, I think the previous example itself, we saw that its period was pi by 2. This period is pi by 2. So your answer will be LCM of pi and pi by 2 which if I am not mistaken is pi. But again you are dealing here with functions which are interchangeable. So it is always advisable to check at pi by 2. So just replace your X with X plus pi by 2 and see whether you are getting the function back. When you do that, I believe that sin X will become cos X but cos X will become a minus sin X. Okay. Nothing will happen to the denominator. So if the numerator will become sin X minus cos X in that case which no longer remains the same. So yes, you will stick to this answer. The answer is actually pi. Right? So better be safe, better be safe. Okay. So there is a high probability that if there are interchangeable functions, your period may be reduced to half. So do a quick check. It doesn't harm anybody to do a quick check. Any questions, any concerns? Okay. Let's look at the D part. Okay. Param Vikas. Very good. Okay, Chaitanya. Okay, Rajdeep. Mahit. Richard. Okay, here. We'll see. We'll see. Very good. I'm very glad to see responses coming from everywhere. See, when you have tan pi by 2 GIF of X. Okay. And you want to find its period. So let's say the period is T. Okay, let the period be, let the period be T. So tan pi by 2 GIF of X plus T is giving you tan. By the way, this is some angle. Okay. So don't forget it is some angle. So it is giving you something like this. One tan is equated to the other tan. So how are their angles related? So you'll say, sir, obviously we have seen this in our last year technometric equations chapter. That your angle would be n pi plus one of them. Right. So they will be what happened to my n. Okay. So there will be something like this. Okay. Now, if you see it very closely, you realize that X has come out of the GIF, not come out. It has actually taken the GIF. In fact, T has come out of the GIF. Okay. If T is coming out of the GIF, that means T must be an integer. See, if you observe here, then there has been a split of this term into GIF of X plus something, right? That's why this pi by 2 GIF of X has made its appearance. And it has also resulted into a multiple of pi. So that means T must come out of the GIF. And it has also resulted into a multiple of pi. So that means T must be an integer. Correct. So first thing you would admit that T must be an integer. So those who have written the answer as an irrational number, pi or pi by 2 or 2 pi, what not, their answer is straight away wrong. Okay. Okay. If this is an integer, that means of course I'll get this and I'll get T pi by 2. By the way, I hope everybody knows this property. We have done this in our theory of equations chapter also. So if N is an integer, then this will break up as this, right? The similar thing is happening here also. Okay. Now, what is the least integer you can think for which N pi can be equated to this where N is an integer, by the way. So you can take N as 1. That's the least integer you can think of. Of course, you can't take on minus 1 because T has to be a positive integer. Remember, T has to be not only an integer, but it has to be a positive integer because period is always going to be a positive real quantity. Okay. So the smallest value that I can think of for N is 1 and hence I can think of a smallest value of T as a 2 because 1 pi is equal to T pi by 2. So T has to be a 2. So period for this function will be 2. It's not that straight forward also. Okay. It's not that straight forward also. So the answer for this question is 2. Could you show the question again? I think I thought it was T pi by 2x into mod x into that fraction x. Not fraction. It's GIF. Yeah. Yeah. Okay. Let's take more questions. Let's take more questions. Sorry. I have a doubt. Yes. Yes. Please ask. So in the last one, like if we get x as like some even like two point something or four point something, the thing comes out to be like 0, 0 and at odd numbers like one point something three point something. It won't be there. So shouldn't like it'll be like a dash dash dash. So shouldn't that be like constant? If there is a, see the best way to justify it. What does the graph say? Okay. Of course you'll not get a chance to plot a graph in your examination hall. Okay. So let's say I want to check pi by 2 or pi times floor divided by 2. Why does. So let's say this value that you have got here. The next value will come over here and that is at a gap of 2. Let me just draw x equal to 2. Yeah. Right. So do you see this, this value here, this value here is repeating again at this point. Okay. Similarly, this value will repeat again at a jump off 2 from here. So this jump will be a jump off 2. Yeah. Okay. Now what was your argument I would like to hear from you. No, but like many points have the same because every time it's zero, the value. Yeah. Every time it's not zero. It's zero for a certain value of x. Right. Yes. Zero. They're not defined. Yeah. Basically you have to see the trend over here. You have to see the trend. In fact, there is a trend in the gap also. So what is the period that it is trying to exhibit? Are you getting my point? Yes. Yeah. Yeah. This is an easy question. I think all of you should be able to answer in this case. The answer is it is not periodic. Yes. We already discussed a case like this. This is a non periodic function because sin x has a period of 2 pi and fraction part has a period of one. So there cannot be an LCM found out between a rational number and a rational number. Okay. Good. Next question. In fact, I'll put it over here itself. Find the period of cos of cos x plus cos of sin x. Okay. Yes. Yeah. There are some instances where you'll see a non periodic functions graph looking like very periodic, but they are not. Their pattern will keep on diminishing or changing after a certain time. Very good. So Richard here has already given the answer. Okay. See the best way here would be, let's say I take a value zero. Right. Now I want to see when does this value repeat again? Okay. So what is the smallest value of t for which it repeats? These are different, different tricks that you can use. So at zero, you know, it's value is going to be one plus one. That is two. And when you put ft, it becomes cos of cos t plus cos of sin t. Right. So when do you think it becomes a two again? Of course, when t is, t is... It's not two, right? It's cos one plus one. Oh, sorry. Yeah. Cos one plus one. Cos one plus one. Yeah. When do you think it will repeat its value again? Cos one, cos one. I think pi by two here if I put it will become cos one. Pi by two here if I put it becomes cos of zero, which is one. Okay. So now it is interchangeable. So you have to be careful. So I think the smallest value that you can, you know, have for the period is pi by two. So that becomes the fundamental period for this. So pi by two is the one which is going to give you a same result back. So pi by two is your answer. Okay. Let's take a few more examples. There is a continuous even periodic function. So there is a continuous periodic function. There is a continuous even periodic function with a period of eight. With a period of eight. Okay. Such that such that f zero is zero. F one is minus two. F three is two. F four is three. Okay. Then find the value of then the value of this function inverse of tan of f minus five f twenty plus cos inverse of f minus ten f seventeen. Okay. So please watch out the brackets carefully is option eight, two pi minus three, three minus two pi, two pi plus three, three minus five. So where does cos inverse end? There's a bracket missing. You must be a computer science student. No sir. I'm a bio student. Normally computer science students have these parenthesis check syntax. Yeah, you're correct. There was a bracket missing from myself from here. Good observation. Should I put the poll on? Yes. So don't need f of six for this question. The question doesn't provide that. No question doesn't provide that. 20 can be obtained from four. No, no. F six is not required. You can manage with this only. I think you're ignoring the fact that it's an even function. I think not. Yeah. No, that's visible. Very clearly. It's an even function. Don't move. It's periodic. That's fine. It's even also. So it's a mix of ITF. It's a mix of even concept. It's a mix of periodicity. So three concepts are involved. Three people have responded so far out of 24 of you. It's already going to be two minutes. Okay. A last 15 seconds. Five, four, three, two, one. Go. Okay. So let me end the poll. Very few people have voted so far. Those who want to vote, please do so very quickly, quickly. Some of you don't want to vote at all. Very mixed response. Almost equal amount of votes have been given to BC and D. Okay. Let's discuss. So first of all, we need to take care of the part which is within it. F of minus five and F of three would be the same. So this will be two. Yes, sir. So isn't there some value missing from here? We need some value. You need some value. Okay. Let's say F 20 will be same as F four. Why? Periodic with eight. So if you add a 16, that should give you the same value back. Now F of minus 10 F minus 10 is same as F minus two. And F minus two is same as F of two because it's an even function. But we don't have. Oh, F two was given. F two was actually one. It's a F 23. Sorry. F 20 would be three. F 20 would be same as F four. Oh yeah, it would be three. Yeah, sorry, my bad. F two was given in the question. I think I lost that information. F two was one. Okay. So cost inverse one. Sorry about that, but it's fine. Even if you have not got it, my mistake. And cost 17 is same as F. I'm sorry. F 17 is same as F one. So which is minus two. So basically you realize this becomes a zero. You'll realize you're finding tan inverse of tan three. If you recall the graph of tan inverse tan theta, this is the graph of tan versus and theta. This is your pie value. Three is very close to pie. That means you are almost here. So you need to seek the equation of this part of the line. This part was equation is why is equal to X minus five. So your answer for this will be three minus five, which is option D. I'm sure you would have done this question, but I think because of that loss of information, you could not do it. All right. What are the fundamental period of this function? Nine X plus sign three X upon cost X plus cost three X. I don't want to give you the options because if I give you the options, everybody will get it right. Okay. Oh, yeah. I just saw your message. I should have seen it before. This is a very interesting question. If I give the options to most of you, I'm sure would mark a wrong option. Okay. Let me give you an option. What is the fundamental period? Not the period fundamental period. Let me see on the poll. Let me see your poll results. Not, not the period only fundamental periods, least T value for which the function repeats itself. That's the trick because that's the trick actually. Okay. 11 of you have voted. I'll close the poll in exactly three minutes. If you have voted, I'll close the poll in exactly 45 seconds from now. Five, four, three, two, one, and a poll. Surprisingly, yes. The answer that you people have given is correct. Now see what is the trick here? Answer is option B. Now see what happened is many of us obviously would try to apply simplification on this. Okay. We can do that. So if you apply your transformation formula, the numerator will become 2 sin 2x cos x. The denominator will become 2 cos 2x cos x. It actually give you tan 2x. It actually gives you tan 2x, right? Now, now looking at tan 2x, people comment that the period is going to be pi by 2 because tan x is having a period of pi. So tan 2x should have a period of pi by 2. And they also see that it is the least of these numbers. So they mark option A. But let me tell you this is not right. Why? Because it violates one of the conditions of periodicity. If you see F0 will be defined. F0 will be 0. Check it out. Correct? So F pi by 2 should also give me 0. Right? But it is not. Because F pi by 2 will give you 0 by undefined. I mean denominator will become 0 there. Are you getting my point? Okay. Even when you put x as x plus pi by 2, you would realize that the function will change its structure. So pi by 2 cannot be your answer. Even though it appears from this that pi by 2 would be your answer. Are you getting my point? So in this case we will check for the higher values. We will check for pi. Of course sin x plus pi will give you minus sin x. Sin 3x plus 3pi that will also give you minus this thing. This will also give you minus. So your function will repeat itself. So pi would be the answer in this case. But had I not given you the options, I am sure most of you would have gone for pi by 2. Okay. So this is undefined in this case. Just a second guys. Is that fine? Any questions? Any questions in this? So what if you take like, not using transformation, if you just take the individual periods of each of the functions. Then you will get 2 pi. That is not the least period. That is what happens. See if you go by the regular property, you will always get higher versions of the answer. Basically they try to see when does this function repeat itself, when does sin 3x function repeat itself like that. But there could be an internal changing of these functions. Sin function, cos function and all of that are interchangeable. Okay. So in that case we cannot claim 2 pi to be your answer. We have to see for a lower value. Preferably check it all half the intervals. Okay. Try this one. See it's an easy concept but many people make mistakes. If there is an option, life is very very easy in case of periodic functions. But if they don't give you options, that means they make it as a, you can say integer type question. Then probably you will feel the pain. Signal function of gif of x plus gif of minus x. Okay. What is the fundamental period for this function? Okay. The fundamental or just period of the function. Fundamental period of the function. Okay. Did you start a poll? No, not yet. I'll start. Okay. Last 30 seconds. Five, four, three, two, one. Okay. Most of you have voted. Maximum people have said A and D. Either one or doesn't exist. Okay. Good. See, we have all seen this in our theory of equations chapter. That negative gif, a negative of x gif is negative gif of x. Or you can say negative x if x is an integer. Okay. And it is minus one minus this if x is not an integer. Correct. So now with this function, I'll define my function in two ways. If x is an integer, they both will cancel each other out and you'll get signum zero. Signum zero is zero. Okay. And when x is not an integer, you will be left with signum minus one. Signum minus one is minus one. Correct. Now, just a look at the graph of this function tells you that at every integer, it will be zero. Then it will be at minus one. Again, next integer will be zero. Then again, we had minus one. Next integer will be zero. Again, it will be minus one. So this is a trend of the graph. This is a trend of the graph. Okay. If you look at this graph. What is the periodicity of this function? Anybody can say it is periodicity will be one. Okay. So answer is option A. Again, remember the definition. It is the value of T. It is the value of the T for which the function will repeat itself for all x in the domain of the function for all x in the domain of the function. Okay. So here the deciding, the deciding factors are the values of x which are at the integer points. So you're getting this point back again after a jump of one. I'm getting my point. These are not the deciding points. Of course, they will also repeat itself. But out of the two, the decision maker is an integer point because they should repeat themselves also. Everybody should be happy. It is not only about checking one or two points. Got the point here. So the answer here will be option A. That is one. Any questions here? Okay. We'll take a few more questions and then we'll move on to the concept of functional equations. And we are natural numbers. This function sine of under root A square minus 3x plus cos of under root B square plus 7x is periodic with a finite fundamental period. Needless to say that it's just periodic. Okay. Period of f of x is option A pi 2 pi 2 pi times under root A square minus 3 plus under root B square plus 7 pi under root A square minus 3 plus under root B square plus 7. This x is outside the bracket outside the under root. Okay. So don't treat x to be within. So x is outside. There's somebody who is in a very hurry to answer. Okay. Last 15 seconds. Please vote. Okay. Five, four, three, two, one. Mixed response again. None of you have gone for option A. It's either B, C or D. Almost equal responses have gone to but most people have voted for C. Okay. Let's check. If you realize that this function is periodic. Okay. Now normally the period of this function is 2 pi by under root of A square minus 3 and the period of this function is 2 pi by under root of B square plus 7. Correct. Now there is an LCM existing for these guys. There is an LCM existing for these guys. Okay. Can I say for an LCM to exist for this guys and of course A and B are natural numbers. These two must be such that there is an integer coming out from there. Can I say that? Yes. Correct. If that is an integer then A square minus 3 must be a perfect square. Okay. And same goes with B square plus 7. The only possibility when A square minus 3 is a perfect square is when A value is 2 and B value is 3. There is no other value for which A square minus 3 A being natural number and B also being natural number gives you a perfect square. Yes or no? Yes sir. Correct. Yes sir. So A has to be 2 and B has to be 3. That means these values have to be 2 pi and 2 pi by 3. What is the LCM of 2 pi and 2 pi by 3? LCM of 2 pi and 2 pi by 3 is 2 pi. It is 2 pi by 2 but it is still the same. Sorry? 2 pi divided by 3. Oh yes. So your answer is option number B. I think in the poll yeah B got second highest vote. Most of you went for option C. Yeah you can do that this thing because you can do that scamming also. Okay one last question we will take and then we will close this idea of periodic functions. One I wanted to take involving functional equation like this. If f of x is such a function which satisfies this functional equation f of x plus f of x plus 4 is equal to f of x plus 2 plus f of x plus 6 for all x. Prove that f of x is periodic and find its period. This period need not be the fundamental period. Okay Param. Okay Richard. Good try. Okay. Keith then. Okay Raul. Okay last 15 seconds and then we will discuss it. Very good. Alright so let's stop it. Let's discuss. I think I have seen responses for most of you. Some of you have said 2. Some of you have said 8. Okay. Now if you are saying f of 2. If you are saying 2 is the period that means you are trying to claim f of x is equal to this. Correct. If you are trying to claim and you are trying to claim that f of x plus 4 will be equal to f of x plus 2. Okay. What is the evidence in the question which says that or for that matter this will become x plus 6. How are you sure that this function becomes this and this only becomes this. There is no evidence like that just because the sum of these two is equal to sum of these two we cannot make do a one on one mapping. Okay. So saying 2 would be a risky game. Okay. So what is the right way to do this question? Let's check this out. Let's check this out. We already have a functional equation which is f of x plus f of x plus 4 is equal to f of x plus 2 plus f of x plus 6. Replace your f of x or replace your x with x plus 2. That will give you f of x plus 2 plus f of x plus 6 is equal to f of x plus 4 plus f of x plus 8. Now add these two. If you add these two you would realize this will get cancelled with this. This will get cancelled with this. This will get cancelled with this and one solid proof that we have over here is that f of x plus 8 is equal to f of x. That means for sure it will have a period of 8 but period of 2 cannot be confirmed. 8 is definitely a period because I am getting f of x is equal to f of x plus 8. Now there may be a lesser period but for that we do not have any confirmation. So if you can show that there is a value lesser than 8 for which the function f of x will repeat itself then probably we can accept that as a period or we can accept that as a fundamental period but the question is not fundamental period. The question is actually a period. So 2 cannot be claimed as a period because there is no evidence that f of x is equal to f of x plus 2 but 8 has an evidence which you can see here. So 8 can be claimed as a period. Are you getting my point here? Yes sir. Fine. So I think we have taken enough questions on period. So what we will do is...