Jo dil se lage, usse kehdo hai hai hai hai Jo dil na lage, usse kehdo bye bye bye bye To dil bhi lagega aur maza bhi aega Love you my army Swagat hai aapka another beginners course ke session me Jahaan pe functions jisse majority bachchon ko dharr dharr dharr lakta hai Believe me agar aaj aap 70-75 minute is video can invest coletio Sarah Darbhaga Jai Gatika the book called basic malo to make function of FB nia Tana zero levels is start to the way basics domain and range hulka-hulka carrying a the swagger they are a kibachi kya yaha par Joe children a apka 21 days 21 sessions car challenge just k under hum foundation by foundation at the building core ready career so let the magic begin in the chat section TK and for the entire playlist you can click here to watch all these sessions moreover, any child who is preparing for JEE all the PYQ sessions, NTA sessions, for main and advance you can check out tinyurl.com.jee with nehamam link is in the description box and DPP of these sessions will be shared as always in the telegram group of mathematically inclined where you go to file section then you will definitely get all the DPPs and yes let's see here what all chapters we are doing what all things we are doing so with introduction basic ordered pairs vertical line test which is very strong domain and range how many kids are here write in chat section who are afraid of finding domain and range I will tell you such golden rules that all the air will come out as a question so mother that looking algebra functions different types of functions and their graphs will call basic valley hulk a pool K and few mixed problems so yeah yeah I'm big okay her Jiggy paper my boss up a function questions and it is not necessary exclusive function guys about me like a bar coffee Taraaki chapters make history paki o T here the sure what I did and we're hurrying about chase a and may this video k comment chai yeah hundred minutes hundred minutes basics hundred minutes Kadele does basics functions done take a DP BB done hoja a guitar lick a batana function who take a machine a machine function kuchini a machine a machine a knee kick which under Jai goto which Bahar I got correct ganaga juice PA to be keep the budget jelda with our juice PA to the gunna under jata a Bahar say mazada juice Bahar at a to gunna a knee kick input and juice a knee kick output for example many happy X key values baby up go to f of X Yannick a output function job car a big function of older out What is the output? For that, look here. And like should keep on increasing.
So, F of minus 2, means wherever X is, pick it up and replace it. So, if your F of X is 2X plus 1, quickly answer it. What will I get? F of minus 2 means, what does it mean?
Put the value here, 2 into minus 2 plus 1, means minus 4 plus 1 minus 3 answer is given. I am sure that this is a big surprise for all of us. You are in 11th, 12th, preparing for JEE, board, any competitive level, this session will take you higher and higher. Believe me, foundation is going to be set like this today. F of minus 1, same way, 2 into minus 1 plus 1, so this will be minus 1. Similarly, keep adding all the things, so you will get the full F of X.
why a bowl set there take a up to the function goes some of the day Hannah the basic of a definition we're gonna make the coach a lot the reggae say Jenga abhisamjo why bullet to output he can F bullet or job he function can am I just assign it's put a hit them say X plus one put a exponential function which I love function put check yeah put a ticket scan am I F and yeah I can input variable just go a below the independent variable yeah means whatever value you want to give, it is dependent on this. Because whatever value you gave of x, it will come out and accordingly value will come in output. Till now things are very simple.
Right? Simple, right? Come on.
This question. I want you to show me quickly quickly and yes you will get my hand written pdf in the description of the video but here I want to see at least 1500 likes before the session ends ok now function f of x equal to this is given to us. Abhi tak nahi, hum aamni definition, nahi batai function ki. Samajh lo, dheere dheere leke ja rahi hoon, ye cheeze tumhe aati hai, isle level badha rahi hoon. Find the value of f of 2, toh aapko pata hai ki, ji aisa hai dosto.
So, you will leave your house by giving 2 square minus 3 into 2 and taking answer minus 2. Right? Okay, if I say, do the second part and show me. f of x equal to 4. Means, x square minus 3x equal to 4. Now, we can solve quadratic equations since childhood. If you do middle term split, you know, right?
Should I write direct factors in this? x minus 4 and x plus 1. Correct? Correct? So, it means either x will be 4 or minus 1. End of story.
Drink water. Right? Why are you saying 2 and what? It will be 4 and minus 1. It means one thing, Neha ma'am, is shining on me. All the children here, write down if you too have a shine.
That this function, right? Function. Here, let's say I have given anything.
f of x is equal to x cube plus sin x. Okay? I have given something like this.
Now, I am asking you. ask f of? You want heart. Let your heart beat.
So, what is this? Understand. X is a symbol.
Whatever you put inside it, accordingly your output will come out. Right? So, what will you do here? Heart cube.
Oh, plus sine heart. Oh, 2 is constant. It doesn't matter. Understood? So, never love f of x.
Understand. X is a symbol. Correct?
Everyone? Yes. very good now let's understand that Neha ma'am all this is done now tell me what is the function if you read this, you will get confused ok he says that there is a relation of this type which is on one set A and on other set B there are some elements which connects them every element in A has a mapping every element in A has a unique mapping nothing is different You didn't have to read anything, just explain it properly Neha maam.
So now you are going to understand from Neha maam's style. Assume, what is this example? Here it is children's truth. So this is a group of children. And what is this?
Assume this is a group of women. Okay? And what I am saying is that I am making a connection between these two.
Is a child of. Is a child of. And believe me this is a trick that will make the kids reach college level and I don't know where. and never forget this trick. What is the trick?
I have made 1,2,3,4,5 1,2,3,4,5,6 like this 1,2,3,4,5 these are the names of the children And this is the name of the girls. Okay? So much of shine.
Now tell me one thing. Now tell me one thing. This example is my favorite. I have taken it many times. I said, every child will have a mother.
Right? Every element in set A has a mapping. Has a mapping in, I should write B.
That means, it will be connected to some element. So, understand this psyche. Okay? Every child will have a mother and that mother will be unique.
Right? Every element in A has a unique mapping in B. It won't be like this that the child asks number 2, tell me where is your mother?
Then he will say, A can also be A and D can also be A. Someone's mother is not sitting near, right? Tell me and let me know that a flying slipper has come here. Do you understand?
It can be A or B or D. a jeep see but car the correct so is one example say I hope up BB but love company be fully power case cheese cook correct so as I told honey Satya me company do cheese a chama can equally care every element in the first set should be connected with the second set guy element and this connection has to be unique so much real I say yeah but I'll yet tell the guy yeah but I am onto Chontu and Chontu both went to B. Is it possible?
Yes or No? Quickly spam in the chat section. Quickly.
Yes or No? Spam in the chat section once that one went to B, one went to B and two went to B It's absolutely possible Boss, there can be two children of one mother Right? There can be two children of one mother Leave two, three, four, send all the children to her So you get the idea So if you have this spark top to me yeah I'm a little examples pay yeah I'm a little key set a co-yard hum domain get it there's a three B skin the elements a in koala the TAM do me and say to be a jitani be elements call them co-domain.
I am increasing the level slowly. Now look at it carefully. First tell me whether this function is there or not.
I am saying it quickly. Write in the chat section whether this function is there or not. Quickly. Quickly.
Quickly. Yes. So what will happen?
Whether it is there or not. So you will say 1, 2, 3, 4 all the connections. connected at point number one justified. Is it uniquely connected? 1 is near x, 2 is near y, 3 means 1 element is not near 2. Correct?
So yes, this is a function. Now I asked domain, you said in domain, see whole set A came. In codomain, see whole set B came.
So I can write A and B too. Ok? If we say range, range is called range. set of images samajh lo dhyan se set of images images matlab mummiya mummiya yaani ki yaar zaruri toh nahi hai na ab yeh maine kaha yeh aisi woman hai jo mummi nahi hai that is ok hoongi na aise ladkiya bhi correct toh yeh kya hai yeh wo mummiya hai jo connect ho chuki hai toh mai bolungi iss mese jis bhi element se connected hai wo output rate range kehlata hai. Wo output mera range kehlata hai.
I guess yeh cheez mujhe cham cham cham ki. Hai na? Yeh cheez mujhe cham cham cham ki. Chalo. Aage badte yeh cheez toh sort ho chuki hai ji.
Ab. I think yeh child aur woman wala bhi example sabko chamak gaya hoga hai. Yeh tumhai reference ke liye na.
Idhar laga diya. Apni mummy ho jaake dikha dena. Chalye. Okay.
Ab yahaan pe yaar zara main aapke saath ek game khel nu. So, all the students will tell me one by one whether there is one function or not. So, before giving your answer, write one and then write yes or no.
Come on. First one, whether there is a function or not in the chart section. Fast, fast, fast.
Come on, come on. Quickly, quickly. Wow, absolutely right.
Yes, there is. Absolutely there is. Okay. Absolutely there is this function. Yes.
Come on to the second one. Second wale pe aajao, uri baba ye function toh fara khan nikla, bhaiya 3 bache, 3 ho ek hi mammi pe map ho rahe hai, correct toh function toh ye bhi hai, ok, function toh ye bhi hai, hai na, bilkul sahi Ab zara third wale pe aao, function hai ya nahi hai, jaldi batao, nahi, ye nahi hai function, kyun bichare si C is left. C is not given by your mother. If you left it here, it will work.
Leaving it here is a sin. And if one of the two points of the function is violated, then that function is not there. So the answer of third is no.
My heart is getting hardened. On such a thing, hearts should come and likes should increase. Like your heartbeat.
Come on. In the fourth one. Fourth one is not there because it is saying A. I have two mothers. I have two mothers who are not...
It won't work, boss. It won't work. Correct?
I hope I have understood the function. So Neha ma'am, is there only one way to know the function? I am teaching you another graphical method. So the things which you should know from the basics. A graphical method to check if the curve is a function or not.
What is it? It is called Vertical Line Test. It means you have been given a graph. You draw a vertical line.
badiya hai hum hai na haan. Thike perpendicular lines maro. Ab agar toh wo at a time sirf ek hi point pe cut karti hai graph ko. Tab toh badiya hai. If it cuts at more than one point then it is not a function.
Example lo. Example lo. So nahi ham apne ko nahi chamka. Toh aao chamkate hain. Example number 1, example number 2, example number 3 Now we will do one by one, I am not showing second and third, focus on first Focus on first, this is a function, I said we are not vertical line, we are perpendicular will cross the line of the perpendicular if he cut the graph only once at one time means if he goes to the next point then obviously he will cut somewhere else if he goes to the third point then obviously he will cut somewhere else only once but no see this is very strange If he cut here and This is not it either.
This is not it either. Come to the third one. It is the third. Because you can make perpendicular lines as much as you want. Every time it is just cutting your graph once.
So yes. So yes. So one or two is no. The third one is yes.
Did you understand? If you understood, then clap your hands. Clap your hands, my monkeys.
Come on, my army. Well done. Superb yaar. Issi tana se na, aapke saare poore syllabus ke chapters ko dheere dheere dheere karke base banayenge aapka.
Taaki jaa jaa jai level ke saawal solve karo, so there is absolutely no problem that you face. Chalye, which graph is... Not a function.
Now look at this monster carefully. Which one is not a function? Which one is not a function?
Look at this. If I make a line here, then it is cutting in two places. If I make a line here, then it is cutting in two places. So, this is not. Right?
This, yes, if I hit it like this, it is cutting it once. Here also, as much as I make it perpendicular, it is cutting the graph once. So, graph number 1. Right?
This is not because it is cutting at more than one point. Is it fun or not? Is it fun or not?
Come on, very good. Now, tell me which is a function in this? Okay, I have played one more thing with you. In the last one, I asked not a function. I am asking about the function in this.
So, look very carefully at the options. You have stood in front of it, how can we recognise it? This is the function. Yes, it is cutting only once. This is there, this is not there.
See this. Every time, wherever you draw perpendicular, it cuts at multiple points. It cuts at 2 points.
This is also cutting at 2 points. Did you understand? See here, it is cutting here also.
So, you must have understood. That means, there is also answer 1 in this. Yes, graph number 1 shined to everyone. Very good.
Very good. Come on. Important note. Note was so important that it came out of the box.
Isn't it? This is a little harsh sin with us, friends. We will have to punish this sin.
Wait. What kind of box is this? It came out of the box.
This point was so important. Are you seeing, Vinod? What is happening to what, son?
At least come in the box, why are you so angry? Why are you so angry, friend? This is not from the box Yes, come on We have made the box bigger You also make your heart bigger Okay Now he said there are two sets A and B Which contain P and Q elements So, in set A, brother, there is P element Means, children are sitting in P Right? And women are sitting in Q Now he said number of functions possible from set a to set B skis but deafened derivation many jangi but obvious watch low better to me mommy is a rich are about to own a such a without such a without the was I say a director look yeah set B value a jo mummy a bonita child is kisser pitch at our way yes or no s I you turn up yeah this is what the mummy may it is car a common me but the mirror sir perot day the buses sir pair of can I up name a meeting take care yes to chalo a whatever take question caro jab I got a man career at the way I got ticket question again I a jay number of functions of set of three elements to set of four elements Mojito spam car do well I'm Mojito spam car spam the answer come on come on quickly quickly quickly yes exactly this is how it happens actually i also do this come on tell me exactly he said set of 3 elements in first one there are 3 elements in second one there are 4 elements this is your This is your child and this is your mother.
So then we will sit on top of mother's head. 4 cube exactly 64. Mistake alert son, mistake alert. Here no child will say 3 raised to power 481 by mistake.
Set B comes first. Set A comes later. I was very happy to know that majority of kids are on their mom's head.
Okay? Okay? Very good son.
Amazing. Okay. Next question is number of elements.
This is two here. This is number of functions. How many will be there? Oh man.
I will tell you. But read this, from which set to which set you have asked. Set A to set B you have asked.
So, you have to climb on mom's head, you have to do chatter. Correct? Very good.
Now comes the most dangerous thing. I don't know how to remove domain reen. Now see, how we are going to make it simple for you.
But first understand that whatever acceptable inputs. When this brother is a machine to make a domain reen, then it is difficult. and then you will put pumpkin in it.
This is a very strange thing. You understand, it is a machine to make a pumpkin and you will put pumpkin in it then it will not work. So there are some inputs that can go. Or you instead of paper printer you say that it is the season of mango then you put mango in it.
It will not accept. Suppose this is your paper printing machine you put mango of your season in it. Ok, you put mango.
It will not accept. He said that the acceptable inputs will go and will be called domain Correct? And what will come out?
What will come out? Brother, if you had put sugarcane or mango, then mango shake will come out After putting mango, you will not say that it will come out as dal Do you understand? So what does this mean?
The output that will come out will be called dal range get a yes Joe output I got comes code range get a high score find get a gun I would hold a he may be may I get birthday yeah I suppose I'm a double say bahar or a boy about me don't have now go so much the job she ever was a cheese there okay hey so I'm going to put the tower domain I put a no first set is me so to coach who Tini's acta henna and range get a jizz jizz case are connected a open ticket So, I have a question for you. Tell me the answer of domain and spam it. Spam it.
You are adding mangoes and saying that the dal will come out. It won't happen. You will get what you sow. Keep this thing. What can be the input of domain?
If you ask the range, it will tell you this. Did you understand? If you ask domain, it will tell you this.
If you ask range, it will tell you this. Correct? Amazing.
Superb. Superb yaar. Bohot hi simple sa sawal tha. Ab zara dhyan se suno ek naya term introduce kar rahi hu tumhare saamne question ke zarye.
Theek hai? And imaandari se tumhi log baag mein likh ke jaoge ki haan maam aise humne functions nahi pada tha aaj tak. Chalo.
Agar yehi same way rakhi hai which of the following is true? To is the image of Q. Ab yaad rakho jo bachche hai theek hai jo man ke bat jo man ke sakche bache hai, wo pre image kehlaate hai and jo mummy hai wo image kehlaate hai samajh rahe ho?
Samajh rahe ho? Mummy hai wo image kehlaate hai, tum jo kaand karte ho wo image beta tumhari fir mummy pe nikal ke aati hai, samajh rahe ho na, bas waise hi samajh lo kuch, toh to is the image of Q hundred percent correct, hundred percent correct, achcha. R is the pre-image of 3. So, 3 ki R image.
Pre-image hai. Pre-image hai. Hanji.
1, 2, 3 is the image of F. 1, 2, 3 is the image of F. Hanji.
Jisko aap range bhi bula sakte hoon. Toh, yaani iske andar, main bolungi, sare ke sare hi options correct hai. Theek hai? Sare ke sare hi options correct hai. Samajh mein aara hai?
Chal ye. Very very good children. Let's move ahead.
Algebra of functions. How to add two functions. How to subtract.
How to multiply. How to divide. You don't have to do anything.
Just do whatever is there. Just do it. Just do it.
So. Addition is simple. If I say.
F plus G of X. Means. F of X plus G of X. Same minus.
Multiplication. Division. Yes. Division.
Remember that g of x cannot be equal to 0. It is not possible that your denominator is 0. You don't have to define 0. This c is a constant. If you multiply it, what difference does it make? I will tell you that this is a function.
Listen, 2 sin x. It is the same thing. What difference does 2 into sin x make? Right? Now, let's do a small question and apply it to see how it works.
Right? The line is a little up and down. The line is a line.
People can hit it anywhere. Do you understand? You have understood. Let's go. A is given to you, function is also A, f of x is defined like this, g is defined as x.
Okay. You have to find f plus g. What does f plus gx mean?
f of x plus g of x. You just find it. Hit the range, you just find it. Okay, now do step 1. So what you have to do is nothing, you have to take two functions and add them. So what happened?
3x plus 2. Understood? It was very easy. Correct. If I tell you to do f minus g, then what is it?
Hold both the functions, pick them up and subtract them. So, you will say x plus 2f. Okay? Multiply means multiply.
Divide means write like this. F upon G I have already asked you what you will do You will tell me right So now keep this much We will find range range later We will find range range later And here friends this is 4th part In 3rd part we have asked this and this Ok no problem So write this also X into 2X plus 2 Was there anything in this? Was there anything in this? No, it was very simple.
Let's see how we can find it in the range. Now presenting to you, S. Dhakad hai, Dhakad hai, aisey Dhakad hai Rules of finding domain Agar tumne yeh do chamakte sitare Yad kar liye Yad matlab samaj liye, ratta nahi maana koi You will remove domain of any function Even God will not be able to stop you Means The expression given to me by 1 upon That expression should not be 0 Yes That expression should not be zero.
This is the first point my lord. Second thing, if there is an expression in square root. We are talking about real numbers.
So the expression of square root is greater. Greater than equal to 0. Remember in childhood you used to do like this. B square minus 4AC. Discriminant.
Oho. This is root minus 20. You used to say not possible. Do you understand? However, when we do complex numbers in next chapter.
This will also be possible. But for now, in real numbers we used to write like this. Say yes at least once. Heart will feel good.
Don't be stingy. Okay, so means whatever will come inside the square root, it should be greater than or equal to 0. Do you remember these two rules? Exactly, they are like two golden rules.
I am not sharing the PDF, I am not seeing 1000 likes, so do this work quickly. Do I remember these two golden rules? Yes or no? Hey, once write yes or no.
And this is a small thing that if two domains can be given of the function, then if two functions are given of the function, then it is a simple function. track kardu to unki domain jo bhi iska output aaya aur iska output aaya uske common terms hongi. D1 intersection D2 matlab common terms. Tikey?
To ispe bhi aungi dhere dhere. Yad hoi ki nahi? Yad bhi ho gaye?
Aay hai bade smarto. Aao fir. Yaar ye kya hua?
Hare baba. See this. This will be a grave sin. If I have given f of x is equal to 1 by x.
Now find out what will be its domain. What will be its domain? I want to see how many children answer this. Look boss, we assume our domain as real numbers. Correct?
I have told you only two rules, not much. I said, there is something in denominator, it will not be 0. And there is something in square root, it will not be negative. Okay. There is something in denominator, it will not be 0. 0 is not possible. So what can be my input?
Full real line minus singleton 0. Many students make mistakes by writing interval here. Understand, whatever number you put in your curly brackets or braces, only that specific number is subtracted. One mistake which many students make, I will write here, mistake alert.
I will write here, mistake alert. What do you do? Now understand the difference between this and this I write this I write this And I write this What is the difference between these? Understand this I know the part of this question but this mistake will be a lot. That's why.
This means, on my number line, this was also there in the last class, all the infinite numbers from 0 to 1, just this is included. Because it is an open interval Understood? Okay Closed means All the numbers between 0 and 1 Correct?
And Endpoints also And this time endpoints will also be included 0 and 1 What does this mean? I am talking about only 2 numbers 0 and 1 This is not an interval Only 2 numbers are there 0 and 1 Understood? So I don't want just 0. Rest, anything will work for us.
So you will write its answer like this. Okay? You will write like this.
This should shine to everyone. Come on. Come to the next question.
f of x is equal to root x. What does this mean? Whatever is inside this. You will say that x should be greater than or equal to 0. Negative. It cannot be.
Right? If it cannot be negative, then what does it mean? Now you can write it in interval form. 0 to infinity.
You cannot keep infinity or minus infinity. So 0 will be included up till infinity. Now it will be fun.
Now it will be fun. How many children have read the wavy curve? How many wavy? Ya denominator zero nahi ho sakta.
Aur square root ke andar jo hai. Wo negative nahi ho sakta. Correct. Toh yehi cheez.
Main bolu achcha. Aap bolu ki yaar. Ye jo square root ke andar hai dost.
Ye negative nahi ho sakta. Now make some factors of this. So what will the factors be?
1 minus x into 1 plus x is greater than or equal to 0. Okay? Now what I want is the coefficient xk positive. What is the wavy curve method?
What does this series say? We have done this part of the series You can find the playlist in the description box You can see What does the wavy curve method say? Its coefficient is positive It is not positive So you will do it from minus 1 So what will happen with this?
Inequality will change. Correct. Inequality will change. Now you remember how we used to do it on number line.
Minus 1, 1. We used to start from here. Plus. And because both of these have the power odd. Plus, minus, plus. What will be the answer?
Tell me in your ears. Jaan, doon. You want the negative one.
So, minus 1, say 1. Now, don't put curly brackets here. Because I don't just want minus 1 and 1 numbers. No, I don't want all the numbers between them infinitely.
How many children have got this? How many children have got this? Play the clap. Did you understand? If I am cheating you a little bit with this question If this same question was like this Same question If I say you can write its direct answer This is the domain of this function What is the domain of this function?
What do you think? What difference does it make? Neha ma'am You said that whatever is in square root Whatever is in square root negative ni ho sakta. Lekin Neha maam ne yeh bhi to ka ki denominator zero ni ho sakta.
Hai na? To wo dono ke dono star rules jo hai. Wo mix ho gaye hai.
Yani is bari aap minus 1 se 1 logi. Kyo? Kyo ki minus 1 liya to phir Then denominator 0. If you take 1, then denominator 0. If you take 1 by 0, then you will make it not defined.
It would have worked here because it is in the numerator. Did you understand? Did you understand? Oh, right. Very good.
Proud of you, my army. It is shining, right? Oh, very good, son.
Very good. Very good. Okay. Now, take out their domain too. One by one, up, up.
one by one, one by one come on and do solve today's DP also I am telling you come on, keep making hearts friends ok what is here in the first part I said what is domain so you will say that domain is such that denominator can't be zero here square root is not there so you directly say that from real line 2 delete kardo hana haan yaha pe ab dekho kai bache garbad kya karte hai x greater than 2 bhai kyun sirf 2 number nahi aasakta baaki toh kuch bhi aaye vo plus o minus o kya frak padta hai aarey bohot sahi bohot saare bache aabe jawab derey hain mujhe dekhe bohot acha lag raha hai very good beta proud of you theek hai ab agle wale pe bhi ajo thodi proud proud aur feeling hogi Same thing, see what I did in the last one. Same, that x square minus 5 should be greater than equal to 0. But this time this thing is positive. So I can break it.
Play root root. x minus root 5, x plus root 5 is greater than equal to 0. Brother, it is the grace of God that both the coefficients of x are positive. So in this case, what should I do? No, don't dance.
But if you are happy, then this dance is taken out. What will happen? Do minus root 5 here, do root 5 here, play plus minus plus and these extreme answers will come. How will you write domain?
How will you write? Minus infinity to minus root 5. Should I include or not? Do it, Neha ma'am. It will be at least 0. It will be fine.
Union. Root 5 to infinity. See, minus infinity to infinity is an independent bird. Purrr. You can never put a close interval to this.
Okay? Okay? Understood?
Are you enjoying? Yes, that's the thing. That's the thing. Okay, tell me this.
Should I give you these notes which I am writing with my hand or should I give you these empty ones? Tell me quickly. Do you want your notes too or these empty ones?
These typed ones. give me this only if you want my video then why don't you like it come on guys many new kids from this channel they think that I always say like why do you keep saying like Because as I have told you many times before, this channel is a passion for me. This is not my business. Where I show you freaky lollipop and say, come on friends, buy this course from me. Whatever is on this channel is for you.
It is complete and I try to give maximum things here only. Meaning whatever is, everything is here. So for that, Guru Dakshina is something, right boss?
You will show love, right? You will do a comment lovingly, right? Now, just like that. Come on, come on.
Okay. Find the domain of this function. Oh my God, this is very difficult.
Is it really tough? Is it really tough? Listen carefully, the golden rule is the same. The question, no matter how dangerous it is, two stars should shine in your mind. Denominator cannot be zero.
Okay, denominator cannot be zero. So, how will it be? A grab is zero. If you add middle term split in it, then these two factors will come. So, what domain tells me is that put anything, boss, don't put 2 and 6. Now, see, I have removed 2 and 6 exclusively from the real line.
That's why I have used curly brackets. Many children get confused in this. They sometimes put open, sometimes square, sometimes curly. If I tell you the truth, I was in grade 11-12.
So, in the beginning, I also didn't understand. So, I used to say, what does it mean? teacher come on just like a he may call you be bracket like a Jati a Joe mood with a logic with a hair mood in Utah cello great great great I have a taxi okay find the domain of the following functions a key to function a beta tell you yeah baby do you and See, twist in the story I am going to tell you the next session very soon Which is that? Okay, F of X is this Rule is that these two cannot be zero So, simple thing is You will delete 0 and 1 from the real line And will move ahead in life Yes or no? Are you chumking or not chumking?
Done? Your question is done, my love Do you understand? Okay Oh my God!
This is very difficult Neha ma'am, you say that this is a basic session and you ask such a dirty question in between How do I do this? We are stuck here. No, we are not stuck.
The rule is that the function inside the square root will not be negative. So, x into x minus 1 into x minus 2 is greater than equal to 0. Wavy curve zindabad. Wavy curve zindabad.
What will you do? Roots are already 0, 1, 2. And the thing is that all the coefficients are positive. Of x.
Okay. So, you will go from plus. All are linear factors.
That means odd power ones are plus, minus, plus, minus. You want greater than equal to 0. So, domain has come, friends. Domain has come.
0 to 1. Union 2 to infinity. I have included 0, 1, 2. Because at best they will make the function 0. Which will work for us. It is not in denominator. Right?
It is not in denominator. Yes or no? Tell me once, has this thing become clear or not? Has this thing become clear or not? Okay.
Very good. Okay, okay. Neha ma'am, you are increasing the level of the question. Now how will we solve this? Nothing is there in your square root.
First of all, because your x3 cannot be. Right? Because it will make denominator 0. Okay. And x-2 upon x-3 greater than 0. zero equal to zero yes it can be but denominator 3 should not be there denominator 3 should not be there rest is fine so what is there in this I told you that it is the same thing.
Wavy curve. I told you that wavy curve is a big thing. What will you do?
2 and 3. Positive, negative, positive. These extreme ones will come but 3 will not come. So what will be the domain?
Minus infinity to 2. See, if you have any doubt, put that number and see. So as soon as you put 2, it becomes 0. It will work. Square root of 0 is 0. It will work. Okay? But here union 3 is infinity.
We cannot include 3 here. Because 3 will make denominator 0. Ok. 3 is a dirty fish. Don't do 3. 3 is a dirty fish.
Ok. Let's do this question. Come to the ground. I had said one thing that if domain comes as addition of two functions or subtraction of two functions, then separate its domain, let's say call it D1, separate its D2 and in the end, whatever is common. That will be your answer.
So, look at this. This says that x minus 2 is greater than or equal to 0. That means x is greater than or equal to 2. Okay. This says that x minus 3 is greater than or equal to 0. That means x is greater than or equal to 3. So, in both, common.
You can see it from common sense also. One goes from 2 to infinity and one goes from 3 to infinity. So what is common?
This is it. So what is the final answer? D1 intersection D2 which goes from 3 to infinity. Intersection of domains seems to be very high fun.
But this question is solved in a very fast way. I am not enjoying it, children. I am not showing love.
Bacche, pyaar nahi dikha rahe? Ab, yeh chaar prakaar ke question tumhare saamne hai. Dur dur, dur dur, dur dur.
Harek type, yaha pe koi bhi constant ho sakta hai. Thik hai, koi bhi. It can be constant.
Why did 4 jump? No problem. We will bring it up.
Okay? Come on. How to solve them? Listen carefully. Now, instead of 4, we can put 9, 16, 25, any perfect square.
That is, root of 4 minus x square, root of x square minus 4, 1 upon root of x square minus 4, or 1 upon root of 4 minus x square. How to deal with this? Now, look at it carefully. How are you going to do it? First of all, look at it.
There is no rocket science. You will say 4 minus x square will be greater than or equal to 0. Brother, keep x square in mind. You want the coefficient positive.
That means x square minus 4. is less than equal to 0. That means, x minus 2, x plus 2, is less than equal to 0. What does the wavy curve say? Minus 2, 2. In linear, you will come to plus, minus, plus, right to left. You want negative. So, you will put 2 from minus 2. How much time can you put to do this?
How much time is it taking to solve this question? Tell it quickly. Yes or no?
Hey, not the one with the spark. Now, along with this, the second part which is this one, 1 upon root of 4 minus x square. In this, you will say that this should be strictly greater than 0. So, from the first part only.
Take it. The logic of first. You can use the logic and tell me what the answer is. The answer which you got here will be the same in the domain. Just your interval will be open.
Why? Because it cannot be equal to 0. Right? It cannot be equal to 0. We have just hit two arrows at once. Okay, now come to the next one. Come here.
Don't come out of the screen, pay on your notebook. Okay, now see here what we are doing. Oh, oh devil.
X square minus 4 is greater than equal to 0. So X minus 2 into X plus 2 is greater than equal to 0. So you also know that mind is mind. Mind is mind. Why are you smiling so much?
You are telling everyone that you are a domain You are taking an extreme interval Right? This time you can include minus 2 and 2. Okay? But if I take this function 1 upon root of x square minus 4. Then you also know. Nothing. While running the logic of second.
Friends. You know that you can't include minus 2 and 2. Rest everything is fine. Rest everything is fine.
Just running. You have broken all the questions together You have broken all the questions together Did you enjoy it? Yes or no? Okay?
Let's go There were two stars Here are three golden balls Beautiful balls The shiny balls range range now understand what is procedure range of function to find the output do some work directly na hi nikal para. Aap y ko f of x ke terms me. Aap jo bhi f of x ho ga isko y likh lo.
Express x explicitly in terms of y. Nahi samajh me aaya. Kya bola?
Yaani ke x ko y ki terms me likho. Okay? Okay?
No, I mean okay. Write x in the function of y. Number 2. Now try to find possible values of y. The procedure of domain that we were applying.
Apply that for range. And think like you are taking domain in terms of y. Okay? Okay?
Domain in terms of y. Feel as if you are finding the domain in the terms of y. Okay?
Finally, actually your answer will come here only. But sometimes you have to do this that you have to eliminate some values. I will give you a hint here also, where you will do.
Whenever there will be any square root function, then in this case, you have to reject or eliminate negative values. Because if you see square root function, always you have to... answer data a small put you here a channel to demand Maya many more X square equal to 25 sorry sorry much a job of the X key value key on the jelda with our much a day can I have a check a job they pay a cello cello baby hello cello but Joe Yes, so you will say that x is plus or minus 5. Okay, x is plus or minus 5. Correct, you write this only. Now the question is different. I asked x is equal to root 25. So now what will be the value of x?
Now what will be the value of x? Now the value of x will be only 5. What is the difference? If you have never thought about this, think about it today. This is a square function whose output is plus or minus.
kuch bhi ho sakta hai ye square root function hota hai toh abhi na me function ke bohot deep me nahi ja rahi hai aaj toh basics ka session hai toh halka phulka cheeze bata rahi ho samjhe hai ni Yes, right? Yes. Okay. So, you will eliminate negative values in this case. Yes?
You had not thought like this till today, right? Now, think. Come on, let's take a question, Neha ma'am.
So, I said, you will remove domain from this. Okay? You will remove domain in your dream also. Let's talk about range. And I will tell you those three steps how to use it.
And let me tell you, if you want this handwritten PDF, I want to see at least 1100 likes before the session ends. Let's talk about range, so what steps will you follow? Understand, step number 1 says that this F of x de rakhaya hai, aise y likto. Step number 2 kehta hai ki x ko y ke terms me likto.
So x y plus 3 y equal to 1. Yani x kya kehalaega 1 minus 3 y upon y. Ab forget anything. If your function was this and I tell you to find its domain then tell me can it take all the answers or is there anyone who can't take them?
Can it take all the answers or is there anyone who can't take them? Yes. It can take all the answers except y. So, it means your domain has come. Sorry, not domain, we will call it range because y actually gives you range.
And any other square root function So, it will be R minus singleton 0. My friends, this is the answer. We will take anything except 0. This is the range. This is the answer.
How was that 3 steps? Amazing. Okay.
How was that 3 steps? Amazing. So today, your range also says Do something so amazing that I become yours I am someone else's for now Do you understand?
Means till now we didn't understand that someone else's was enough But today after understanding these three steps, I also feel that I am enough to do this To find the range Right? Okay Wonderful Now come here Find the range of this Think about it Means tension What is he saying? f of x is 2x plus 3 upon x.
Okay. So, what is the range? Step 1, y is equal to 2x plus 3 upon x. Okay. You have cross multiplied.
Step 2, yx is equal to 2x. You have to write x in terms of y. What will you do? Take x common here. So y minus 2 is equal to 3. What is x?
3 upon y minus 2. Now tell me quickly. What would have to be the domain of this? Because that will be its range. Possible values of y.
This will give you range. So quickly tell me quickly. Tell me quickly kids Exactly!
Wow! Did you enjoy? Did you enjoy it? Superb! Okay, find the range of this now Now the game is increasing slowly slowly slowly slowly The game is coming up a little Okay?
Let's go Now think here, you have to find the range of steps, that is where you put y as root x minus 2. Putum putting karo. Kya karoge ab isme? X ko y mein kaise karoge?
Ab square both sides kar doge na usse se ho paega. Yani ki y square equal to x minus 2. Yani ki x is equal to y square plus 2. Ab agar ab khali isko dekho to aap bologe y to koi bhi value le sakta hai. Isme se kuch bhi delete ni karna.
Srarf y square plus 2 ko dekho. Aap bologe nahi na kuch denominator mein zero ho raha hai na kuch square root mein hai to y to koi bhi value le sakta hai. Leave. But, Sorry.
Yes. You understood, right? I told you that when square root comes, then you should understand the meaning of this.
That I cannot give negative number, that output. What is y, son? y is your output.
So, if I took output as real, you think for yourself. It is possible that square root of x minus 2 is minus 1. We are playing in real. Okay?
We are playing in real. Okay. Now, the questions which I had asked for you with domain, we are asking the same questions with range.
So, are you ready? Are you ready? Let me see if the like and shike is increasing or not.
Only then I will believe that you are ready. Negative can't come exactly because square root output is never negative. Okay, so first of all let's talk about this.
Okay, so first of all let's talk about this child. We are talking about its range. We have taken out the domain in the last slides. We raised our hand so much that it went away from the screen.
Okay, it came back. So what will you do for range? We will raise our hand.
equal to f of x. Y is equal to root of 4 minus x square. Now, we have to do the same.
We have to do x in terms of y. We have to do x in terms of y. So, what will you do for that? You will do square both sides.
Y square equal to 4 minus x. x square? Okay. That means x square will be 4 minus y square. Listen carefully people of the world.
Tell us one thing. Can I say that x square is always greater than or equal to 0? Will it be?
Will it be or not? Tell me quickly in the chat section. I am reading your chats.
X square greater than equal to 0 will be or will not be? Tell me quickly. Will be. Obviously, X is a real number.
Negative cannot be. That means 4 minus Y square will always be greater than equal to 0. Got it? What? Remember the wavy curve?
Right? Remember the wavy curve? That means Y square minus 4 less than equal to 0. Y minus 2 into Y plus 2 less than equal to 0. Now what is it? Now my... Minus 2, 2. This will be your answer.
Why will belong to minus 2, 2? But. It's a square root function, my dear.
If it's a square root function, then here the golden ball should come in the spinning mind. What? What will be the range? There will be no negativity in the range. So all the negative numbers have gone.
Here the children make mistakes, so this is a mistake alert. Don't give the range from minus 2 to 2. Give 0 to 2. Give 0 to 2 closed interval. Yes or no? Did you understand?
Everyone? Now see this. You are doing the same thing. You said y is equal to Root of x square minus 4. Okay. I have squared both sides.
So what came? y square is equal to x square minus 4. That means x square came y square plus 4. Listen. This is how it is. You can say x square is greater than or equal to 0. No difference. because y square plus 4 is not greater than or equal to 0. So, here there is nothing.
Here the range is written like this for you. See, the problem could have come in y square minus 4. What is there in plus 4? y square is positive as it is.
And this is also positive. So, according to everyone, positivity is what is needed. This is a question of that level which is not easy for kids to answer. But boss, we are taking this challenge towards the end. Right?
Let's do the square root thing. y is equal to 1 upon root of x square minus 4. If you square both sides, then what will you get? y square equal to 1 upon x square minus 4. Do the same thing in terms of y. x square y square minus 4y square equal to 1. So, here what will come? x square 1 plus 4y square upon y square.
Now you know this is 1 plus This has to be positive There is only one thing which cannot be This is not equal to 0 This cannot be 0 Then what will be the range? Open interval 0 to infinity You don't have to do anything If it is plus then what will you break up? Neha ma'am Look, there is one thing On one side no one will take the name of break up This is sorted?
Yes or no? Why can't the range be negative? See, actually your range can be negative.
Just remember one thing. Just remember one thing. If square root is involved, then square root's output is not negative. You just remove it there. Okay?
Let's take this last thing. Square both sides karoge. y square is equal to 1 upon 4 minus x square.
Phishe cross multiply kia. Okay, so this is 4y square minus 1 upon y square is equal to x square. Here the matter will be a little serious. y 0 will not happen, it will work.
But x square greater than equal to. to 0. I mean this thing. Because it is minus, then I will have to see.
So 2y minus 1 into 2y plus 1 is greater than equal to 0. Now baby likes wavy. So this is a little challenging question. It's not like that, it's just a wavy curve.
So, plus minus plus. This one will be done. But remember one thing, friends.
What? You will give the answer. But because it's square root, it won't be negative. That means your final answer will be half to infinity.
Yeah? But one thing is, what will happen on 0, I will have to see that. 0, because one more thing is, x squared greater than or equal to 0 and half a.
Only here a little thing can come and stick to you. Can you take 0 from this? Means my output can come 0, let's check that thing. So if I take this, then it will cross multiply and come 0 equal to 1, no. So, things will come from half to infinity.
Yes or no? Did it shine? Did it shine or not?
Come on, amazing, superb. Now, it is like this. You can... you can try out on this.
There is nothing, you have to do the same thing. Take it like this, square both sides, as done earlier. x square is y square plus 16, no, 16 minus y square, reverse it.
Okay, greater than equal to 0 means y square minus 4 square less than equal to 0. So, you know that roots will be. This answer will come. But if it is square root, then it cannot be negative.
So your answer will come from 0 to 4. It will come from 0 to 4. That's it. That's it, son. In the previous one, closed half will be there.
Closed half. Yes, you can take closed. Okay, you can take closed. Okay?
Great, great. Okay, let's move ahead. Now these are some types of functions.
Okay. There is nothing else. In this, your basic graphs are given.
Like linear function which is straight line. I hope everyone knows how to plot. So this is a random example.
example like a car the a which be a sector y equal to X a Tokyo y equal to X plus 1 to X plus 3 y equal to 0 yes a which we have an asset to quadratic function is prokaryote given conic sections carrying it to my score detail I will do it in the same way. This can happen, this can happen, this, this. Now we are writing in terms of y. So, this is an example given. Cubic graph is always made like this.
Remember, whatever is your odd number polynomial, like 3, its one real root is minimum. That means, if you plot x raised to the power 5, then it will cut your x axis minimum. Modulus function, which I will do separately. Reciprocal, polynomial, anything can happen.
Exponential and here log function. And exponent and log are of each other. What is it?
It is inverse. But we will go into depth of all this later. That is all for today. Tell me honestly, did your physics or functions scared you a little bit?
Tell me honestly, did your physics or functions scared you a little bit? I mean, we don't feel that crazy. Right?
This is your 21 days 21 chapters promise where we are setting a foundation a base for you If you want to ask J-level questions then check out the playlist in the description box yeah well a PDF tiny world comm slash J with the Neha ma ticket is for car couple PDF the Jaya jaha pay sorry sorry chapter wise up go J P Y Q K targeted sessions and T abhiyaas key questions aga aga jahe to mock papers then J advance key questions sub kuch mil jayega and escape sauce You will also get JEE advanced material. Everything for free. Only on YouTube.
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No problem. Okay, so if you guys have opened up quickly, then I want to see minimum 100 likes right now. And as soon as the session reaches 1000 likes, I will share my handwritten PDF.
So here is your pinned comment section for the next session. So quickly reach here, like this video and set reminder. Turn on notification.
We are doing complex numbers first and very easily you will find these things very easily. Come on, come on my army I want to see some excitement. Let's see this.
Say it fast. I did it. Say it fast.
I did it. So that I can share DPP with you. So I will show you the plan. The whole thing. Which we are targeting you by doing all the chapters.
Right. And this is the whole playlist. You will get all the sessions here.
Okay, everyone come on, I have to do this work right now for everyone. After liking me quickly, say done. After liking me quickly, say done. Sir, it is like this, in hand... PDF, Telegram, DPP, basically the ones that are going to be shared are standing.
They are saying, do 100 likes and I will share it right now. And yes, all the kids will definitely comment below this video. 100 minutes basics.
What? Functions done. And the other comments that you want to do because that pumps me up.
This is my motivation. Right? This channel.
is this channel is not a business for me Barbara a key cheese Katie who QQ key it is very a son here is sorry sessions go combine turkey course me down the other eye up for body a giant but just to keep this channel with the same vibe and matlab just get a new cease park is our continue pernick a leah make sure key you are subscribing up in a junior scuba tower and escape salsa Make sure that you like and comment if you like this session. Right? 100 minutes of basics and functions done.
And make a hard chart. It looks good. Right everyone?
Come on. This should be done quickly. Ma'am, DPP is on 100 likes only. I just saw a guy answer carry a saha DPP to madam I saw yes is current session disco de cruz cobi-coro and ugly valley go be for the first jacket done caro-taki I'm aga Yes my army?
Make sure that you tell me in the comment section what you liked in the video today. What was the part? Any feedback that you want to share?
Please make sure to do that. always not only like your comments but may koshish karte hu zyada se zyada comments ko jawab dene ki bhi. Right? Everybody come on bachcho.
I want to see this Josh in the chats. Please make sure that you tell me in the comment section what are the areas of improvement. So that we can do this series best for you. And work hard for you in the right direction. Yes or no?
Are you doing this work or not? Everyone, come on. Make sure to do that. And I am going to see you guys very very soon again.
But you have to do this work. DPP will be sharing with you very soon. And these two...
here your DPP will be shared on the telegram group of Mathematical Incline let me tell you it is very safe because all your details are hidden here and as you go to the file section you can see all the sessions DPPs are there with great respect right everybody let's go so thank you so much for being here my army see you again Go to the pinned link and like the session first. Right now. If you want DPP. I will not see 100 likes minimum there. I am not playing DPP.
Do this work quickly. Write it down. What a goon you are.
When you are working so hard, you deserve this much. Yes or no? Come on.
Bilkul graphs and transformations is also a part of this series only. So don't worry about it. Chaliyye.
Great. So, chalo fir, puri ki puri jo aapko playlist hai session ki, wo yaha pe click karoge mil jayegi. Waise description box me bhi you will find the same.
Cool? Awesome. Chalo bachcho then, milte hain kal.
Firse again, we will make complex numbers super duper easy for you. Alright, thank you so much my army for being here. Bye. You are doing the same right? I was doing it You were doing it?
I was doing it I was troubling my children