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00:00:00The most important thing is the trigonometric function domain is the trigonometric function of the trigonometric function
00:00:05This is all about the trigonometric function
00:00:07This is all about the trigonometric function
00:00:08Today, sin inverse x is the domain
00:00:10So, minus 1 to 1
00:00:12Range
00:00:14Minus pi by 2
00:00:16Pi by 2
00:00:18Okay
00:00:19Cos inverse x
00:00:21Well, domain is the domain
00:00:23Minus 1 to 1
00:00:24But now range is 0 to pi
00:00:26I still think that this is going to be a very much learning
00:00:32I am asking you something
00:00:34Any few is a small piece of calculus
00:00:36Chemistry or Math or English
00:00:38I want to say that it is very difficult
00:00:42But I am not going to do my own
00:00:46But I am not going to do my own
00:00:48I am going to do my own
00:00:50Just wait
00:00:51You are working hard for me
00:00:53Why will you go to 10 inverse x?
00:00:55all rail numbers or minus infinity to infinity
00:00:59and range is minus 5 by 2
00:01:02y is equal to cot inverse x
00:01:06all rail numbers or minus infinity to infinity range 0 by 5
00:01:10and cosec inverse x minus 1 by 1
00:01:13and range minus 5 by 2 by 5
00:01:17minus 0
00:01:20and y is equal to seg inverse x
00:01:24so domain minus 1 سے لے کے 1 تک range
00:01:29domain یہ نہیں دیکھو نا
00:01:3110 اور cot کی سی ہمیں
00:01:33sin اور cos کی سی ہمیں
00:01:35seg اور cosec کی سی ہمیں
00:01:37تام range جو ہے وہ الگ الگ آ رہی ہیں
00:01:39یہ 0 pi سے لے کے minus 5 by 2 تک
00:01:43so range سب سے اوپر والی یہ کس کی تھی
00:01:45جی sin کی inverse x
00:01:47پھر cos inverse x
00:01:48یہ 10 inverse x
00:01:50یہ cot inverse x
00:01:51یہ cos second inverse x
00:01:53اور یہ second inverse x
00:01:54so یہ آپ کو بھرحال یاد کرنی ہے
00:01:56trigonometric functions کی domain
00:01:58اور range کیا ہوتی ہے
00:01:59میں بھی دیکھ لیچے
00:02:00sin x کی domain
00:02:03تمام rail number
00:02:04cos x کی بھی تمام rail number
00:02:06یہ دونوں تو سب سے آسان ہے
00:02:08اور range بہت آسان ہے
00:02:10minus 1 سے لے کے
00:02:13plus 1 تک ہی سارے
00:02:14سارے کے لئے
00:02:15یعنی زیرو بھی آتا ہے
00:02:16اور sin اور cos
00:02:17کا گراب بھی دیکھیں
00:02:18تو minus 1 سے لے کے
00:02:191 تک ہی ہوتا ہے
00:02:20تو میرے بھائی
00:02:21بہت سیمپل
00:02:22ویری ویری سیمپل
00:02:23sin اور cos
00:02:24کا ماملہ
00:02:25sin اور cos کی دونوں چیزیں
00:02:26ڈزہ میں پوچھتے ہیں
00:02:27domain بھی پوچھتے ہیں
00:02:28range بھی
00:02:29اب باد یہ
00:02:3010 اور quad کی
00:02:31domain بہت مشکل ہے
00:02:32اور ڈزہ میں پوچھتے بھی نہیں
00:02:34یہ ان کی range پوچھتے ہیں
00:02:35x x such that x belongs to
00:02:38all real number
00:02:39n does not equal to
00:02:402n plus 1
00:02:41upon pi upon 2
00:02:43n n integer
00:02:45اسی طرح x x belongs to r
00:02:47and x does not equal to n pi n n integer
00:02:53اور ان کی range کیا ہوتی ہے
00:02:55real number
00:02:57سارے real number
00:02:5810 اور quad کی range بہت important ہے
00:03:00range ہی پوچھتے ہیں
00:03:01all real number
00:03:02سارے گھر کوسیک کی بات کریں
00:03:05سارے گھر کوسیک کی بات کریں
00:03:06سارے گھر کوسیک کی ڈومین x belongs to x
00:03:08x does not equal to 2
00:03:09n plus 1
00:03:10pi upon 2
00:03:11n is n integer
00:03:13اور کوسیک کی بات کریں
00:03:14x belongs to x
00:03:15x does not equal to n pi n n integer
00:03:19So they also have tension.
00:03:21Cosec and secs are also important.
00:03:25Y greater than or equal to minus 1.
00:03:29Y greater than or equal to 1.
00:03:31Y less than or equal to minus 1.
00:03:33This is the second sec and the next sec is the Cosec.
00:03:35This is a very dangerous question.
00:03:37Range.
00:03:39Square root is 9 minus x square.
00:03:41This range is very important.
00:03:43First of all, square root is equal to y square.
00:03:47If x is equal to y square,
00:03:51x square is equal to 9 minus y square.
00:03:53Then we say x square is greater than or equal to 0.
00:03:57Why don't you listen?
00:03:59If x's value is 0,
00:04:02then it should be positive.
00:04:05The result is 9.
00:04:07And if this is less than 0,
00:04:13then the result goes to imaginary.
00:04:17So remember,
00:04:18if the cost number is equal to equal to 0,
00:04:21then it will be greater than or equal to 0.
00:04:23If this is greater than or equal to 0,
00:04:25then 9 minus y square will be greater than or equal to 0.
00:04:29Because both of them are equal to 0.
00:04:31Now we will do the same way.
00:04:33How can we write it?
00:04:343 minus y, 3 plus y.
00:04:35Now, those who have gone to the last class,
00:04:37they will see more than or less.
00:04:39Those who have gone to the last class,
00:04:40they will also see.
00:04:41Now, see.
00:04:42We call our timeline.
00:04:43Minus infinity to minus 3.
00:04:45Minus 3 to 3.
00:04:47And 3 to infinity.
00:04:48In values.
00:04:49In values.
00:04:50Because if we solve here,
00:04:52we have two values.
00:04:54Minus 3 and 3.
00:04:55So we check this out.
00:04:57Minus infinity to minus 3.
00:04:59Let's suppose minus 4.
00:05:01Let's suppose minus 4.
00:05:02If we keep minus 4,
00:05:04then it will be plus and minus.
00:05:07Plus and minus.
00:05:08If we make minus,
00:05:10then our answer is minus infinity to minus 3.
00:05:13Minus 3 to 3.
00:05:15We can do anything.
00:05:160, 1, 2.
00:05:17The most easy is 0.
00:05:18If we keep 0,
00:05:20then the two are plus.
00:05:21The yellow part is plus.
00:05:22Plus plus plus.
00:05:23Plus plus.
00:05:24This is our answer is minus 3 to 3.
00:05:26Okay, dear.
00:05:27Let's see.
00:05:29Then, let's see.
00:05:31From 3 to infinity,
00:05:34we keep 4.
00:05:35We keep 4.
00:05:36We keep 4.
00:05:37The yellow part is negative.
00:05:38This is positive.
00:05:39Negative.
00:05:40Positive.
00:05:41Negative.
00:05:42So it means,
00:05:43that it doesn't matter.
00:05:44Why should minus 3 to 3?
00:05:45Now, we have a twist.
00:05:47We have already mentioned that x square is greater than or equal to 0.
00:05:57So this means,
00:05:58what does it mean?
00:05:59That minus 3 value will not be possible.
00:06:01Well,
00:06:02minus 3 to 3 will not be possible.
00:06:05What do we need?
00:06:07Yes,
00:06:080 or 0 from above.
00:06:10This means,
00:06:11minus 3 and 3 will not be possible.
00:06:13If you have 0,
00:06:140 will be possible.
00:06:15Value will be 0 from 3.
00:06:16It's just that we have more trouble.
00:06:18So,
00:06:19we need to do it.
00:06:20So,
00:06:21range will be 0 from 3.
00:06:22If you have a problem,
00:06:23f of x equal to x minus 1 divided by x plus 4 equal to y.
00:06:27And,
00:06:28if you have a problem,
00:06:29if you have a problem,
00:06:30then you have a problem.
00:06:31So,
00:06:32the domain will be very easy.
00:06:33What is the real number?
00:06:34What is the real number?
00:06:35What do we need?
00:06:36What do we need?
00:06:37It doesn't need to be minus 4.
00:06:38And,
00:06:39this will be my answer.
00:06:40If you have a problem.
00:06:41But,
00:06:42if you have a problem,
00:06:43what do we need to do?
00:06:44What do we need to do?
00:06:45That means,
00:06:46x plus 4
00:06:47x plus 4
00:06:48x plus 4
00:06:49y
00:06:50equal to x minus 1.
00:06:51Now,
00:06:52what do we need to do?
00:06:53This minus 1 is plus 1.
00:06:55This minus 1 is plus 1.
00:06:56This minus xy is minus xy.
00:06:58Here,
00:06:59x is common.
00:07:00So,
00:07:01x is common.
00:07:02So,
00:07:03x is equal to 4y plus 1,
00:07:041 minus y.
00:07:05Then,
00:07:06what range will be?
00:07:07Yes.
00:07:08All value,
00:07:09exception of 1.
00:07:10Because,
00:07:11y is equal to 1.
00:07:12So,
00:07:13now,
00:07:14my answer will be the range.
00:07:16And,
00:07:17the domain has already told you.
00:07:18A very important topic is
00:07:20even and odd function.
00:07:22What are the even and odd functions?
00:07:25What are the even functions?
00:07:26If,
00:07:27f minus x,
00:07:28fx,
00:07:29then,
00:07:30even function is even.
00:07:31But,
00:07:32if,
00:07:33f minus of x,
00:07:34minus of x,
00:07:35then,
00:07:36this is odd.
00:07:37So,
00:07:38even function,
00:07:39first,
00:07:40x is given even power.
00:07:41Then,
00:07:421 upon x,
00:07:43even,
00:07:44then,
00:07:45then,
00:07:467,
00:07:47or,
00:07:48constant number.
00:07:49Even,
00:07:50even,
00:07:51even.
00:07:52Even,
00:07:53even.
00:07:54Even,
00:07:55even.
00:07:56Even,
00:07:57even.
00:07:58Even.
00:07:59Even.
00:08:00Even.
00:08:01Even.
00:08:02Even.
00:08:03Even.
00:08:04Even.
00:08:05Even.
00:08:06Even.
00:08:07Even.
00:08:08Even.
00:08:09Even.
00:08:10Even.
00:08:11Even.
00:08:12Even.
00:08:13Even.
00:08:14Even.
00:08:15Even.
00:08:16Even.
00:08:17Even.
00:08:18Even.
00:08:19Even.
00:08:20Even.
00:08:21Even.
00:08:22Even.
00:08:23Even.
00:08:24Even.
00:08:25Even.
00:08:26Even.
00:08:27Even.
00:08:28Even.
00:08:29Even.
00:08:30Even.
00:08:31Even.
00:08:32Even.
00:08:33Even.
00:08:34Even.
00:08:35Even.
00:08:36Even.
00:08:37Even.
00:08:38Even.
00:08:39Even.
00:08:40Okay, now if some even function multiply or divide even function, then what will happen?
00:08:47Well, example, look, x power is 6, x square, both even power are 6 and 2.
00:08:54You know that the basis of power is 6 plus 2 is 8.
00:08:58This is even.
00:08:59Now, if we divide and divide, then we subtract from 6 minus 2 to 4.
00:09:03This is even.
00:09:05This is even.
00:09:06Now, if we multiply the odd function and divide the odd, then it will be even.
00:09:11Like x5 and x3.
00:09:13If we multiply, then we think that 5 plus 3 is 8.
00:09:17And 5 minus 3 is 2.
00:09:20It will be even.
00:09:22Now, we have talked about this.
00:09:25Okay, now we have talked about x square minus 9 is equal to y.
00:09:32Now, what is going to happen?
00:09:37x square is equal to 9 plus y square.
00:09:39If x square is greater than.
00:09:41And equal to 0, then 9 plus y square.
00:09:43Now, these are not factors.
00:09:45Both values are plus.
00:09:47Both values are plus.
00:09:48Then the range is 0 to infinity.
00:09:50This is an easy shortcut.
00:09:53Yesterday, x square minus 25, x square minus 16.
00:09:57No format is equal to square root and x square is equal to constant.
00:10:02Now, if you say, if it is minus 3,
00:10:04which is no square.
00:10:06Why not?
00:10:073 root 3 is equal to square.
00:10:08The answer always is 0 to infinity.
00:10:11Well, remember, when it was 9 minus x square,
00:10:14the answer was 0,3.
00:10:16Yes, if it is 25 minus x square,
00:10:20the answer is 0,5.
00:10:22If it is 16 minus x square,
00:10:24the answer is 0,4.
00:10:26I'm telling this slit trick,
00:10:28it's easier.
00:10:29Line line line system,
00:10:31it's not the same.
00:10:32I'm telling everything.
00:10:33I'm telling everything.
00:10:34I'm telling everything.
00:10:35I'm telling everything.
00:10:36I'm telling everything.
00:10:37Y equal to f of x.
00:10:38Y equal to x minus 1.
00:10:39If it is 0,1.
00:10:40If it is 0,1.
00:10:41Let me tell you.
00:10:42What do we do?
00:10:43Range and domain.
00:10:45Well, domain.
00:10:47We did a lot of detail.
00:10:48If it is 0,1.
00:10:49If it is 0,1.
00:10:50If it is 0,1.
00:10:51And f of x is 0,1.
00:10:53Then we always do it.
00:10:54We always do it.
00:10:55We always do it.
00:10:56If it is 0,1.
00:10:57If it is 0,1.
00:10:58If it is 0,1.
00:10:59If it is 0,1.
00:11:00If it is 0,1.
00:11:01x equals to y plus 1.
00:11:02Where y belongs to rail number.
00:11:04Just like if it is 0,1.
00:11:05If f of x is 0,1.
00:11:07x minus 2 is 0,3 minus x.
00:11:08Then we take y.
00:11:09If it is 0,1.
00:11:10If it is 0,1.
00:11:11If it is 0,1.
00:11:12If it is 0,1.
00:11:13It is 0,1.
00:11:14It is 0,1.
00:11:15We say that all rail number.
00:11:17Except minus 3.
00:11:19Because if it is minus 3.
00:11:21What happens to infinity?
00:11:22It becomes infinity.
00:11:23But when we take a range.
00:11:25What do we do?
00:11:26We have to take x to one side.
00:11:27So you cross multiply.
00:11:29You can do it.
00:11:303 minus x.
00:11:31Do it.
00:11:32From y.
00:11:33Cross multiply.
00:11:34Take x to one side.
00:11:36You can do it.
00:11:383y plus 2.
00:11:39Divide by 1 plus y.
00:11:41Well, now y.
00:11:42What happens.
00:11:43What happens.
00:11:44We say.
00:11:45All rail number.
00:11:47Except minus 1.
00:11:48Why?
00:11:49Because if it is minus 1.
00:11:51So 1 minus 1.
00:11:53Infinity.
00:11:54So now.
00:11:55From this method.
00:11:56The same.
00:11:57The same.
00:11:58We will take range.
00:11:59The same trick.
00:12:00We will take range.
00:12:01The difference.
00:12:02The question.
00:12:03When we give x.
00:12:04We will convert it.
00:12:05In y.
00:12:06In one side.
00:12:08Then x.
00:12:09We will take a range.
00:12:10Like we have not done.
00:12:11And the final answer.
00:12:13We have to.
00:12:14The rail number.
00:12:15Minus minus 1.
00:12:16The range.
00:12:17All rail number.
00:12:18Minus minus 1.
00:12:19Let's see.
00:12:20Here.
00:12:21Some other.
00:12:22Short trick.
00:12:23Short trick.
00:12:24Short trick.
00:12:25So.
00:12:26All time.
00:12:27I will take.
00:12:28Long trick.
00:12:29Then.
00:12:30Now.
00:12:31I will take.
00:12:32The answer.
00:12:331 upon square root.
00:12:35Then.
00:12:36After.
00:12:37Minus minus.
00:12:38We will also.
00:12:39Minus.
00:12:40Minus.
00:12:41Minus.
00:12:42Minus.
00:12:43Minus.
00:12:44Minus.
00:12:45Minus.
00:12:46Minus.
00:12:47Minus.
00:12:48Minus.
00:12:49Minus.
00:12:50Minus.
00:12:51Minus.
00:12:52Minus.
00:12:53Minus.
00:12:54Minus.
00:12:56no
00:12:59here
00:13:09so
00:13:10so
00:13:16it
00:13:23close bracket with union 3 to infinity close bracket so
00:13:28so I set a chart and put it in the mind so don't have to do all of this
00:13:33very dangerous thing. This is a very dangerous thing. This is also going to play.
00:13:37X square minus 9 was given in the numerator. Now what did you do in the denominator?
00:13:42Let's leave that thing. We know that when the denominator is root
00:13:47then X square minus 9 is greater than 0. When it is in the numerator
00:13:51we say greater than or equal to 0 and we say greater than 0.
00:13:55A square minus B square formula looks like A minus B plus B greater than 0.
00:14:01We will call the timeline minus infinity to infinity to infinity to minus 3
00:14:06and plus 3. The other thing is minus infinity to minus 3 minus 4.
00:14:11The two brackets are minus and minus plus. The whole calculation is like this.
00:14:16Then we keep it from 0. When you keep it from 0 to 2 then it will be minus 1 plus
00:14:22and plus. The overall minus is not included. The overall minus is not included.
00:14:25When you keep it from x square then it will be plus and plus.
00:14:28Zono plus plus. Now what is that?
00:14:31The answer is minus infinity to minus 3 and minus 3 is infinity.
00:14:35Now let's see how this is, which is greater than and equal to at the time.
00:14:43Then the answer is greater than.
00:14:49Now let's see how this is.
00:14:50but in minus 3 and plus 3, which bracket is also round bracket
00:14:54which is minus infinity or infinity
00:14:56Why is that? It's greater condition
00:14:58Second last question, we have done this
00:15:02It's like the old one
00:15:04Remember the domain ID is minus 3 and 3
00:15:08But it's close bracket
00:15:10What time it's round bracket
00:15:12Please do this
00:15:14x square minus 9
00:15:16f of x
00:15:18So square root we call greater than no equal to 0
00:15:20A square minus b square
00:15:22formula
00:15:24It seems like this
00:15:26But answer is very different
00:15:28x minus 3 x plus 3 greater than 0
00:15:30What do we do?
00:15:32Number 9
00:15:34Minus infinity to minus 3
00:15:36Minus 3 to 3
00:15:38Minus infinity to minus 3
00:15:40This is an easy number
00:15:42Now keep minus 5
00:15:44Keep anything
00:15:46Well, when we put here
00:15:48Minus 4
00:15:50Minus minus 3
00:15:52Minus 7
00:15:54Minus 7
00:15:56Minus 7
00:15:58Minus 7
00:16:00Minus 3
00:16:02Minus infinity to minus 3
00:16:04Well, now
00:16:18Miami
00:16:23Minus 8
00:16:24Minus 9
00:16:26Minus 19
00:16:32infinity. So how will it be?
00:16:34It will be like this.
00:16:36We have made a bracket of minus infinity to minus 3.
00:16:38Union 3 to infinity.
00:16:40Now one more question.
00:16:42Minus infinity and infinity round bracket.
00:16:44Minus 3 and plus 3 are in close bracket.
00:16:48I have two questions.
00:16:50I will give you two questions.
00:16:52I will do it now.
00:16:5416 minus x square and x square minus 16.
00:16:57One is just like this and one is just like this.
00:17:02So what do we say?
00:17:06Yes.
00:17:08When I said numerator and square root of expression
00:17:12greater than or equal to zero.
00:17:149 minus x square greater than or equal to zero.
00:17:17This is a square minus b square.
00:17:19Which is a minus b, a plus b.
00:17:213 minus x into 3 plus x greater than or equal to zero.
00:17:27Well, when this situation comes,
00:17:31what do we define?
00:17:33Time line define.
00:17:35Minus infinity to minus 3.
00:17:38Then minus 3 to 3.
00:17:403 to infinity.
00:17:41So minus infinity to minus 3 minus 4.
00:17:44Put here.
00:17:46Here minus 4.
00:17:48Then plus 7.
00:17:49Then minus minus here.
00:17:51Plus minus minus here.
00:17:53Minus infinity to minus minus 3.
00:17:55Minus infinity to minus 3.
00:17:56Then minus 3.
00:17:57I will try to type things.
00:17:59Well, minus 3 to 3.
00:18:00There is no number in the middle.
00:18:022, 1.
00:18:03It is easier to keep that.
00:18:04It is easier to keep 0.
00:18:051, 0.
00:18:06Then, let's keep this plus 3.
00:18:08Here, here, here, and here.
00:18:09There is plus 3.
00:18:10If both plus then, there is plus or not.
00:18:12Here, it is a domain.
00:18:143 to infinity, 4 to infinity, it's easy to find 4.
00:18:193 minus 4 minus 1, 4 plus 3 minus 7, minus 1 plus 1 minus, this is not possible.
00:18:26So the answer between minus 3 to 3.
00:18:28Now look, here less than or greater is not a problem.
00:18:31So minus 3 plus 3, which bracket will be square bracket or rectangle bracket?
00:18:37Round bracket is when it comes to infinity minus infinity or less less.
00:18:42So this is the square bracket.
00:18:44So it's possible to give them the option.
00:18:46Minus 3 to 3.
00:18:482 square brackets, 2 square brackets, 2 round, 1 square, 1 round.
00:18:53So this concept is also good.
00:18:55In NERS, FAST, ADD, ADD, so many times.
00:18:59Square root x minus 2 and here 3 minus x domain.
00:19:04Okay, one thing first.
00:19:06Denominator should not be zero.
00:19:08X does not equal to zero.
00:19:10The expression in square root must be greater than or equal to zero.
00:19:15These are our basic things.
00:19:16Are or not?
00:19:17Not at the beginning.
00:19:18I haven't told you before.
00:19:20Now, some people know what to do.
00:19:23Allah will forgive.
00:19:24Allah will forgive.
00:19:25We will forgive this.
00:19:27Zero from zero.
00:19:29To x, y, no.
00:19:30This is not equal to.
00:19:32It is greater than or equal to.
00:19:34This is not equal to.
00:19:35Now, my child, tell us greater than zero,
00:19:36my child, come to the end.
00:19:37It is greater than 0.
00:19:38It is greater than 0.
00:19:39i can tell us that greater than 0.
00:19:40This is greater than 0.
00:19:41�� is negative.
00:19:42It is negative.
00:19:43I can prove each other thing.
00:19:44The fact is worth it.
00:19:45I can prove every before.
00:19:47To prove it.
00:19:48To prove it.
00:19:49To prove it.
00:19:50To prove it.
00:19:51It is greater than 0, equal to 0, less than 0.
00:19:54so I will do a job
00:19:56denominator in 3-x
00:19:58or not
00:20:00denominator and numerator
00:20:02both 3-x multiply
00:20:04this is a big key step
00:20:06now what is it?
00:20:08I have done it
00:20:10now what is it?
00:20:12any kind of square is positive
00:20:14minus 7 square
00:20:16minus 9 square
00:20:18why did I do this?
00:20:20because I had to say
00:20:22it is
00:20:24must be greater than
00:20:26or equal to 0
00:20:28now this is a whole thing
00:20:30but one more thing
00:20:323-x
00:20:34where was it?
00:20:36root
00:20:38root
00:20:40root
00:20:423-x
00:20:44greater than 0
00:20:46x
00:20:48equal
00:20:50this must be greater than
00:20:520
00:20:53now
00:20:54we have to go
00:20:56gravy
00:20:58why did we do it?
00:21:00here we have two things
00:21:02one is a
00:21:043-x
00:21:06what should we do?
00:21:08greater than or equal to 0
00:21:10okay?
00:21:12well
00:21:13when we see
00:21:14this type of domain
00:21:15question
00:21:16so now
00:21:17the first question
00:21:18which we don't have to do
00:21:19number 9
00:21:20is also in months
00:21:21I will tell you
00:21:22what to do
00:21:23a little brain
00:21:24use it
00:21:25those kids
00:21:26don't understand
00:21:27that it's not
00:21:28any relationship
00:21:29before
00:21:30it is
00:21:31scalar vector
00:21:32this is
00:21:33typical
00:21:34important
00:21:35this topic
00:21:36repeat
00:21:37here
00:21:39minus
00:21:40infinity
00:21:412
00:21:42line
00:21:431
00:21:442
00:21:453
00:21:461
00:21:473
00:21:48infinity
00:21:49we see
00:21:50where is
00:21:51valid
00:21:52well
00:21:53minus
00:21:54infinity
00:21:552
00:21:560
00:21:570
00:21:580
00:21:59plus
00:22:003
00:22:01minus
00:22:02minus
00:22:03minus
00:22:04minus
00:22:052
00:22:06positive
00:22:07positive
00:22:08positive
00:22:09positive
00:22:10positive
00:22:11yes
00:22:12answer
00:22:14how can we
00:22:15between
00:22:162 and 3
00:22:17between
00:22:182 and 3
00:22:192 and 3
00:22:20keep the value
00:22:212.5
00:22:22x
00:22:232.5
00:22:242
00:22:25minus
00:22:262
00:22:27positive
00:22:28positive
00:22:29positive
00:22:30positive
00:22:31yes
00:22:32answer
00:22:33between
00:22:34two
00:22:35three
00:22:36two
00:22:37infinity
00:22:38domain
00:22:39or
00:22:40no
00:22:41so
00:22:42three
00:22:43four
00:22:44five
00:22:45can
00:22:46check
00:22:47four
00:22:48four
00:22:49put
00:22:50x
00:22:51minus
00:22:52four
00:22:53plus
00:22:54minus
00:22:55minus
00:22:56two
00:22:57two
00:22:58three
00:22:59two
00:23:00three
00:23:01square
00:23:02bracket
00:23:03two
00:23:04two
00:23:05three
00:23:06two
00:23:07three
00:23:08three
00:23:09two
00:23:10three
00:23:11round
00:23:12two
00:23:13one
00:23:14minus
00:23:15two
00:23:16zero
00:23:17zero
00:23:18zero
00:23:19zero
00:23:21zero
00:23:22zero
00:23:23zero
00:23:24zero
00:23:25so two is valid so two we will use closed bracket so my brother
00:23:33further we will discuss this number line in the system so we will not have a lot of questions
00:23:39f of x is equal to 1 upon 1 minus x so we are saying for fx to be defined now see
00:23:45when I told you that square root view in the denominator then what will be greater than 0
00:23:52one minus is greater than 0 now see a big emphasis when this is greater less
00:23:57condition is not equal to do it in a big way so the difference is that x minus
00:24:02if we take it from the left we can take it from the left we can take it from the left
00:24:05of x less than 1 so this means value is small from 1 from 1 to 1 to 1 from 1 from 1 to 1
00:24:17minus infinity where can I get from minus infinity? Well, from minus infinity.
00:24:23If I get from minus infinity, then I will make one to infinity.
00:24:29Now my domain will make minus infinity.
00:24:32Which bracket will be round or square?
00:24:36Square bracket is when the number is included.
00:24:39If it is equal to 2 or less than or equal to 2, I will make square bracket.
00:24:45But it is less than 1.
00:24:48If it is less than 1, then we will make round bracket.
00:24:52And infinity or minus infinity will always make round bracket.
00:24:56Keep this in mind.
00:24:58So, this is your answer.
00:25:00Very, very interesting question.
00:25:02x squared plus 2x plus 1 divided by x squared minus 8x plus 12 domain.
00:25:10What will I talk about?
00:25:12First of all, I will be denominated.
00:25:14Denominator.
00:25:15Denominator.
00:25:16Denominator.
00:25:17Denominator.
00:25:18Denominator.
00:25:19Denominator.
00:25:20X squared minus 8x plus 2.
00:25:21Now the factors are very easy.
00:25:23Matrix.
00:25:24From the class.
00:25:25Those who don't have to know.
00:25:26Tell them.
00:25:27The factor.
00:25:28The factor.
00:25:29The factor.
00:25:30The factor.
00:25:31The factor.
00:25:32The factor.
00:25:33The factor.
00:25:34The factor.
00:25:35The factor.
00:25:37The factor.
00:25:38The factor.
00:25:39The factor.
00:25:40The factor.
00:25:41The zero factor.
00:25:42The factor.
00:25:43I received my six value.
00:25:44I just would say x-6 multiplied by x-2.
00:25:46does not equal to zero
00:25:48meaning domain
00:25:49all rail numbers
00:25:51can be used
00:25:52which one should be used
00:25:54x minus 6
00:25:56does not equal to zero
00:25:58x does not equal to 6
00:26:00x does not equal to 2
00:26:016 and 2
00:26:02all numbers can be used
00:26:05which one cannot be used
00:26:07which one cannot be used
00:26:08so we can say domain
00:26:10how easy it is
00:26:12now this domain is released
00:26:14square root
00:26:16square root
00:26:17what trick did you do
00:26:18that the expression is greater than
00:26:20or equal to zero
00:26:21okay
00:26:22so mass rule
00:26:24this is minus 2
00:26:25here is plus 2
00:26:26we can say x is greater than
00:26:28or equal to zero
00:26:29meaning domain
00:26:30where can we take it from 2
00:26:32from infinity
00:26:33from infinity
00:26:35so we have to have very strong
00:26:37concept
00:26:382
00:26:38square bracket
00:26:41infinity round bracket
00:26:43why is that
00:26:44infinity
00:26:46if there is no number
00:26:47then we have to assume
00:26:48what is happening
00:26:49so we have to assume
00:26:50what is happening
00:26:51here we have to write it
00:26:52round bracket
00:26:53infinite
00:26:54infinity
00:26:55where is going to infinity
00:26:56again
00:26:57look at the top
00:26:58big simple
00:26:59denominator
00:27:00like x minus
00:27:01minus 6
00:27:02and x minus 2
00:27:03what should we do
00:27:04what should we do
00:27:05what should we do
00:27:06what should we do
00:27:07that
00:27:08denominator
00:27:09does not equal to 0
00:27:10x minus 6
00:27:11does not equal to 0
00:27:12x does not equal to 6
00:27:13x does not equal to 6
00:27:14to 2
00:27:16and here
00:27:17we can see
00:27:18value
00:27:19from 2
00:27:20infinity
00:27:21we will make
00:27:22square bracket
00:27:23and here
00:27:24infinity
00:27:25round bracket
00:27:26here
00:27:27domain
00:27:28cut
00:27:29we
00:27:30talk about
00:27:31domain
00:27:32what is
00:27:33basically
00:27:34first
00:27:35domain
00:27:36range
00:27:37here
00:27:38when we talk about
00:27:39inter
00:27:48domain
00:27:49inside the function
00:27:50and fx
00:27:51we say all values of x
00:27:52for which fx is defined
00:27:53what does
00:27:54mean
00:27:55first
00:27:56f of x
00:27:57x
00:27:58q minus 3
00:27:59now
00:28:00any value of x
00:28:01has no valid answer
00:28:03that will be our domain
00:28:05in this case
00:28:06we will say
00:28:07all the real number
00:28:08because
00:28:09any real number
00:28:10has no result
00:28:11imaginary
00:28:12should not come
00:28:13right
00:28:14domain
00:28:15one thing
00:28:16y equal to
00:28:18f of x
00:28:19should not
00:28:20denominator
00:28:21zero
00:28:22another
00:28:23square root
00:28:24if there is
00:28:25an expression
00:28:26then
00:28:27equal to
00:28:28zero
00:28:29or greater than
00:28:30zero
00:28:31what
00:28:32should not
00:28:33denominator
00:28:34zero
00:28:35if there is
00:28:36square root
00:28:37equal to
00:28:38or greater than
00:28:39zero
00:28:40well
00:28:41if
00:28:42denominator
00:28:43and expression
00:28:44square root
00:28:45one thing
00:28:46must
00:28:47be
00:28:48greater than
00:28:49zero
00:28:50because
00:28:51equal to
00:28:52zero
00:28:53is
00:28:54greater
00:28:55here
00:28:56until
00:28:57this
00:28:58is
00:28:59very
00:29:00awesome
00:29:01trick
00:29:02y
00:29:03is
00:29:04equal to
00:29:05x
00:29:06plus
00:29:07y
00:29:08minus
00:29:098
00:29:10g of z
00:29:11z
00:29:12z
00:29:13square
00:29:14is
00:29:15z
00:29:16square
00:29:17left
00:29:18now
00:29:19where
00:29:20i
00:29:32into
00:29:33x
00:29:34So, I have given the value of 4 to Z. So, I have to put the value of 4 to Z. So, let's do it.
00:29:41So, let's do it. G of 4, 4 square is 16.
00:29:45Now, I have to get out of 16.
00:29:48Now, I have to get out of 3, G, 4.
00:29:52So, well, let's see here, 3x plus 2y minus 8.
00:29:55So, it's 3 written.
00:29:56What will we do in the place of x?
00:29:59Yes, we will write 3.
00:30:013, 3s are 9.
00:30:03Plus 2.
00:30:04Yes, G.
00:30:05See, 2y was 2y.
00:30:07So, y was 16.
00:30:12I put 16.
00:30:14Minus 8 has a test.
00:30:16So, now, 3, 3s are 9.
00:30:192, 16s are 32.
00:30:21Minus 8.
00:30:22Everything is 33.
00:30:24Well, it's a big problem.
00:30:26fx, y, z,
00:30:28equal to x square plus 2y square minus 2z.
00:30:31x is minus 1 put,
00:30:33y is 1,
00:30:34z is minus 1.
00:30:35And now, the answer will come.
00:30:36Now, the question is,
00:30:37f of x equal to x minus 1,
00:30:39h of x equal to 5 upon x.
00:30:42The question is,
00:30:43g of x equal to 3x.
00:30:47I am saying,
00:30:48g of x is minus 1,
00:30:50z has 3.
00:30:52At this point,
00:30:53y is schools from x north.
00:30:55You can use 8 part of x minus 1 over x.
00:30:57So, steps by steps.
00:30:58Someßen also truths,
00:30:59this can beidel example.
00:31:01H is the first placecess.
00:31:02So, the point of x!
00:31:04The point of x?
00:31:05That means 4 axis is the person,
00:31:065 upon 5?
00:31:07So, I am the equivalence to x.
00:31:08So, g of x is our power.
00:31:10we have to get 1
00:31:12here
00:31:141 put 3
00:31:163
00:31:18g h of 5
00:31:203
00:31:22now
00:31:24me here
00:31:26here
00:31:28x-1
00:31:30which is not known
00:31:32a x plus b upon c
00:31:34this is where the two
00:31:36cx minus b upon a
00:31:38x minus 1 upon 1
00:31:401
00:31:42where is it?
00:31:44denominator
00:31:45what happens
00:31:46this is the sign
00:31:48here
00:31:49here
00:31:50here
00:31:51x plus 1
00:31:52this sign should be
00:31:53plus b
00:31:54minus b
00:31:55minus 1
00:31:56here
00:31:57f inverse x
00:31:58we have to get out
00:32:00f inverse
00:32:01g h of 5
00:32:02g h of 5
00:32:03value
00:32:04so
00:32:05valid here
00:32:06here
00:32:07x plus 1 upon 1
00:32:08we have to get this value
00:32:093 put
00:32:103 plus 1
00:32:12which is 4
00:32:13and this is our answer
00:32:15so
00:32:16how many types of questions
00:32:17are
00:32:18very detailed
00:32:19here
00:32:20all topics
00:32:21in theory
00:32:22in aptitude test
00:32:23every topic
00:32:24get needed
00:32:25fx
00:32:26e
00:32:27power
00:32:287x
00:32:29plus root
00:32:302
00:32:31gf
00:32:32x
00:32:33and this is gx
00:32:34well
00:32:35ln 2x minus root 3 upon 7
00:32:382x minus root 3 upon 7
00:32:402x minus 3
00:32:41So what will we do?
00:32:43We will do this place
00:32:46In option
00:32:47We will put something here
00:32:48Now think about
00:32:50Question in E is there
00:32:52Exponential
00:32:53Exponential can be done
00:32:55When ln
00:32:56Because both are reciprocal
00:32:58Is there any option
00:33:00Which ln is in the first one
00:33:02So it can be the first one
00:33:04If ln is in the first one
00:33:05I will not think about it
00:33:07But because ln is in the first one
00:33:09We will put it
00:33:11Now we will put it
00:33:13When ln and e will die
00:33:16Now what is ln?
00:33:182x minus root 3 upon 7
00:33:20So here with 7
00:33:222x minus root 3 upon 7
00:33:25Plus root 3
00:33:27Like this upon 2
00:33:29Now when we will solve this
00:33:32If we will solve this
00:33:33So this will be 7
00:33:347 cancel
00:33:35Or not
00:33:362x minus root 3
00:33:38And here with plus root 3
00:33:39And here with plus root 3
00:33:40And here with minus root 3
00:33:41And plus root 3
00:33:42And 2 to cancel
00:33:43Of course
00:33:44Of course x
00:33:45To keep this
00:33:46So this will be the option
00:33:47If this will be the option
00:33:48If ln 2 options
00:33:49Then we will see
00:33:50If ln is the option
00:33:51Then it will not be the option
00:33:52So this will not be the option
00:33:53This will be the option
00:33:54So this will make the option
00:33:55To be the option
00:33:56And the option
00:33:57To be the option
00:33:58To be the option
00:33:59Of course
00:34:00x is equal
00:34:01Plus bx plus c
00:34:03And f minus 3
00:34:040
00:34:05And f1
00:34:060
00:34:07So b plus c
00:34:08we need to see who we need to see
00:34:11target
00:34:13express news
00:34:14slogan
00:34:15we need to see target
00:34:18b plus c
00:34:19now
00:34:20one thing
00:34:22y
00:34:23is
00:34:24equal to case
00:34:25x
00:34:26x
00:34:27x
00:34:28x
00:34:29x
00:34:30x
00:34:31x
00:34:32x
00:34:33x
00:34:34x
00:34:35x
00:34:36x
00:34:37x
00:34:38x
00:34:39x
00:34:40x
00:34:41x
00:34:42x
00:34:43x
00:34:44x
00:34:45x
00:34:46x
00:34:47x
00:34:48x
00:34:49x
00:34:50x
00:34:51x
00:34:52x
00:34:53x
00:34:54x
00:34:55x
00:34:56x
00:34:57x
00:34:58x
00:34:59x
00:35:00x
00:35:01x
00:35:02x
00:35:03x
00:35:04x
00:35:05x
00:35:06I don't need to do something else.
00:35:09f of x gives 3x minus 1 upon 2, g of f of x gives 2x plus 1 upon 3, 2x plus 1 upon 3.
00:35:19You get more questions online, and this is the advantage.
00:35:24And I'm very willing to do this, because I think it's good practice.
00:35:27And if you practice good, then you can do it.
00:35:30What is the problem?
00:35:31So here we put something like this, where x is the result of x.
00:35:38Now give a option, 2x plus 1 upon 3, 2x plus 1 upon 3.
00:35:42Now we put all the time together, so it's time.
00:35:45It's so important that there is no chance of which 1 upon 3.
00:35:49Like 1 upon 3, if I put 1 upon 3, then there will not be x.
00:35:53I don't see 2x plus 1 on 2, but here there is a lot of chance.
00:35:59So while put it, here we put 2x plus 1 upon 3, and see that there is x.
00:36:05Let's do it and see.
00:36:07The first step is done.
00:36:10Okay.
00:36:112x plus 1 minus 1 upon 2, this is also the answer.
00:36:15This is also the answer, the answer is x.
00:36:17So the answer is the option.
00:36:19So the answer is the answer.
00:36:21I have some graphs of the videos that I have added.
00:36:23I have some graphs of the group.
00:36:25I have some graphs of the group.
00:36:27Let's see.
00:36:28Let's see.
00:36:29Now f of x is equal to 2 upon x plus 5.
00:36:30And g of x is equal to x.
00:36:32How much will x is equal to x?
00:36:35Is it 2 upon x minus 5?
00:36:37Is it 7?
00:36:38Is it 3x?
00:36:39Is it x square?
00:36:40I have some question about that, but this is the trick.
00:36:43This is f of x.
00:36:44Here we put what we put here?
00:36:47What do we put here?
00:36:48gx.
00:36:49What we put here?
00:36:50What should we put here?
00:36:51x.
00:36:51In fact, 2 will be 2 upon 2, plus 5.
00:36:53This will be just the option.
00:36:55Well, I will give you an answer.
00:36:57Make sure everything is good.
00:36:58x square will not come.
00:37:017 will not come.
00:37:04That is constant number is not 3x.
00:37:06Well, I have seen anything from this.
00:37:08Let's confirm.
00:37:09we do this idea.
00:37:10Now what is this?
00:37:13Let's see here.
00:37:142 is like this.
00:37:15Plus 5.
00:37:162x minus 5.
00:37:19This is my brother.
00:37:22So put it.
00:37:24This is cancelled.
00:37:26This is cancelled.
00:37:27I will go down the lower things.
00:37:30And I need to get this.
00:37:33So what will my answer be?
00:37:35This is the same year.
00:37:37And I'm going to do it.
00:37:39I'm going to do it too.
00:37:40Well, let's see.
00:37:42Let's see.
00:37:44F of the fxdi.
00:37:46The question is 4x minus 5.
00:37:49And the gxdi is 3x.
00:37:51Today we have learned a lot.
00:37:52Now we need fg of 2.
00:37:54So it doesn't matter.
00:37:56It doesn't matter.
00:37:57It doesn't matter.
00:37:58It doesn't matter.
00:37:59It doesn't matter.
00:38:00It doesn't matter.
00:38:01It doesn't matter.
00:38:02It doesn't matter.
00:38:04It doesn't matter.
00:38:06Now we need fg of 2.
00:38:09So let's see.
00:38:10F of x is equal to 4x minus 5.
00:38:13What can I put in this place?
00:38:16This is g of 2.
00:38:18This is the g of 2.
00:38:19This is the g of 2.
00:38:20Do it.
00:38:21Do it.
00:38:22While this is done.
00:38:23The phone is inside 36.
00:38:245 minus.
00:38:25It's 31.
00:38:27Now if we can.
00:38:29If we can.
00:38:30If we can.
00:38:31If we can.
00:38:32We can.
00:38:33Fg of x.
00:38:34Which is what happens.
00:38:36So.
00:38:37Where we put.
00:38:38What will we put in x.
00:38:40This.
00:38:413 power x.
00:38:42Now.
00:38:432 should.
00:38:44So we will put here.
00:38:46I will put it in three square nine nine four thirty six minus five thirty one
00:38:51I have to go to the two methods.
00:38:52What do I have to do with the answer?
00:38:54Okay, now a big important question is f of x equals x square minus four.
00:39:00And g of x, I don't know.
00:39:02I have to get out of it.
00:39:03And fg of x equals 4x square minus 8x.
00:39:07Okay, check.
00:39:10Now, how will we get out of it?
00:39:13Now, g of x value.
00:39:16We are going to get out of it.
00:39:18These are four options.
00:39:19x, 4x, x square, 2x minus 4.
00:39:22And today we are going to get out of it.
00:39:23One is f-99.
00:39:25This is a way of question.
00:39:27We will get out of it.
00:39:28We will get out of it.
00:39:29We will get out of it.
00:39:31So, f of x equals x square minus 4.
00:39:35I don't know.
00:39:37fg of x equals 4x square minus 8x.
00:39:41Now, we have given out of this type of question.
00:39:44What should happen?
00:39:46Look at this.
00:39:49This is f of x.
00:39:51x square minus 4.
00:39:55x should be such value.
00:39:57What should happen?
00:39:584x square minus 8x.
00:40:00Now, let's see here.
00:40:024x square minus 8x.
00:40:04I will keep the value of x.
00:40:07So, how can this happen?
00:40:09Can this happen?
00:40:10Can this happen?
00:40:11Can this happen?
00:40:12If x will come here,
00:40:13x square minus 4.
00:40:14It's not going to happen.
00:40:15Can this happen?
00:40:16Can this happen?
00:40:17Can this happen?
00:40:18Let me tell you.
00:40:20No.
00:40:214x square to 16x square.
00:40:23It will not happen.
00:40:24It will not happen.
00:40:25It will not happen.
00:40:26It will happen.
00:40:27It will happen.
00:40:28x square square to x square.
00:40:29No.
00:40:302x minus 2.
00:40:31Yes.
00:40:32It looks like something.
00:40:33Let's see.
00:40:352x minus 2.
00:40:36Put it.
00:40:37It will come.
00:40:38A minus b.
00:40:40All square square.
00:40:41All of this.
00:40:424x square minus.
00:40:438x plus 4 minus 4.
00:40:45Plus 4 minus 4.
00:40:46Cancels.
00:40:474x square minus 8.
00:40:48And this is now.
00:40:50The answer is our way.
00:40:51This is what we should do.
00:40:52Now, let's see.
00:40:54Let's see.
00:40:55So, tomorrow we will have many questions.
00:40:58And hear new kids.
00:40:59Scheduling is like a function.
00:41:02Chapter is starting.
00:41:03Tomorrow we will complete it.
00:41:05Dominant range limits.
00:41:06These are chapters.
00:41:07We will have now.
00:41:08English class.
00:41:10English class.
00:41:11Morning.
00:41:12It is all.
00:41:13It is all.
00:41:14At the time of the time.
00:41:15And tomorrow,
00:41:16Inshallah,
00:41:17I have done a chapter of chemistry.
00:41:19Properties of matter,
00:41:21gas, liquid, solid.
00:41:22I will also do revision.
00:41:24And well,
00:41:26I will also do physics.
00:41:27We will take a small topic tomorrow.
00:41:29So, boys.
00:41:30This is our entire story.
00:41:32Inshallah,
00:41:33This is our answer.
00:41:34Now, let's take a look.
00:41:35F of x is equal to x square plus 3x minus 4.
00:41:38So,
00:41:39g of x, x minus 4.
00:41:40f g of x,
00:41:41g of f of x,
00:41:42equal is equal.
00:41:43Then,
00:41:44what will be x?
00:41:45So,
00:41:46f g of x,
00:41:47g of x,
00:41:48and then,
00:41:49x of x,
00:41:50and then,
00:41:51x of x,
00:41:52and then,
00:41:53x of x,
00:41:54x plus 3x,
00:41:55g of x,
00:41:56x square.
00:41:57And,
00:41:58for what value of x,
00:42:00f g of x is equal to g of x.
00:42:03So,
00:42:04f of x is equal to g of x.
00:42:06So,
00:42:07well,
00:42:08f g of x,
00:42:09is equal to x plus 3x,
00:42:11and then,
00:42:12x of x is equal to x square.
00:42:13So,
00:42:14if I put here,
00:42:16x of x is equal to x square.
00:42:18So,
00:42:19if I put here,
00:42:20x of x is equal to x square.
00:42:22So,
00:42:23g of x is equal to x square.
00:42:25What will happen?
00:42:26This is x plus 3.
00:42:27Or,
00:42:28x is equal to x square plus 3.
00:42:31Which concept?
00:42:32We have a lot of questions.
00:42:34Then,
00:42:35g of x is equal to x square.
00:42:37While,
00:42:38here,
00:42:39f of x is equal to x plus 3.
00:42:43Take this square.
00:42:45Now,
00:42:46we are saying that x is equal to x,
00:42:48when f g of x,
00:42:49g of x is equal to x.
00:42:50So,
00:42:51x square plus 3 is equal to x plus 3 square.
00:42:54Left,
00:42:55x square plus 3.
00:42:56Here,
00:42:57open,
00:42:58x square plus 6x plus 9.
00:42:596x is equal to 6.
00:43:01x is equal to 6.
00:43:02x is equal to x.
00:43:03What will happen?
00:43:04Minus 1.
00:43:05This is a very important topic.
00:43:07This chapter is the composite function.
00:43:08F of x is equal to x plus 3.
00:43:11G of x is equal to x square minus 1.
00:43:14We are saying f g of 2.
00:43:17This is the question of blue color.
00:43:20It's the question of black.
00:43:21Okay?
00:43:22And the two tricks.
00:43:23One,
00:43:24one,
00:43:25one,
00:43:26one,
00:43:27one,
00:43:28one,
00:43:29one,
00:43:30one,
00:43:31one,
00:43:32one,
00:43:33one,
00:43:34one,
00:43:35one,
00:43:36one,
00:43:37one,
00:43:38one,
00:43:39one,
00:43:40one,
00:43:41two,
00:43:42one,
00:43:55one,
00:43:56one,
00:43:57one,
00:43:59This is the first time I put FG of 2 and FG of X
00:44:05So F of X is equal to 2X plus 3
00:44:08X is equal to GX
00:44:11GX is equal to X square minus 1
00:44:14Now I open the bracket
00:44:162X square minus 2 plus 3
00:44:18minus 2 plus 3
00:44:201 is equal to 2X square plus 1
00:44:22Now I put 2 on the bracket
00:44:262 square 4
00:44:272 is equal to 8
00:44:288 plus 1 is 9
00:44:29This is our answer
00:44:30Now we have some magical trick
00:44:32F of X is equal to X7
00:44:34minus 97 X6
00:44:37minus 199 X5
00:44:39plus 99 X4
00:44:41minus 2X
00:44:43plus 199
00:44:45F99
00:44:47How do we do that
00:44:49without a calculator
00:44:50So now my magic starts here
00:44:52The blue color is written
00:44:54The whole question is
00:44:56So now we are magic
00:44:58Magic
00:44:59F99 should be
00:45:01Synthetic division
00:45:02Part 99
00:45:03X is equal to 7
00:45:05What is the option?
00:45:06Nothing happens
00:45:071 is equal to
00:45:08X is equal to 6
00:45:09minus 97
00:45:10X is equal to 5
00:45:11minus 199
00:45:12X is equal to 4
00:45:1399
00:45:14X is equal to 99
00:45:15X is equal to 99
00:45:16So they are equal to 0
00:45:17I will write 0
00:45:18X is equal to 0
00:45:19X is equal to 1
00:45:20minus 2
00:45:21And then it is
00:45:22that it is 190
00:45:23this is 190
00:45:24Okay
00:45:25This is 190
00:45:26Okay
00:45:27Now what will happen?
00:45:29Synthetic division
00:45:30rule of the system
00:45:31First value 1
00:45:33hazardous
00:45:34This one
00:45:36multiply
00:45:3790
00:45:38This is
00:45:39minus 97
00:45:40plus 99
00:45:41How much
00:45:422
00:45:43This
00:45:442
00:45:45multiply
00:45:46198
00:45:47While
00:45:48minus 199
00:45:49and 198
00:45:50minus 1
00:45:51Since
00:45:52minus 1
00:45:5399
00:45:54multiply
00:45:5599
00:45:5699
00:45:57and minus 99
00:45:580
00:45:59Now
00:46:002
00:46:01Then
00:46:02minus 2
00:46:03multiply
00:46:0499
00:46:05So
00:46:06minus
00:46:07198
00:46:08Now
00:46:09190
00:46:10and minus
00:46:11198
00:46:12and
00:46:13this is
00:46:14question
00:46:15answer
00:46:16So
00:46:17as
00:46:18as
00:46:19as
00:46:20as
00:46:21as
00:46:22as
00:46:23as
00:46:24as
00:46:25as
00:46:26as
00:46:27as
00:46:28as
00:46:29as
00:46:30as
00:46:31as
00:46:32as
00:46:33as
00:46:34as
00:46:35as
00:46:36funciona
00:46:37with
00:46:39x
00:46:40rea
00:46:42to
00:46:4310 inverse hyperbolic inverse x 1 upon 2 ln 1 plus x 1 minus x
00:46:51so in the cod inverse 1 upon 2 ln is 1 upon 2 ln
00:46:54the actual x plus 1 is ok
00:46:58below 1 minus x then x minus 1 is going to be
00:47:01and the cosec hyper in inverse x
00:47:07ln 1 plus x square upon x seg inverse 1 plus 1 minus x square
00:47:14پوری سیم فرمولے بس cosec میں ہے plus ہے
00:47:17sec میں ہے minus ہے
00:47:18اس کے بعد ہمارا بڑا important topic ہے inverse function
00:47:21f of x برابر ہے ax plus b upon c کے
00:47:24یعنی کہ y کے
00:47:26تو f inverse x کیا ہوگا
00:47:29well ہمیں یہاں detail میں نہیں چاہنا
00:47:31short trick سے کرنا ہے
00:47:32short trick ہوتی ہے f inverse x equal to cx minus b upon a
00:47:36یعنی کہ دیکھو یہ 3x plus 4 upon 5 تھا
00:47:39تو ہمیشہ رکھو a اور c کی جگہ تبدیل ہوتی ہے
00:47:42b کا sign تبدیل ہوتا ہے
00:47:44جیسے کہ zem miss والا آیا تھا
00:47:453x plus 4 upon 5
00:47:47تو 3 اور 5 کی جگہ تبدیل ہو چاہے گی
00:47:505x اور 4 کا sign
00:47:515x minus 4 upon 3
00:47:54یہ بچوں فوراں یہ فوراں آنسر آ جائے گا
00:47:56اور empty میں ہمیں ایسے ہی آنسر
00:47:58چاہیے ہوتے ہیں zem میں رکھیں
00:48:00ایک اور پاس سیبر کا بڑا اچھا ہے سمال
00:48:023,2 4,2 3,1 7,1 2,3 میں سے
00:48:07کس کو remove کریں
00:48:09کہ یہ چیز function من چاہیے
00:48:10اب اتنی دوم definition دیکھ چکے ہیں
00:48:13کہ یہ جو پہلے والے ہوتے ہیں
00:48:14بچے ایک بچہ 3
00:48:15ایک بچہ 4
00:48:16ایک بچہ
00:48:17ہے پھر 3 آ گیا
00:48:19ایک بچہ 7
00:48:22ایک بچہ 2
00:48:23ایک منٹ
00:48:244 ایک بار آ رہا ہے
00:48:267 ایک بار آ رہا ہے
00:48:272 ایک بار آ رہا ہے
00:48:27تو پنگا یہاں تو کچھ نہیں ہے
00:48:29ٹھیک ہے چھے
00:48:30پنگا یہاں تو کچھ نہیں ہے
00:48:32یہاں تو یہاں ہیں
00:48:34یہاں یہاں ہیں
00:48:34کیونکہ 3,2 کے ساتھ بھی
00:48:363,1 کے ساتھ بھی ہے
00:48:37پھر وہی دو دو اممائے
00:48:38ایسا کیسے ہو سکتا ہے
00:48:40تو ساہرے ان میں سے کسی ایک کو ہٹھائیں گے
00:48:42اب کس کو
00:48:43سوال یہ پیدا ہوتا ہے
00:48:453,1 یا 3,2 تو دیکھو
00:48:47آپشن میں کیا 3,1 موجود ہے
00:48:49نہیں نا
00:48:50فالی کیا موجود ہے
00:48:513,2 تو اس کو ہٹا دو
00:48:53فنکشن من جائے گا یہ
00:48:54سال سر نسٹ میں آتے ہیں
00:48:56باگی کہ نہیں آتے ہیں
00:48:57تو باگی سب ٹینشن نہ لیں
00:48:58کہ بھائی
00:48:59g of x
00:49:00equal to x
00:49:01square
00:49:01اور یہ x کی value
00:49:02گہ رہے ہیں
00:49:03greater than or equal to zero
00:49:05less than or equal to two
00:49:06پھر ایک اور اس میں
00:49:07split function ہے
00:49:08دو پارٹ میں
00:49:083x
00:49:10x is less than or equal to ten
00:49:11اب اس کے حوالے سے
00:49:13پوچھ رہیں گے
00:49:14یہ کیسے ہوگا
00:49:15تو آپ دیکھو
00:49:16یہاں تھوڑا سا
00:49:17مسئلہ ہے
00:49:18تھوڑا سا پنگا ہے
00:49:19اس پنگے کو
00:49:20مسئلے کو ہلکی سے
00:49:21کریں گے
00:49:21دیکھو پہلے تو
00:49:22اگر آپ g of x
00:49:270 سے لے گے
00:49:272 تا پوائنٹ ہے
00:49:280.5 square
00:49:302 upon 3 square
00:49:31اور 2 کا
00:49:31square 4
00:49:32کیا ہے کہ
00:49:33ہر ایک کا
00:49:34آنسر
00:49:34ڈیفرنٹ آنا چاہیے
00:49:35اگر ہر ایک
00:49:36آنسر
00:49:37ڈیفرنٹ آ رہا ہے
00:49:37پوائنٹ پوٹ کر کے
00:49:38تو تھو فنکشن ہے
00:49:39اور کبھی
00:49:402 کا آنسر
00:49:41unique solution
00:49:44جیسے کہ
00:49:45value دیکھو
00:49:450 رکھنے سے
00:49:470.5 سے
00:49:482 رکھنے سے
00:49:482 کا
00:49:49square 4
00:49:49ٹھیک ہے
00:49:50اب زرہا
00:49:52ہی ہاں دیکھو
00:49:523x
00:49:57اب یہ فنکشن ہے
00:49:59یا نہیں
00:49:59دیکھو
00:49:59پنگا کیا ہوا
00:50:00کہ 2
00:50:02پوائنٹ یہاں بھی
00:50:02موجود تھا
00:50:032 پوائنٹ یہاں بھی
00:50:04موجود تھا
00:50:05یہاں پوٹ کرنے سے
00:50:062 کا
00:50:07square 4 آ رہا ہے
00:50:08آنسر
00:50:08یہاں پوٹ کرنے سے
00:50:092
00:50:093
00:50:10سا 6 آ رہا ہے
00:50:11تو پھر وہی
00:50:12دو آنسر
00:50:12کیسے ہو سکتے
00:50:13یعنی وہی
00:50:13دو امائیں
00:50:14بات ہو گئے
00:50:15تو یہ فنکشن
00:50:16نہیں ہو سکتا
00:50:17اب
00:50:17میس میں
00:50:18اگر کوئی ایسا
00:50:19سوال پیپر میں آئے
00:50:20پہلی پوپ
00:50:20تو ایسے آتے نہیں
00:50:21میں ہٹ رہا
00:50:22آ رہا ہوں
00:50:23تم لوگ
00:50:23اگر آ جائے
00:50:24تو پورا چیک
00:50:25مت کرو
00:50:26تم دیکھو
00:50:26فنکشن
00:50:27سپلٹ
00:50:27کہاں پہ
00:50:27ہو رہا ہے
00:50:282 پہ
00:50:282 کامن ہے
00:50:29صرف 2 کو
00:50:30چیک کر کے
00:50:30دیکھو
00:50:30یہاں پوٹ کر کے
00:50:31یہ سب کھڑنے کی
00:50:32چیز ہوتی ہیں
00:50:322 کا
00:50:33square 4
00:50:333
00:50:332
00:50:346
00:50:34الگ الگ
00:50:35آنسر آ رہے ہیں
00:50:35تو فنکشن
00:50:36نہیں ہے
00:50:36لیکن
00:50:49x کی ویلیو
00:50:50ٹو پٹ کرنے سے
00:50:51اور یہاں ٹو پٹ کرنے سے
00:50:52سیم آ رہا ہے
00:50:53تو اس کا
00:50:54لپے فنکشن
00:50:54کیا مطلب ہوا
00:50:55میں زیرو سے لے گئے
00:50:572 تک میں سے
00:50:572 کو
00:50:58چیک کروں گا
00:50:59کیوں 2 یہاں بھی
00:51:00موجود ہیں
00:51:01یہاں بھی
00:51:01یہاں کا
00:51:01ہے
00:51:02یہاں بھی
00:51:02جب ایسا سال
00:51:04آئے تو
00:51:04بچھو بہت آسان
00:51:05بات ہو گئی
00:51:05اپنے
00:51:05f
00:51:07x
00:51:07برابر
00:51:083x
00:51:08square
00:51:08plus 4
00:51:09g
00:51:10x
00:51:102
00:51:11g
00:51:12h
00:51:13x
00:51:19f
00:51:20g
00:51:20h
00:51:20یا سارے
00:51:21دیکھو یہ
00:51:22تو
00:51:22فنکشن ہو گئی
00:51:23ہوگا کیونکہ
00:51:23کوئی بھی
00:51:24پرابلم نہیں
00:51:25آ رہی
00:51:25اگر یہ
00:51:26فنکشن کا
00:51:27square
00:51:27اور plus
00:51:27کوئی ایسی
00:51:28چیز تو نہیں
00:51:29ہے کہ
00:51:29دو
00:51:30elements
00:51:31اگر ہوں
00:51:31تب ہمیں
00:51:32چیک کرنا
00:51:33تھا کہ
00:51:33all elements
00:51:33انہیں
00:51:34اور بھی
00:51:34خالیس
00:51:35طرح کا
00:51:35فنکشن
00:51:35ہی ہمیشہ
00:51:36فنکشن
00:51:37ہی ہوگا
00:51:38جب دیکھو یہ
00:51:38آپ کے پاس
00:51:39ایک پاور
00:51:40بھی اور یہ
00:51:40ایڈیشن
00:51:41کانسٹن نمبر
00:51:42بھی فنکشن
00:51:42ہوتا ہے
00:51:42یہاں
00:51:43چیک کرنا
00:51:43پڑے گا
00:51:44اگر یہ
00:51:451,2,3
00:51:46میں سے
00:51:46کوئی ایک چیز
00:51:47دو دفعہ
00:51:48دو
00:51:49ڈیفرنٹ
00:51:49نمبر کے ساتھ
00:51:50ہمارے
00:51:50ایسا نہیں
00:51:51ہے نا
00:51:51تو یہاں
00:51:511,2
00:51:522,1
00:51:533,2
00:51:54کوئی بھی
00:51:55ریپیٹ نہیں
00:51:55ہو رہا
00:51:561 کی
00:51:56اما 2
00:51:572 کی
00:51:57اما 1
00:51:583
00:51:58اما 2
00:51:58سب کی
00:51:59اما 2
00:51:59اما 2
00:51:59فنکشن
00:52:00ہے
00:52:00تو
00:52:00all
00:52:00of
00:52:00them
00:52:013
00:52:01کے
00:52:013
00:52:02فنکشن
00:52:02ہے
00:52:02بھئی
00:52:02ایسا
00:52:03یہ
00:52:03کہ f
00:52:03x
00:52:04equal to
00:52:05cos x
00:52:05x
00:52:05g
00:52:06of
00:52:06x
00:52:06z
00:52:06sin
00:52:06f
00:52:07x
00:52:08function
00:52:08g
00:52:09x
00:52:09or
00:52:09both
00:52:09ہوں
00:52:09تو
00:52:10answer
00:52:10both
00:52:11trigonometric
00:52:12ratios
00:52:12زیادہ
00:52:12کرو
00:52:13ہمیشہ
00:52:13function
00:52:14ہوتے
00:52:14ہیں
00:52:15جتنے
00:52:15بھی
00:52:15trigonometric
00:52:16ratios
00:52:16ایسے
00:52:17ہی ہے
00:52:17کہ f
00:52:17x
00:52:18equal to
00:52:18cos
00:52:19x
00:52:19x
00:52:19z
00:52:19g
00:52:19x
00:52:20z
00:52:20sin
00:52:20x
00:52:20f
00:52:21x
00:52:22function
00:52:22g
00:52:22x
00:52:23or
00:52:23both
00:52:23ہوں گے
00:52:23تو
00:52:24answer
00:52:24ہے
00:52:24both
00:52:25trigonometric
00:52:26ratios
00:52:26ہمیشہ
00:52:27function
00:52:28ہوتے ہیں
00:52:29جتنے بھی
00:52:29trigonometric
00:52:30ratios
00:52:30ایک اور
00:52:31پاس
00:52:31سیور
00:52:31کا
00:52:31بڑا
00:52:31اچھا
00:52:32سمال
00:52:323,2
00:52:334,2
00:52:343,1
00:52:357,1
00:52:362,3
00:52:37میں سے
00:52:37کس کو
00:52:39remove
00:52:39کریں
00:52:39کہ یہ
00:52:40چیز
00:52:40function
00:52:40من
00:52:40چاہیے
00:52:41اب
00:52:41اتنی
00:52:42دوم
00:52:42definition
00:52:42دیکھ
00:52:43چکے ہیں
00:52:43کہ
00:52:43جو
00:52:43پہلے
00:52:44والے
00:52:44ہوتے ہیں
00:52:44بچے
00:52:45ایک
00:52:45بچہ
00:52:453
00:52:45ایک
00:52:46بچہ
00:52:464
00:52:46ایک
00:52:47بچہ
00:52:48ہے
00:52:49پھر
00:52:493
00:52:49آگیا
00:52:50ایک
00:52:51بچہ
00:52:527
00:52:52ایک
00:52:52بچہ
00:52:522
00:52:531
00:52:54من
00:52:544
00:52:55ایک
00:52:55بار
00:52:55آرہا
00:52:567
00:52:56ایک
00:52:56بار
00:52:57آرہ
00:52:572
00:52:57ایک
00:52:57بار
00:52:57آرہ
00:52:58تو
00:52:58پنگا
00:52:58یہاں
00:52:59تو
00:52:59کچھ
00:52:59نہیں
00:52:59ہے
00:52:59پنگا
00:53:01یہاں
00:53:01تو
00:53:01کچھ
00:53:02نہیں
00:53:02ہے
00:53:02یہاں
00:53:03تو
00:53:03یہاں
00:53:04ہے
00:53:04یہاں
00:53:04ہے
00:53:05کیونکہ
00:53:053
00:53:062
00:53:06کے
00:53:06ساتھ
00:53:063
00:53:061
00:53:07کے
00:53:07ساتھ
00:53:072
00:53:082
00:53:082
00:53:082
00:53:08ایسا
00:53:09کیسے
00:53:10ہو سکتا ہے
00:53:10تو
00:53:11ساہرہ
00:53:11ان میں سے
00:53:12کسی
00:53:12ایک کو
00:53:12ہٹھائیں گے
00:53:13کس کو
00:53:14سوال
00:53:14یہ پیدا ہوتا ہے
00:53:153
00:53:161
00:53:16یا
00:53:163
00:53:172
00:53:17تو
00:53:17دیکھ
00:53:17آپشن
00:53:18میں
00:53:18کیا
00:53:183
00:53:181
00:53:19موجود
00:53:19نہیں
00:53:19نا
00:53:292
00:53:30کے
00:53:30برابر
00:53:30ہے
00:53:30اور
00:53:31g
00:53:31h
00:53:32ہے
00:53:32یہ بھی
00:53:33یہ برابر
00:53:34ہے
00:53:351,2
00:53:352,1
00:53:363,2
00:53:37ان میں سے
00:53:37کون سا
00:53:38فنکشن
00:53:38ہے
00:53:38f
00:53:39g
00:53:39h
00:53:39یا
00:53:40سارے
00:53:40تو
00:53:41یہ تو
00:53:41فنکشن
00:53:41ہو گئی
00:53:42ہوگا
00:53:42کیونکہ
00:53:42کوئی بھی
00:53:43پرابلم
00:53:43نہیں آرہی
00:53:44اگر یہ
00:53:45فنکشن
00:53:45کا
00:53:45سکوائر
00:53:46ہے
00:53:46اور
00:53:46پلس
00:53:46کوئی
00:53:47ایسی
00:53:47چیز
00:53:47تو
00:53:47نہیں
00:53:48ہے
00:53:48دو
00:53:49elements
00:53:49اگر
00:53:50ہوں
00:53:59کونسٹن
00:54:00نمبر
00:54:01بھی
00:54:01فنکشن
00:54:01ہوتا ہے
00:54:01یہاں
00:54:02چیک
00:54:02کرنا
00:54:02پڑے گا
00:54:02اگر
00:54:04یہ
00:54:041,2
00:54:043
00:54:05میں سے
00:54:05کوئی
00:54:05ایک
00:54:05چیز
00:54:06دو
00:54:06دفاع
00:54:06دو
00:54:08ڈیفرنٹ
00:54:08نمبر
00:54:09کے ساتھ
00:54:09ہمار
00:54:09ایسا
00:54:09نہیں ہے
00:54:10کوئی
00:54:101,2
00:54:112,1
00:54:113,2
00:54:13کوئی
00:54:14بھی
00:54:14ریپیٹ
00:54:14نہیں
00:54:14ہو رہا
00:54:151
00:54:15کی
00:54:15اممہ
00:54:162
00:54:162
00:54:16کی
00:54:16اممہ
00:54:161
00:54:173
00:54:17کی
00:54:17اممہ
00:54:172
00:54:17سب کی
00:54:18مہ
00:54:18الگ
00:54:18الگ
00:54:18فنکشن
00:54:19ہے
00:54:19تو
00:54:19all
00:54:19of
00:54:19them
00:54:203
00:54:20کی
00:54:20تین
00:54:20وں
00:54:20فنکشن
00:54:21ہے
00:54:21بھائی
00:54:21سال
00:54:22سر
00:54:22نسٹ
00:54:22میں آتے ہیں
00:54:23بھاگی
00:54:23نہیں آتے ہیں
00:54:24تو بھاگی
00:54:24سب
00:54:24ٹینشن
00:54:24آ لیں
00:54:25کہ
00:54:26g
00:54:26of
00:54:26x
00:54:27equal
00:54:27to
00:54:27x
00:54:28square
00:54:28اور
00:54:28x
00:54:29value
00:54:29greater
00:54:31than
00:54:31or equal
00:54:31to
00:54:320
00:54:32less
00:54:32than
00:54:32or equal
00:54:33to
00:54:332
00:54:33پھر
00:54:33ایک
00:54:33اور
00:54:34اس
00:54:34میں
00:54:34split
00:54:34function
00:54:35ہے
00:54:352
00:54:35part
00:54:35میں
00:54:353
00:54:36x
00:54:36x
00:54:37is
00:54:37less
00:54:37than
00:54:37or equal
00:54:38to
00:54:3810
00:54:38اب
00:54:39اس کے
00:54:40حوالے
00:54:40سے
00:54:40پوچھ رہیں
00:54:41کہ
00:54:41یہ
00:54:41کیسے
00:54:42ہوگا
00:54:42تو
00:54:42یہاں
00:54:43تھوڑا
00:54:43مسئلہ
00:54:44ہے
00:54:45تھوڑا
00:54:45سا
00:54:45پنگا
00:54:46ہے
00:54:46اس
00:54:47پنگے
00:54:47کو
00:54:47مسئلے
00:54:47کو
00:54:47حل
00:54:48کیسے
00:54:48کریں
00:54:48پہلے
00:54:49تو
00:54:49اگر
00:54:49آپ
00:54:49g
00:54:49of
00:54:50x
00:54:50دیکھو
00:54:50کہ
00:54:50x
00:54:51کی
00:54:510
00:54:52پٹ
00:54:52کر
00:54:52ہو
00:54:52یہاں
00:54:53پہ
00:54:53square
00:55:05پٹ کر کے
00:55:05تو
00:55:05تھے
00:55:06function
00:55:06اور
00:55:07کبھی
00:55:072
00:55:07کا
00:55:08answer
00:55:08unique
00:55:10solution
00:55:11جیسے
00:55:11value
00:55:12دیکھو
00:55:120
00:55:13رکھنے
00:55:13سے
00:55:140.5
00:55:14سے
00:55:152
00:55:15رکھنے
00:55:15سے
00:55:152
00:55:15کا
00:55:15square
00:55:164
00:55:16ٹھیک
00:55:16ہے
00:55:16اب
00:55:18سرحہ
00:55:19یہاں
00:55:19دیکھو
00:55:193
00:55:20x
00:55:20value
00:55:212
00:55:21سے
00:55:2110
00:55:21تک
00:55:211
00:55:22رکھا
00:55:223
00:55:222
00:55:23رکھا
00:55:236
00:55:243
00:55:24رکھا
00:55:249
00:55:25اب
00:55:25یہ
00:55:25function
00:55:25ہے
00:55:26یا
00:55:26نی
00:55:26دیکھو
00:55:26پنگا
00:55:272
00:55:29پوائنٹ
00:55:29یہاں
00:55:29بھی
00:55:29موجود
00:55:30تھا
00:55:302
00:55:30پوائنٹ
00:55:31یہاں
00:55:33پوٹ کرنے
00:55:33سے
00:55:332
00:55:334
00:55:34آرہا ہے
00:55:34یہاں
00:55:35پوٹ کرنے
00:55:35سے
00:55:362
00:55:363
00:55:376
00:55:37آرہا ہے
00:55:37تو
00:55:38پھر
00:55:39دو
00:55:39answer
00:55:39کیسے
00:55:40ہو سکتے
00:55:40یعنی
00:55:40دو
00:55:41اممائے
00:55:41بات
00:55:41ہو گئی
00:55:42تو
00:55:42یہ
00:55:42function
00:55:43نہیں
00:55:44ہو سکتا
00:55:44اب
00:55:44mass
00:55:45میں
00:55:45اگر
00:55:45کوئی
00:55:46احسا
00:55:46solve
00:55:46paper
00:55:47میں
00:55:47آئے
00:55:47پہلی
00:55:47پپ
00:55:47اگر آجائے
00:55:51تو
00:55:51پورا
00:55:52check
00:55:52مت
00:55:52کرو
00:55:53تم
00:55:53دیکھو
00:55:53function
00:55:53split
00:55:54ہاں
00:55:54پہ
00:55:542
00:55:552
00:55:55common
00:55:55ہے
00:55:55صرف
00:55:562
00:55:56کو
00:55:57چیک
00:55:57کر کے
00:55:57دیکھو
00:55:57یہاں
00:55:57پوٹ کر
00:55:58یہ سب
00:55:58کرنے
00:55:59کیسے
00:55:592
00:56:004
00:56:003
00:56:002
00:56:012
00:56:016
00:56:01الگ
00:56:01الگ
00:56:01answer
00:56:02آرہے
00:56:02تو
00:56:02function
00:56:02نہیں
00:56:03لیکن
00:56:03first
00:56:04کرو
00:56:04اوپر
00:56:04والے
00:56:05line
00:56:05تو
00:56:05ہی
00:56:05ہوتی
00:56:052
00:56:062
00:56:064
00:56:07اور
00:56:07نیچے
00:56:07بجائے
00:56:073
00:56:082
00:56:09x
00:56:10کیا دیا ہوتا
00:56:11تو
00:56:11جب ہم x
00:56:112
00:56:12پوٹ کرتے ہیں
00:56:122
00:56:122
00:56:123
00:56:134
00:56:13جو
00:56:13کہ
00:56:14سیم
00:56:14answer
00:56:14آگر
00:56:15یہاں
00:56:16x
00:56:162
00:56:17پوٹ کرنے
00:56:18سے
00:56:18سیم
00:56:19آرہا
00:56:20تو
00:56:20اس
00:56:20کلپ
00:56:21function
00:56:21کیا
00:56:22مطب
00:56:22ہوا
00:56:22میں
00:56:230
00:56:23سے
00:56:232
00:56:24تک
00:56:24میں
00:56:242
00:56:25کو
00:56:25چیک
00:56:26کروں
00:56:26کیون
00:56:262
00:56:27یہاں
00:56:27بھی
00:56:27موجود
00:56:27ہے
00:56:28یہاں
00:56:28یہاں
00:56:28کھر
00:56:28یہاں
00:56:29جب
00:56:30ایسا
00:56:30children, auntya, children, auntya
00:56:35so now another question
00:56:39x is 1, 2, 3, 4
00:56:42and y is auntya
00:56:451, 5, 9, 11, 15, 16
00:56:481, 1, 2, 3, 4
00:56:51now which function is not
00:56:53so first check this
00:56:55that all elements need to be
00:56:57no missing, 1, 2, 3, 4
00:57:01all right
00:57:03so not
00:57:05so not
00:57:07y is not
00:57:09y is not
00:57:10y is not
00:57:11y is not
00:57:12y is not
00:57:14y is not
00:57:16y is not
00:57:18y is not
00:57:20now this is one
00:57:22this is one
00:57:24y is not
00:57:25y is not
00:57:27.
00:57:51Okay, this is a function.
00:57:53F2, see.
00:57:55You guys tell me what function is or not?
00:57:571,1, 2,7, 3,5.
00:57:59Function is?
00:58:01No, brother.
00:58:03Each element should come.
00:58:05This 4 is the mother's mother.
00:58:07No, mother.
00:58:09Function is not.
00:58:11Now, F3.
00:58:131, 2, 3, 4.
00:58:15What is this function?
00:58:171 with 5.
00:58:192 with 9.
00:58:213 with 1.
00:58:233 with 5.
00:58:254 with 5.
00:58:271 with 5.
00:58:294 with 5.
00:58:31But what is this?
00:58:332 with 11 and 9.
00:58:35This is our children.
00:58:37How can it be?
00:58:392 with 1 and 2 with 11.
00:58:41You can't have function.
00:58:43F1.
00:58:45The first class of our online class.
00:58:49This is the first class.
00:58:51This is the course revision again.
00:58:53Attitude test.
00:58:55We start from 2nd year maths.
00:58:572nd year maths.
00:58:592nd year maths.
00:59:012nd year maths.
00:59:032nd year maths.
00:59:052nd year maths.
00:59:072nd year maths.
00:59:09Then we start from.
00:59:112nd year maths I have basically done about it.
00:59:13Then that will be the course complete.
00:59:15And when our secondary maths is finished then again,
00:59:17then then 1st year, then 1st year.
00:59:18Then vaat Lovely.
00:59:19Don't bother about it!
00:59:20Before the topic of function.
00:59:22What is function?
00:59:23The reason why is function often?
00:59:24After that,
00:59:25is a connection between 2 sets A and B.
00:59:29Such that.
00:59:30all elements in A are associated to some elements in B
00:59:35this association is unique
00:59:37it's a very good definition
00:59:394 kids have been 1, 1, 2, 1, 2, 1
00:59:44and 6 kids have been 1, 2, 3, 4, 5, 6
00:59:49they have been Roman alphabet
00:59:51so what is that all elements in A are associated to some elements in B
00:59:57A is that all elements in B should be associated with any element
01:00:02all elements can be associated with all elements
01:00:04all elements can be associated with all elements
01:00:05but all elements should be associated with all elements
01:00:07what is that?
01:00:08if these are children, then children will be
01:00:11some mother, without mother, then no child will not come in the world
01:00:16first, the mother of Olive is 1, B is 1, P is 3, and T is 4
01:00:23all elements in A are associated to some elements in B
01:00:28all elements in A are associated to some elements in B
01:00:30all elements in A should be associated with all elements in B
01:00:34all elements in A should be associated with all elements in B
01:00:36all elements in A should be associated with all elements in B
01:00:38all elements in A should be associated with all elements in B
01:00:40all elements in A should be associated with all elements in B
01:00:42all elements in A should be associated with all elements in B
01:00:44all elements in A should be associated with all elements in B
01:00:46all elements in A should be associated with all elements in B
01:00:48that all elements in A are associated to some elements in B.
01:00:52Everyone needs to be associated with each other.
01:00:55Now listen, listen.
01:00:58Alif, M, I, A, B, M, A, 2, P, 3, A, 4.
01:01:03It's okay.
01:01:04All four kids have become one mother.
01:01:07Let's go.
01:01:08First, let's do this.
01:01:10This was the function.
01:01:12Alif and we both have two kids.
01:01:15No two kids, no five kids, no six kids.
01:01:18Now this is the function.
01:01:20Now this is the function.
01:01:21If there are some elements,
01:01:23if the element of B is missing, then it goes.
01:01:26The element of B is missing, then it goes.
01:01:28And this is also possible.
01:01:30There are two, five, six.
01:01:32There are three aunty.
01:01:34One, three, four.
01:01:35Aunty one of the child for me.
01:01:37Aunty three of the child for me.
01:01:38Aunty four of the child for me.
01:01:40Aunty four of the child.
01:01:41Okay.
01:01:42Now all of the elements from B is missing.
01:01:45Let's go.
01:01:46In the definition,
01:01:47there are some elements that need to be equal.
01:01:49And if there are some elements that need to be equal,
01:01:51then it will be the function.
01:01:52B is the same.
01:01:53Okay.
01:01:54Now this is the next definition.
01:01:56This association is unique.
01:01:57What is this?
01:01:58Let me tell you.
01:02:00Now this is a big problem.
01:02:03We take auntie two too.
01:02:07Aunty five and auntie six too.
01:02:10Aunty five and auntie six too.
01:02:12Aunty five and auntie six too.
01:02:17One child for three months.
01:02:19It can't be real months.
01:02:20It can't be real months.
01:02:21It can't be real months.
01:02:22So brother, three months.
01:02:25No, no.
01:02:26It can't happen.
01:02:27It's not a function.
01:02:28What do you need to do?
01:02:29What do you need to do?
01:02:30You don't need to do.
01:02:31One child for one month.
01:02:32Why do you need to do this?
01:02:33Why do you need to do this?
01:02:34Because they have to stay three days.
01:02:35So if you remove them,
01:02:37now this is right.
01:02:38Let's do two elements.
01:02:39This is the basic definition of function.
01:02:43I will tell you.
01:02:44I will tell you.
01:02:45Let me tell you.
01:02:47What is this?
01:02:49Oh shit.
01:02:50Sine hyperbolic x
01:02:54plus e g r minus x divided by 2.
01:02:59Cos hyperbolic x
01:03:01minus e g r minus x upon 2.
01:03:04While 10 is equal to sin upon cos.
01:03:08So e g r x plus e g r minus x
01:03:11e g r minus x.
01:03:13Cos hyperbolic x
01:03:14cord hyperbolic x
01:03:16e g r x minus e g r minus x.
01:03:17Is it right?
01:03:18So c g r x plus x.
01:03:20And here,
01:03:23cos x it is sin.
01:03:24It is a sin principle.
01:03:25This means 2.
01:03:26It's a sin.
01:03:27So e y r minus x.
01:03:28The sin would be minus x.
01:03:29The sin would be minus x.
01:03:30The sin would be minus x.
01:03:31It's only a sin would be minus x.
01:03:32The sin would be minus x.
01:03:34And this is a sin.
01:03:35The sin would be minus x.
01:03:36X plus EGR minus X nilche ho jayegah.
01:03:39Yarni,
01:03:39EG plus EGR minus X upon 2.
01:03:42Sine EGR X plus EGR minus X upon 2
01:03:46or the cost.
01:03:47EGR X minus EGR minus X upon 2
01:03:50Baaghi kama to khut-bakhut ho jayegah.
01:03:53Ten ab nikal lo que sign upon cost se court
01:03:55Zahir 10 ka posit ho jayegah.
01:03:57Cos X sign ka posit ho jayegah.
01:04:00Or sec cos x ho jayegah.
01:04:01MCQs kaisa a sakti ha?
01:04:02Ten hyperbolic X
01:04:04Pooch layengi kis cake wall hai?
01:04:05Yeah,
01:04:06Yeh yeh dee deenghe
01:04:062 EGR X plus EGR minus X
01:04:09Kis cake wall hai?
01:04:10Ab option bhout saare hoon gaye
01:04:11Toh X cake wall hai nai shahe?
01:04:13Cos X hyperbolic X cake wall hai nai shahe?
01:04:16MCQs is a chih khashi bach ho aake.
01:04:18Basic concept,
01:04:19Domain,
01:04:20Range
01:04:20Yeh sab bhotta kya hai?
01:04:22Reekah eger
01:04:22Domain e ABC
01:04:24Range e 1, 2, 3, 4
01:04:25Reekah chahi.
01:04:27Ab agar
01:04:28Domain range
01:04:29Hap detail meh am kon sa pahd hai hai?
01:04:32Domain ka ek member range ke
01:04:33Ek member ke saad
01:04:33Domain ka ek member range ke
01:04:35Okay, range ka koi member bhaj bhaj bhaj se koi fad nai pahd ta
01:04:41Yeh bilgul function ho ga ji?
01:04:43Okay, chahi?
01:04:44Or one to one function ho ga.
01:04:45Q one to one function ho ga?
01:04:47Ek domain, ek range, ek domain, ek range, ek domain, ek range.
01:04:50Range ke ek member khalye pahd hai hai?
01:04:52Koi maslala nai.
01:04:53Es function ko one to one function kehet hai.
01:04:55Doosan naam is ka injective function ho ga.
01:04:57Okay, chahi?
01:04:58Ab eek ho ta hai on to function.
01:05:01A joh hai, woh one ke saad.
01:05:05B2 ke saad, C bhi one ke saad.
01:05:07Yarni domain ke do member range ke kisii
01:05:09Eng member ke saad ho sakti hai.
01:05:11Okay, chahi.
01:05:12Onto ya, surjective function ho ta hai.
01:05:13Kyun kye atas ko surjective range ka her member bharawa ho.
01:05:17Her member bharawa ho,
01:05:18Tho woh on to function ho ga.
01:05:20Dekhi, yaa bharawa nai tha.
01:05:22Function phir bhi hai.
01:05:23Function phir bhi hai.
01:05:26Okay, chahi.
01:05:27Ab yuh ho gaya on to function.
01:05:29Ab yuh function ho ta hai,
01:05:30Juh kya all rounder function ho ta hai.
01:05:32One to one bhi, or on to bhi.
01:05:33Isil ehem isko kye ta hai,
01:05:34Koon surjective?
01:05:35By ejective.
01:05:37Kyun?
01:05:37Ye injective bhi hai,
01:05:39Or surjective bhi hai.
01:05:40Dekho, injective one to one.
01:05:41Har chiz one one.
01:05:42Phrir range ka her member poora ho raha hai.
01:05:45Bilkul ji poora ho raha hai.
01:05:46Isil eh ke laika by ejective ya one to one,
01:05:49Or on to function do no.
01:05:50Dekho, aysa tabhi ho sakta hai,
01:05:51Jab domain or range ke member equal ho.
01:05:54Okay, ab isko bada asan,
01:05:56Hume nye samehna hai.
01:05:57Bhi, range ka koji member khalhi ho gaya,
01:05:59It's okay, tab bhi function ho ga.
01:06:01Lekin agar kabhi domain ka member khalhi ho jai,
01:06:04Tho function nahi ho ga.
01:06:07Range ko samjho ma hai,
01:06:09Domain ko samjho bachche.
01:06:10Ye auntie hai,
01:06:11Ye bachche hai,
01:06:12Okay, chai.
01:06:13Ye chom auntie hai, range me,
01:06:15Auntie number four,
01:06:16In ka koji bachche nigh hai,
01:06:17Thi ki?
01:06:19Ya range me,
01:06:20Chom auntie one hai,
01:06:21In ka dho bachche hai,
01:06:22A or C.
01:06:23It's okay.
01:06:24Ye haan tino auntie ho ga,
01:06:26Ek ek bachche hai, range auntie bachche.
01:06:27Am, ya haan souchou.
01:06:31Range hai,
01:06:33In auntie ka,
01:06:34In auntie ka,
01:06:34In auntie ka,
01:06:35In auntie ka,
01:06:35In auntie ka,
01:06:35In auntie ka,
01:06:35Is bachche ki ammah kha hai,
01:06:36Haan?
01:06:37CC ammah kha hai,
01:06:38Baai,
01:06:39Baikior ammah ke,
01:06:39Tho bachche nigh ho saksakta na,
01:06:40E esa toho samanga,
01:06:41A
01:06:51A
01:06:51the function is not. Now, I said, what are the domain members?
01:06:55It's a child. And range. Now, in the range, there were two children.
01:07:00A and C, it's okay. A and three are not.
01:07:03So, it means that this function will not be.
01:07:07When a domain member is empty, when a domain member is empty,
01:07:10when a domain member is empty, then there will not be a function.
01:07:15When a domain and a range is empty,
01:07:18when a domain member is empty, one-to-one is in-jective.
01:07:21Range member is complete.
01:07:23First, range member is with domain 2 or 3 members,
01:07:27on-to or surjective.
01:07:29And when a domain member is with range member,
01:07:32then by-jective and one-to-one, on-to function.
01:07:35Now, this is a very sweet question.
01:07:38fx,y is equal to 3x plus 2y minus 8k
01:07:43and gz is equal to z squared.
01:07:47You have to choose f3, g of 4.
01:07:50So, g of 4.
01:07:52Look, first, see g of z.
01:07:55Who is equal to z squared?
01:07:57Z squared.
01:07:58Let's make z right.
01:08:00So, here, z is equal to 4.
01:08:024 squared is 16.
01:08:04G of z squared is 16.
01:08:05That means, g of z,
01:08:06G of 4, and g of 4 is 16.
01:08:11That means, here, I can say that
01:08:13f3 is 16.
01:08:15You know.
01:08:16This is x and y.
01:08:18So, we will put it to the values.
01:08:20So, when we put it, 3x plus 2y minus 8, x will be 3y, 16, 3 3s are 9, 32 minus 8, 32 plus 9, 41, 8 minus, 33 is your answer.
01:08:41So, here function questions, we have to check out the questions.
01:08:45First, let's see the topic of today's topic, explicit function.
01:08:50y always one way, and x always one way, and x always one way, totally on the other side.
01:08:55y equals x squared plus 2x plus 7, y equals ex plus cos x.
01:09:00And implicit function, x and y are always in the same way.
01:09:04x squared plus x squared plus y squared equal to 0, give our y plus x squared, e y equal to 2x plus 7.
01:09:11Y means that x and y are separated, when it's a very difficult task, and it's the same thing, the function is the implicit function.
01:09:19Different pass server, i will question, some things will happen with magical tricks, but when we don't have magical tricks, we will also do it with original method.
01:09:27x squared sin x, power even, even function, sin x is odd function.
01:09:35Even and odd multiplication, I said, odd is odd.
01:09:39So we are asking, odd is odd or neither.
01:09:43So yes, it is even.
01:09:44Now we are using sin x minus cos x.
01:09:47sin x is odd function.
01:09:50Cos x is even function.
01:09:52Odd or even function, if there will be plus or minus, what will be called?
01:09:57Neither even nor odd.
01:09:59So we have option b, neither even nor odd.
01:10:01Option c, we cannot do so easily.
01:10:03Therefore, we have to follow the original method here and we will output f minus x.
01:10:08This is the way it is minus x.
01:10:11We will do this here too.
01:10:14This is upon me 2.
01:10:161 to the power minus x plus 1 to the power x upon 2.
01:10:21This can be changed, I don't see any problem.
01:10:24Plus sin is possible to go back to the house.
01:10:27This is f of x.
01:10:29If f of x is equal to f of x, then what is the function?
01:10:35Even function.
01:10:36Even function.
01:10:37Even here.
01:10:38This is my difficult question.
01:10:4099% from short track.
01:10:42Log 1 minus x, 1 plus x.
01:10:46Let's put minus x.
01:10:48Log 1 minus minus plus x.
01:10:53And here it is minus.
01:10:56Now there is a log property.
01:10:58When you are divided, we can write them differently.
01:11:01And what happens in the middle of the negative sign?
01:11:05Does it happen or not?
01:11:07It happens.
01:11:09It happens.
01:11:10Now let's take a negative comment outside.
01:11:13What do I tell you?
01:11:15Look, there is a negative comment.
01:11:17This is positive or negative.
01:11:19It happens or not?
01:11:22It happens.
01:11:24What is the benefit?
01:11:26We can write the same form.
01:11:29Minus is divided.
01:11:30Log 1 minus x.
01:11:32Divide by 1 plus x.
01:11:35Whatever this form is.
01:11:38But there is a difference.
01:11:39When this was the original f of x,
01:11:43it was plus sin.
01:11:44Now what is minus?
01:11:45What is minus?
01:11:46I mean, f of x is divided,
01:11:48but it is minus f of x.
01:11:49Now let's take a look.
01:11:50If f minus x,
01:11:51minus f of x,
01:11:52minus f of x,
01:11:53then what is function?
01:11:54What is odd?
01:11:55And of course,
01:11:56plus f of x,
01:11:57then function is even.
01:11:58That was just awesome.
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