[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:01.01,0:00:05.16,Default,,0000,0000,0000,,In this video, we'll define the\Ninverse, the function F.
Dialogue: 0,0:00:05.97,0:00:12.52,Default,,0000,0000,0000,,Let's suppose we have a function\NF that sends XY. So I mean F
Dialogue: 0,0:00:12.52,0:00:14.39,Default,,0000,0000,0000,,of X equals Y.
Dialogue: 0,0:00:16.15,0:00:21.43,Default,,0000,0000,0000,,The inverse of F is denoted F to\Nthe minus one.
Dialogue: 0,0:00:22.09,0:00:27.57,Default,,0000,0000,0000,,And it's the function that sends\Nwhy back to X?
Dialogue: 0,0:00:28.21,0:00:32.03,Default,,0000,0000,0000,,For this to be an inverse,\Nit needs to work for every X
Dialogue: 0,0:00:32.03,0:00:33.21,Default,,0000,0000,0000,,that F acts on.
Dialogue: 0,0:00:37.58,0:00:44.27,Default,,0000,0000,0000,,Here's a simple\Nexample of how
Dialogue: 0,0:00:44.27,0:00:49.84,Default,,0000,0000,0000,,to workout an\Ninverse function.
Dialogue: 0,0:00:50.93,0:00:54.50,Default,,0000,0000,0000,,Let's have F of
Dialogue: 0,0:00:54.50,0:00:56.80,Default,,0000,0000,0000,,X. Equals 4X.
Dialogue: 0,0:00:57.32,0:01:04.86,Default,,0000,0000,0000,,Now F of X takes\Nnumber and multiplies it before.
Dialogue: 0,0:01:06.22,0:01:11.72,Default,,0000,0000,0000,,And we want F inverse to take 4X\Nback to X.
Dialogue: 0,0:01:12.67,0:01:17.75,Default,,0000,0000,0000,,Now, this must mean that F\Ninverse of X must divide by 4.
Dialogue: 0,0:01:17.75,0:01:19.71,Default,,0000,0000,0000,,So F inverse of X.
Dialogue: 0,0:01:20.72,0:01:24.77,Default,,0000,0000,0000,,Equals 1/4 of\NX.
Dialogue: 0,0:01:27.24,0:01:34.30,Default,,0000,0000,0000,,Now we can see from this that\Nsince F inverse of X is 1/4
Dialogue: 0,0:01:34.30,0:01:39.84,Default,,0000,0000,0000,,X and one over F of X\Nis one over 4X.
Dialogue: 0,0:01:41.14,0:01:45.67,Default,,0000,0000,0000,,That F inverse and one over F of\NX and not the same thing, even
Dialogue: 0,0:01:45.67,0:01:48.69,Default,,0000,0000,0000,,though the notation makes it\Nlook like they should be.
Dialogue: 0,0:01:49.37,0:01:52.97,Default,,0000,0000,0000,,Because whenever up\Nof X is one over 4X.
Dialogue: 0,0:01:54.00,0:01:59.22,Default,,0000,0000,0000,,But that's not equal to 1/4 X,\Nbecause here The X is on the
Dialogue: 0,0:01:59.22,0:02:02.58,Default,,0000,0000,0000,,denominator, and there the X is\Non the numerator.
Dialogue: 0,0:02:04.77,0:02:11.61,Default,,0000,0000,0000,,Let's workout\Nanother inverse
Dialogue: 0,0:02:11.61,0:02:13.32,Default,,0000,0000,0000,,function.
Dialogue: 0,0:02:14.37,0:02:17.20,Default,,0000,0000,0000,,This time will have F of X.
Dialogue: 0,0:02:17.95,0:02:23.52,Default,,0000,0000,0000,,Equals. 3X.\N+2.
Dialogue: 0,0:02:24.99,0:02:30.36,Default,,0000,0000,0000,,Now what does F of X do we\Nstart with X?
Dialogue: 0,0:02:31.53,0:02:33.31,Default,,0000,0000,0000,,We multiplied by three.
Dialogue: 0,0:02:33.98,0:02:36.21,Default,,0000,0000,0000,,To get 3X.
Dialogue: 0,0:02:36.94,0:02:39.77,Default,,0000,0000,0000,,Then we add onto.
Dialogue: 0,0:02:39.77,0:02:46.90,Default,,0000,0000,0000,,Now, since we want F inverse\Nto take F of X and
Dialogue: 0,0:02:46.90,0:02:49.27,Default,,0000,0000,0000,,give us back X.
Dialogue: 0,0:02:50.18,0:02:53.33,Default,,0000,0000,0000,,To workout if inverse we\Nneed to undo every
Dialogue: 0,0:02:53.33,0:02:54.73,Default,,0000,0000,0000,,operation that ever did.
Dialogue: 0,0:02:55.79,0:02:57.77,Default,,0000,0000,0000,,So if we started with X.
Dialogue: 0,0:02:58.78,0:03:03.55,Default,,0000,0000,0000,,First, we undo the last\Noperation that F did, so we
Dialogue: 0,0:03:03.55,0:03:07.46,Default,,0000,0000,0000,,take away too. That gives\Nus X minus 2.
Dialogue: 0,0:03:08.80,0:03:12.26,Default,,0000,0000,0000,,And before that, we'd multiplied\Nby three. So to undo that, we
Dialogue: 0,0:03:12.26,0:03:18.97,Default,,0000,0000,0000,,divide by three. And that\Ngives us X minus
Dialogue: 0,0:03:18.97,0:03:22.37,Default,,0000,0000,0000,,two, all divided by
Dialogue: 0,0:03:22.37,0:03:29.81,Default,,0000,0000,0000,,three. So F inverse of X in\Nthis case is X minus two, all
Dialogue: 0,0:03:29.81,0:03:31.33,Default,,0000,0000,0000,,divided by three.
Dialogue: 0,0:03:33.14,0:03:40.17,Default,,0000,0000,0000,,Here's one more example of how\Nwe can undo the operations of F
Dialogue: 0,0:03:40.17,0:03:45.04,Default,,0000,0000,0000,,to workout its inverse. Let's\Nhave F of X.
Dialogue: 0,0:03:45.76,0:03:49.31,Default,,0000,0000,0000,,Being 7 minus X cubed.
Dialogue: 0,0:03:50.67,0:03:54.89,Default,,0000,0000,0000,,And I'll rewrite the slightly\Nto make it easier to workout.
Dialogue: 0,0:03:54.89,0:03:58.73,Default,,0000,0000,0000,,Would love F of X written as\Nminus X cubed?
Dialogue: 0,0:03:59.76,0:04:04.46,Default,,0000,0000,0000,,Plus Seven, so now it's a bit\Neasier to see which operations
Dialogue: 0,0:04:04.46,0:04:07.21,Default,,0000,0000,0000,,were doing 2X. We start with X.
Dialogue: 0,0:04:08.30,0:04:10.30,Default,,0000,0000,0000,,We cubit.
Dialogue: 0,0:04:11.62,0:04:13.12,Default,,0000,0000,0000,,That gives us X cubed.
Dialogue: 0,0:04:14.25,0:04:19.78,Default,,0000,0000,0000,,Then we send X cubed minus X\NCube, so we times Y minus one.
Dialogue: 0,0:04:20.28,0:04:24.44,Default,,0000,0000,0000,,And then\Nfinally we
Dialogue: 0,0:04:24.44,0:04:27.57,Default,,0000,0000,0000,,add on 7.
Dialogue: 0,0:04:28.79,0:04:35.62,Default,,0000,0000,0000,,So again to workout\NF inverse we start
Dialogue: 0,0:04:35.62,0:04:39.46,Default,,0000,0000,0000,,with X. And we undo all the
Dialogue: 0,0:04:39.46,0:04:44.99,Default,,0000,0000,0000,,operations of ETH. So we start\Nby taking off 7 to undo the plus
Dialogue: 0,0:04:44.99,0:04:52.52,Default,,0000,0000,0000,,7 bit. And then we\Nmultiplied by minus one. So now
Dialogue: 0,0:04:52.52,0:04:59.85,Default,,0000,0000,0000,,we divide by minus one. That\Ngives us 7 minus X.
Dialogue: 0,0:05:01.05,0:05:03.80,Default,,0000,0000,0000,,And we started off by cubing.
Dialogue: 0,0:05:04.62,0:05:06.20,Default,,0000,0000,0000,,So this time we take cube roots.
Dialogue: 0,0:05:06.79,0:05:14.59,Default,,0000,0000,0000,,That gives us the cube\Nroot of 7 minus X.
Dialogue: 0,0:05:15.26,0:05:18.74,Default,,0000,0000,0000,,So this time F inverse of X.
Dialogue: 0,0:05:19.65,0:05:20.85,Default,,0000,0000,0000,,Is cube roots?
Dialogue: 0,0:05:21.53,0:05:23.51,Default,,0000,0000,0000,,7 minus X.
Dialogue: 0,0:05:25.72,0:05:32.22,Default,,0000,0000,0000,,Now we can also\Nuse algebraic manipulation to
Dialogue: 0,0:05:32.22,0:05:34.65,Default,,0000,0000,0000,,workout in versus.
Dialogue: 0,0:05:35.27,0:05:38.76,Default,,0000,0000,0000,,I'll show you how to do this\Nwith our second example, which
Dialogue: 0,0:05:38.76,0:05:40.52,Default,,0000,0000,0000,,was. F of X.
Dialogue: 0,0:05:41.04,0:05:43.17,Default,,0000,0000,0000,,Equals 3X.
Dialogue: 0,0:05:43.77,0:05:51.08,Default,,0000,0000,0000,,+2. Now remember that\Nwe want to take F of X and send
Dialogue: 0,0:05:51.08,0:05:52.59,Default,,0000,0000,0000,,it back to X.
Dialogue: 0,0:05:53.33,0:06:00.52,Default,,0000,0000,0000,,So if I set F of X\Nequals Y equals 3 X +2.
Dialogue: 0,0:06:01.20,0:06:02.83,Default,,0000,0000,0000,,We want to F inverse.
Dialogue: 0,0:06:03.52,0:06:06.26,Default,,0000,0000,0000,,To take Y and give us back X.
Dialogue: 0,0:06:07.02,0:06:11.39,Default,,0000,0000,0000,,So we need to workout how to get\N2X from Y?
Dialogue: 0,0:06:12.08,0:06:17.58,Default,,0000,0000,0000,,So if I write down again, Y\Nequals 3 X +2.
Dialogue: 0,0:06:17.59,0:06:18.97,Default,,0000,0000,0000,,I can rearrange this.
Dialogue: 0,0:06:19.76,0:06:22.14,Default,,0000,0000,0000,,And I get why minus 2?
Dialogue: 0,0:06:22.76,0:06:29.52,Default,,0000,0000,0000,,Equals 3X. So X\Nequals Y minus two, all
Dialogue: 0,0:06:29.52,0:06:31.39,Default,,0000,0000,0000,,divided by three.
Dialogue: 0,0:06:33.02,0:06:38.49,Default,,0000,0000,0000,,So to get from Y and go to X,\Nyou need to take why take off 2
Dialogue: 0,0:06:38.49,0:06:39.78,Default,,0000,0000,0000,,and divided by three?
Dialogue: 0,0:06:40.87,0:06:44.47,Default,,0000,0000,0000,,This means that if inverse of Y.
Dialogue: 0,0:06:45.22,0:06:47.56,Default,,0000,0000,0000,,Equals Y minus two, all divided
Dialogue: 0,0:06:47.56,0:06:52.82,Default,,0000,0000,0000,,by three. But that's just the\Nsame as saying that F inverse of
Dialogue: 0,0:06:52.82,0:06:56.53,Default,,0000,0000,0000,,X. Is X minus two all divided by
Dialogue: 0,0:06:56.53,0:07:00.39,Default,,0000,0000,0000,,three? So this is how we use\Nalgebraic manipulation to
Dialogue: 0,0:07:00.39,0:07:01.30,Default,,0000,0000,0000,,workout in versus.
Dialogue: 0,0:07:02.96,0:07:09.75,Default,,0000,0000,0000,,Now we can use this\Nmethod to workout some slightly
Dialogue: 0,0:07:09.75,0:07:13.14,Default,,0000,0000,0000,,more difficult in versus 2.
Dialogue: 0,0:07:14.67,0:07:21.43,Default,,0000,0000,0000,,This time will have F of X\Nbeing X over X minus one.
Dialogue: 0,0:07:22.63,0:07:26.03,Default,,0000,0000,0000,,And we have to set X is greater\Nthan one because otherwise
Dialogue: 0,0:07:26.03,0:07:29.42,Default,,0000,0000,0000,,you'll get a zero denominators.\NSo we only look at this function
Dialogue: 0,0:07:29.42,0:07:30.84,Default,,0000,0000,0000,,for X greater than one.
Dialogue: 0,0:07:32.27,0:07:38.81,Default,,0000,0000,0000,,That work at the inverse again\Nwill have Y equals X over X
Dialogue: 0,0:07:38.81,0:07:44.44,Default,,0000,0000,0000,,minus one. Remember again we\Nneed to get to X from Y, so we
Dialogue: 0,0:07:44.44,0:07:48.25,Default,,0000,0000,0000,,need to write X in terms of Y,\Nso will rearrange this.
Dialogue: 0,0:07:49.25,0:07:54.89,Default,,0000,0000,0000,,Or multiply both sides by X\Nminus one and that gives us why
Dialogue: 0,0:07:54.89,0:07:56.63,Default,,0000,0000,0000,,times X minus one.
Dialogue: 0,0:07:56.79,0:08:03.04,Default,,0000,0000,0000,,Equals X. And we want and\NX equals out of this with all
Dialogue: 0,0:08:03.04,0:08:07.45,Default,,0000,0000,0000,,the access on one side, so I'll\Nmultiply it. These brackets you
Dialogue: 0,0:08:07.45,0:08:09.66,Default,,0000,0000,0000,,get YX minus Y equals X.
Dialogue: 0,0:08:10.72,0:08:13.84,Default,,0000,0000,0000,,Then I'll take all the ex is\Nover to one side, so that gives
Dialogue: 0,0:08:13.84,0:08:20.20,Default,,0000,0000,0000,,us. Wax. Minus\NX equals Y.
Dialogue: 0,0:08:21.45,0:08:27.59,Default,,0000,0000,0000,,Factor out the X you get X\Ntimes Y minus one equals Y
Dialogue: 0,0:08:27.59,0:08:32.78,Default,,0000,0000,0000,,and that gives you X\Nequals Y over Y minus one.
Dialogue: 0,0:08:34.66,0:08:39.14,Default,,0000,0000,0000,,So here we have X in terms of Y\Nagain so F inverse.
Dialogue: 0,0:08:39.80,0:08:45.19,Default,,0000,0000,0000,,Of Why? Is Y\Nover Y minus one?
Dialogue: 0,0:08:46.21,0:08:51.11,Default,,0000,0000,0000,,And that's just the same as\Nsaying F inverse of X equals X
Dialogue: 0,0:08:51.11,0:08:56.39,Default,,0000,0000,0000,,over X minus one. So in this\Ncase the inverse of F turns out
Dialogue: 0,0:08:56.39,0:08:59.03,Default,,0000,0000,0000,,to be exactly the same as F.
Dialogue: 0,0:08:59.81,0:09:02.39,Default,,0000,0000,0000,,This example shows how useful\Nalgebraic manipulation is
Dialogue: 0,0:09:02.39,0:09:05.61,Default,,0000,0000,0000,,because it would have been\Nreally, really difficult to try
Dialogue: 0,0:09:05.61,0:09:09.15,Default,,0000,0000,0000,,and get this inverse by just\Nreversing the operations of F.
Dialogue: 0,0:09:09.92,0:09:15.82,Default,,0000,0000,0000,,Not all\Nfunctions have
Dialogue: 0,0:09:15.82,0:09:23.01,Default,,0000,0000,0000,,straightforward inverses.\NLet's look at F of X.
Dialogue: 0,0:09:23.51,0:09:28.35,Default,,0000,0000,0000,,Equals X squared and I'll just\Nquickly sketch you a graph if I
Dialogue: 0,0:09:28.35,0:09:33.58,Default,,0000,0000,0000,,can. Now\Nwhen
Dialogue: 0,0:09:33.58,0:09:37.69,Default,,0000,0000,0000,,we\Nlook
Dialogue: 0,0:09:37.69,0:09:41.79,Default,,0000,0000,0000,,for\Nan
Dialogue: 0,0:09:41.79,0:09:43.84,Default,,0000,0000,0000,,inverse
Dialogue: 0,0:09:43.84,0:09:50.10,Default,,0000,0000,0000,,function. We want to\Ntake the value of F of X.
Dialogue: 0,0:09:51.17,0:09:52.69,Default,,0000,0000,0000,,And send it back to X.
Dialogue: 0,0:09:53.33,0:09:55.75,Default,,0000,0000,0000,,But in this case.
Dialogue: 0,0:09:58.46,0:10:01.44,Default,,0000,0000,0000,,There are two X is we could send\NF of X back to.
Dialogue: 0,0:10:04.47,0:10:10.78,Default,,0000,0000,0000,,That's because F of X is X\Nsquared, but F of minus X is
Dialogue: 0,0:10:10.78,0:10:12.14,Default,,0000,0000,0000,,also X squared.
Dialogue: 0,0:10:13.09,0:10:14.73,Default,,0000,0000,0000,,Now we can't define an inverse
Dialogue: 0,0:10:14.73,0:10:18.44,Default,,0000,0000,0000,,for a function. If there\Nare two things we could
Dialogue: 0,0:10:18.44,0:10:20.00,Default,,0000,0000,0000,,define each value to be.
Dialogue: 0,0:10:21.49,0:10:25.18,Default,,0000,0000,0000,,To get around this problem, we\Nrestrict how much of the
Dialogue: 0,0:10:25.18,0:10:26.52,Default,,0000,0000,0000,,function we look at.
Dialogue: 0,0:10:27.52,0:10:32.62,Default,,0000,0000,0000,,So instead of defining F of X\Nlike this for every X, what we
Dialogue: 0,0:10:32.62,0:10:35.16,Default,,0000,0000,0000,,do is we cut down the graph.
Dialogue: 0,0:10:36.35,0:10:40.06,Default,,0000,0000,0000,,We say F of
Dialogue: 0,0:10:40.06,0:10:42.81,Default,,0000,0000,0000,,X. Equals X squared.
Dialogue: 0,0:10:43.77,0:10:47.36,Default,,0000,0000,0000,,But only look at X greater than\Nor equal to 0.
Dialogue: 0,0:10:48.20,0:10:49.98,Default,,0000,0000,0000,,And that gives us a graph\Nlike this.
Dialogue: 0,0:10:57.12,0:10:59.32,Default,,0000,0000,0000,,Now, since we've cut out all the
Dialogue: 0,0:10:59.32,0:11:02.80,Default,,0000,0000,0000,,values this side.\NEach value of FX.
Dialogue: 0,0:11:04.15,0:11:08.63,Default,,0000,0000,0000,,Only comes\Nfrom One X.
Dialogue: 0,0:11:09.93,0:11:12.81,Default,,0000,0000,0000,,So in this case we can\Ndefine F inverse.
Dialogue: 0,0:11:14.16,0:11:16.28,Default,,0000,0000,0000,,And F inverse of X.
Dialogue: 0,0:11:17.21,0:11:20.20,Default,,0000,0000,0000,,Is plus square root of X.
Dialogue: 0,0:11:21.32,0:11:28.10,Default,,0000,0000,0000,,Now we didn't have to define F\Nfor X greater than or equal to
Dialogue: 0,0:11:28.10,0:11:34.10,Default,,0000,0000,0000,,0. We could have cut out the\Nother half of the graph instead,
Dialogue: 0,0:11:34.10,0:11:38.28,Default,,0000,0000,0000,,so I could have defined F of X\Nequals X squared.
Dialogue: 0,0:11:38.80,0:11:44.35,Default,,0000,0000,0000,,For X less than or equal to 0\Nand that would have given us a
Dialogue: 0,0:11:44.35,0:11:45.46,Default,,0000,0000,0000,,graph like this.
Dialogue: 0,0:11:50.84,0:11:54.78,Default,,0000,0000,0000,,Now again, if a pack of value of\NF of X.
Dialogue: 0,0:11:55.57,0:11:57.70,Default,,0000,0000,0000,,There's only one value.
Dialogue: 0,0:11:58.43,0:12:05.37,Default,,0000,0000,0000,,Of ex. That gives us the F\Nof X, so in this case F inverse
Dialogue: 0,0:12:05.37,0:12:06.57,Default,,0000,0000,0000,,of X defines.
Dialogue: 0,0:12:07.23,0:12:10.48,Default,,0000,0000,0000,,And it's equal to minus root X.
Dialogue: 0,0:12:11.42,0:12:16.61,Default,,0000,0000,0000,,Here's another function where we\Nneed to restrict the domain to
Dialogue: 0,0:12:16.61,0:12:19.44,Default,,0000,0000,0000,,be able to define an inverse.
Dialogue: 0,0:12:20.81,0:12:22.38,Default,,0000,0000,0000,,Would have F of X.
Dialogue: 0,0:12:22.94,0:12:24.37,Default,,0000,0000,0000,,Equals sign X.
Dialogue: 0,0:12:25.99,0:12:28.98,Default,,0000,0000,0000,,And this graph.
Dialogue: 0,0:12:29.68,0:12:31.82,Default,,0000,0000,0000,,Looks a bit like this. I'll try
Dialogue: 0,0:12:31.82,0:12:38.71,Default,,0000,0000,0000,,and sketch it. So for X greater\Nthan zero, does this kind of
Dialogue: 0,0:12:38.71,0:12:41.39,Default,,0000,0000,0000,,thing and carries on forever.
Dialogue: 0,0:12:43.92,0:12:46.73,Default,,0000,0000,0000,,Then repeats itself down this\Nway as well forever.
Dialogue: 0,0:12:47.40,0:12:51.67,Default,,0000,0000,0000,,Now, if a pick a value of\NF of X here.
Dialogue: 0,0:12:55.06,0:12:59.04,Default,,0000,0000,0000,,You can see there's certainly\Nmore than One X giving this F of
Dialogue: 0,0:12:59.04,0:13:02.26,Default,,0000,0000,0000,,X. In fact, there will\Nbe an infinite number.
Dialogue: 0,0:13:05.44,0:13:12.08,Default,,0000,0000,0000,,So we'll certainly need to cut\Ndown the domain of this function
Dialogue: 0,0:13:12.08,0:13:14.29,Default,,0000,0000,0000,,to define an inverse.
Dialogue: 0,0:13:15.60,0:13:19.46,Default,,0000,0000,0000,,What we do in this case is\Nwe look at X for X greater
Dialogue: 0,0:13:19.46,0:13:22.50,Default,,0000,0000,0000,,than or equal to minus 90\Ndegrees and less than or
Dialogue: 0,0:13:22.50,0:13:23.88,Default,,0000,0000,0000,,equal to plus 90 degrees.
Dialogue: 0,0:13:27.36,0:13:30.09,Default,,0000,0000,0000,,And if I block out the rest of
Dialogue: 0,0:13:30.09,0:13:32.53,Default,,0000,0000,0000,,the function. Hopefully.
Dialogue: 0,0:13:33.08,0:13:35.12,Default,,0000,0000,0000,,You'll be able to see that in
Dialogue: 0,0:13:35.12,0:13:39.66,Default,,0000,0000,0000,,this case. For every F of X\Nthere's only one ex giving that
Dialogue: 0,0:13:39.66,0:13:42.94,Default,,0000,0000,0000,,F of X. So for this.
Dialogue: 0,0:13:43.47,0:13:45.02,Default,,0000,0000,0000,,We restrict the domain.
Dialogue: 0,0:13:47.79,0:13:51.68,Default,,0000,0000,0000,,2X is greater than or equal to
Dialogue: 0,0:13:51.68,0:13:56.82,Default,,0000,0000,0000,,minus 90. Less than or equal\N2 + 90.
Dialogue: 0,0:13:58.59,0:14:00.34,Default,,0000,0000,0000,,And the inverse.
Dialogue: 0,0:14:01.05,0:14:06.03,Default,,0000,0000,0000,,Of Cynex Is called arc\Nsine X.
Dialogue: 0,0:14:07.21,0:14:13.91,Default,,0000,0000,0000,,Now this notation can\Nbe particularly confusing
Dialogue: 0,0:14:13.91,0:14:19.63,Default,,0000,0000,0000,,because. F inverse of X. Like I\Nsaid before, is not equal to one
Dialogue: 0,0:14:19.63,0:14:23.26,Default,,0000,0000,0000,,over sine X. But\Nyou'll often see
Dialogue: 0,0:14:23.26,0:14:26.49,Default,,0000,0000,0000,,things like sine\Nsquared of X.
Dialogue: 0,0:14:27.58,0:14:31.49,Default,,0000,0000,0000,,Which means cynex all squared.
Dialogue: 0,0:14:32.02,0:14:34.10,Default,,0000,0000,0000,,Just remember that sign.
Dialogue: 0,0:14:34.90,0:14:38.60,Default,,0000,0000,0000,,Minus one of X is
Dialogue: 0,0:14:38.60,0:14:41.26,Default,,0000,0000,0000,,not. One over sine X.
Dialogue: 0,0:14:42.38,0:14:48.56,Default,,0000,0000,0000,,But the inverse function sign X,\Nwhich is also called AC sine X.
Dialogue: 0,0:14:49.39,0:14:55.12,Default,,0000,0000,0000,,The functions, calls, and turn\Nalso need the domains to be
Dialogue: 0,0:14:55.12,0:14:57.73,Default,,0000,0000,0000,,restricted for us to define
Dialogue: 0,0:14:57.73,0:15:01.09,Default,,0000,0000,0000,,inverses. But we cover\Nthese more fully the trig
Dialogue: 0,0:15:01.09,0:15:01.68,Default,,0000,0000,0000,,functions video.
Dialogue: 0,0:15:03.13,0:15:10.29,Default,,0000,0000,0000,,There are some functions that\Ncounts have inverses even if
Dialogue: 0,0:15:10.29,0:15:13.87,Default,,0000,0000,0000,,we do restrict the domains.
Dialogue: 0,0:15:14.46,0:15:19.04,Default,,0000,0000,0000,,A good example of this is a\Nconstant function and that is
Dialogue: 0,0:15:19.04,0:15:21.72,Default,,0000,0000,0000,,something like say F of X equals
Dialogue: 0,0:15:21.72,0:15:26.34,Default,,0000,0000,0000,,4. The growth of this\Nlooks like this.
Dialogue: 0,0:15:31.34,0:15:36.35,Default,,0000,0000,0000,,Now you can see here that the\Nonly way we could get One X for
Dialogue: 0,0:15:36.35,0:15:41.36,Default,,0000,0000,0000,,each F of X here is to cut down\Nthe domain to a single point,
Dialogue: 0,0:15:41.36,0:15:44.37,Default,,0000,0000,0000,,and this isn't a very useful\Nthing to do.
Dialogue: 0,0:15:45.09,0:15:48.28,Default,,0000,0000,0000,,So in this case, we say that\Nthis function has no inverse.
Dialogue: 0,0:15:49.89,0:15:57.61,Default,,0000,0000,0000,,Now there is a noise easy\Nway of getting the graph of
Dialogue: 0,0:15:57.61,0:16:03.39,Default,,0000,0000,0000,,an inverse function for the\Ngraph of a function.
Dialogue: 0,0:16:04.19,0:16:07.68,Default,,0000,0000,0000,,If I just get you the graph\Nof some random function that
Dialogue: 0,0:16:07.68,0:16:08.85,Default,,0000,0000,0000,,I can think of.
Dialogue: 0,0:16:10.74,0:16:12.43,Default,,0000,0000,0000,,So we have X here.
Dialogue: 0,0:16:13.44,0:16:14.48,Default,,0000,0000,0000,,That's Why.
Dialogue: 0,0:16:15.53,0:16:18.36,Default,,0000,0000,0000,,And I'll just draw some\Nfunction that's appropriate,
Dialogue: 0,0:16:18.36,0:16:19.78,Default,,0000,0000,0000,,so something like that.
Dialogue: 0,0:16:21.32,0:16:23.84,Default,,0000,0000,0000,,Notice that.
Dialogue: 0,0:16:25.01,0:16:27.12,Default,,0000,0000,0000,,A point on this graph.
Dialogue: 0,0:16:27.95,0:16:29.100,Default,,0000,0000,0000,,Has coordinates X.
Dialogue: 0,0:16:31.45,0:16:32.41,Default,,0000,0000,0000,,F of X.
Dialogue: 0,0:16:38.14,0:16:43.73,Default,,0000,0000,0000,,Now since. F inverse sends F\Nof X 2X.
Dialogue: 0,0:16:44.79,0:16:48.88,Default,,0000,0000,0000,,We want the coordinates on\Nthe graph of F inverse to
Dialogue: 0,0:16:48.88,0:16:52.60,Default,,0000,0000,0000,,be the coordinates F of XX.\NSo these two interchanged.
Dialogue: 0,0:16:53.67,0:16:57.79,Default,,0000,0000,0000,,Now for turnover this\Ntransparency so that my old X
Dialogue: 0,0:16:57.79,0:17:03.56,Default,,0000,0000,0000,,axis was, well, my Y axis was\Nand vice versa. Do that for you.
Dialogue: 0,0:17:04.49,0:17:09.71,Default,,0000,0000,0000,,You can see by doing this I've\Ngot precisely that I've got F of
Dialogue: 0,0:17:09.71,0:17:11.85,Default,,0000,0000,0000,,X here. And X here.
Dialogue: 0,0:17:12.69,0:17:13.91,Default,,0000,0000,0000,,So these points.
Dialogue: 0,0:17:14.62,0:17:17.21,Default,,0000,0000,0000,,Must form the graph of F\Ninverse.
Dialogue: 0,0:17:18.71,0:17:24.26,Default,,0000,0000,0000,,So notice that in swapping these\Naxes all I did was I reflected
Dialogue: 0,0:17:24.26,0:17:25.54,Default,,0000,0000,0000,,down that line.
Dialogue: 0,0:17:27.97,0:17:30.52,Default,,0000,0000,0000,,The sort of bottom left, top\Nright diagonal reflecting
Dialogue: 0,0:17:30.52,0:17:33.35,Default,,0000,0000,0000,,down here to get from one\Ngraph to the other.
Dialogue: 0,0:17:36.51,0:17:40.45,Default,,0000,0000,0000,,And this line is precisely the\Nline Y equals X.
Dialogue: 0,0:17:41.27,0:17:42.37,Default,,0000,0000,0000,,So that must mean.
Dialogue: 0,0:17:42.88,0:17:44.76,Default,,0000,0000,0000,,But the graph of F inverse.
Dialogue: 0,0:17:45.73,0:17:50.29,Default,,0000,0000,0000,,Is the graph of F reflected in\Nthe line Y equals X?