Naming alkanes with alkyl groups

WEBVTT
Kind: captions
Language: en

00:00:00.560 --> 00:00:04.870
So far we've dealt with simple
chains and rings of carbons.

00:00:04.870 --> 00:00:06.540
But let's think about what
happens when things get a

00:00:06.540 --> 00:00:07.810
little bit more complex.

00:00:07.810 --> 00:00:10.210
Let me just draw a molecule here
and we can think about

00:00:10.210 --> 00:00:11.470
how we'll name it.

00:00:11.470 --> 00:00:20.890
So let me draw it like that.

00:00:20.890 --> 00:00:23.710
So here we don't have
a simple chain.

00:00:23.710 --> 00:00:25.910
It branches off at some point.

00:00:25.910 --> 00:00:28.870
So what we do in this situation
is we find the

00:00:28.870 --> 00:00:31.370
longest chain in this molecule,
and let's think

00:00:31.370 --> 00:00:33.010
about what the longest
chain is.

00:00:33.010 --> 00:00:37.170
If we start here we get one,
two, three, four, five, six,

00:00:37.170 --> 00:00:39.330
seven, eight carbons.

00:00:39.330 --> 00:00:43.820
If we start here we get one,
two, three, four, five, six,

00:00:43.820 --> 00:00:46.570
seven, eight, nine carbons.

00:00:46.570 --> 00:00:48.060
So what we do is we look
at the longest chain.

00:00:48.060 --> 00:00:49.770
The longest chain is this
one right here.

00:00:49.770 --> 00:00:51.290
Let me do this in a
different color.

00:00:51.290 --> 00:00:56.930
So this is one, two, three, one,
two, three, four, five,

00:00:56.930 --> 00:01:00.920
six, seven, eight,
nine carbons.

00:01:00.920 --> 00:01:04.239
And so this will be-- that will
be kind of the core of

00:01:04.239 --> 00:01:04.840
our naming.

00:01:04.840 --> 00:01:06.510
So it's nine carbons.

00:01:06.510 --> 00:01:15.270
So that right here, longest
chain, has nine carbons.

00:01:15.270 --> 00:01:17.740
You can kind of view this is the
backbone of our molecule

00:01:17.740 --> 00:01:19.830
and also the backbone
of our naming.

00:01:19.830 --> 00:01:22.660
Nine carbons, so we're dealing
with a nonane.

00:01:25.310 --> 00:01:29.600
Remember, nine, non- for nine,
and then -ane, because we're

00:01:29.600 --> 00:01:31.230
dealing with an alkane.

00:01:31.230 --> 00:01:34.860
We have all single
bonds over here.

00:01:34.860 --> 00:01:37.420
Now, what do we do with
this thing over here?

00:01:37.420 --> 00:01:40.140
Well, what's kind of sticking
off of this chain?

00:01:40.140 --> 00:01:44.130
Well, we essentially have one
carbon here that's attached to

00:01:44.130 --> 00:01:44.970
the chain, right?

00:01:44.970 --> 00:01:47.470
This carbon is bonded to
a carbon on the chain.

00:01:47.470 --> 00:01:49.930
If we wanted to draw the entire
molecule, we could draw

00:01:49.930 --> 00:01:53.550
carbon here and it would it be
bonded to three hydrogens, but

00:01:53.550 --> 00:01:54.850
that is all implicit.

00:01:54.850 --> 00:01:56.900
But what is this thing
right here?

00:01:56.900 --> 00:01:58.330
Well, it's one carbon.

00:01:58.330 --> 00:02:01.270
You might be tempted to say
it's methane, but it's not

00:02:01.270 --> 00:02:03.520
methane because it's attached
to other things.

00:02:03.520 --> 00:02:07.550
But it would be right to
use the prefix meth-.

00:02:07.550 --> 00:02:10.340
So this right here, you would
use the prefix meth-, so this

00:02:10.340 --> 00:02:17.020
is one carbon, so you would want
to use the prefix meth-,

00:02:17.020 --> 00:02:21.200
but because it isn't the main
chain, it is added to

00:02:21.200 --> 00:02:24.760
something else, we don't write
methane, we write methyl.

00:02:24.760 --> 00:02:26.210
So let me write this
right here: methyl.

00:02:28.900 --> 00:02:37.360
So this means it branches
off longer chain.

00:02:40.810 --> 00:02:44.070
So this molecule right here
would be called-- and

00:02:44.070 --> 00:02:45.340
actually, there's one
more thing that we

00:02:45.340 --> 00:02:46.390
have to think about.

00:02:46.390 --> 00:02:48.670
This thing could have been--
we could just call it

00:02:48.670 --> 00:02:49.750
methylnonane.

00:02:49.750 --> 00:02:51.250
Let me just write that
down right now.

00:02:51.250 --> 00:02:53.490
So we could just call this
right from the get-go, we

00:02:53.490 --> 00:02:54.740
could call it methylnonane.

00:03:01.990 --> 00:03:04.740
But there's one problem with
just calling it methylnonane.

00:03:04.740 --> 00:03:06.970
This is methylnonane, but so
what's-- let me draw it.

00:03:06.970 --> 00:03:11.930
So the thing I just drew here,
this is, one, two, three,

00:03:11.930 --> 00:03:16.340
four, one, two, three, four,
five, six, seven, eight, nine.

00:03:16.340 --> 00:03:17.610
So that's the nonane.

00:03:17.610 --> 00:03:21.430
And the one I drew here's one,
two, three, the methyl is

00:03:21.430 --> 00:03:22.340
right over here.

00:03:22.340 --> 00:03:23.650
That's what I just drew.

00:03:23.650 --> 00:03:26.250
That is-- let me do it
in the same color.

00:03:26.250 --> 00:03:28.210
So this is what I just drew.

00:03:28.210 --> 00:03:30.420
But maybe, what if it was
something like this?

00:03:30.420 --> 00:03:34.500
What if it was one, two, three,
four, five, six, seven,

00:03:34.500 --> 00:03:35.490
eight, nine?

00:03:35.490 --> 00:03:39.560
And let's say that the methyl
group was, instead of being

00:03:39.560 --> 00:03:42.970
right here, let's say the methyl
group was right here.

00:03:42.970 --> 00:03:45.170
So how would you name these
two things differently?

00:03:45.170 --> 00:03:48.210
This name right now does not
differentiate between the

00:03:48.210 --> 00:03:50.920
methyl group being on this
carbon versus the methyl group

00:03:50.920 --> 00:03:53.740
being on this carbon over here
or this carbon over there.

00:03:53.740 --> 00:03:57.210
And what you do in this case is
you number the carbons on

00:03:57.210 --> 00:04:01.170
the longest chain, on the main
backbone, and you number them

00:04:01.170 --> 00:04:03.820
so that the methyl group is
attached to the lowest

00:04:03.820 --> 00:04:04.570
possible number.

00:04:04.570 --> 00:04:06.730
So there's two possible ways
to number this chain.

00:04:06.730 --> 00:04:08.420
You could start here
as one, two, three,

00:04:08.420 --> 00:04:10.440
four, five, six, seven.

00:04:10.440 --> 00:04:12.710
Then you would say that this
would be that the methyl group

00:04:12.710 --> 00:04:14.170
is attached to the
seven carbon.

00:04:14.170 --> 00:04:16.450
Or you could start numbering
from this end of the chain.

00:04:16.450 --> 00:04:19.779
It would be one, two, three.

00:04:19.779 --> 00:04:21.740
Or so it could also be attached
to the third.

00:04:21.740 --> 00:04:24.420
Three, four, five, six,
seven, eight, nine.

00:04:24.420 --> 00:04:26.440
So you want to number
it so you're closest

00:04:26.440 --> 00:04:27.130
to the methyl group.

00:04:27.130 --> 00:04:28.020
So you want to start here.

00:04:28.020 --> 00:04:36.240
One, two, three, four, five,
six, seven, eight, nine.

00:04:36.240 --> 00:04:39.240
So in this case, this would
not just be methylnonane.

00:04:39.240 --> 00:04:41.406
This thing right here
would be three.

00:04:41.406 --> 00:04:43.690
Let me make this very clear.

00:04:43.690 --> 00:04:46.540
It would be 3-methylnonane.

00:04:46.540 --> 00:04:51.160
Because the methyl group, that
one carbon, that one CH3, is

00:04:51.160 --> 00:04:55.810
attached to the third carbon
on our nonane backbone, so

00:04:55.810 --> 00:04:58.430
that right there is
3-methylnonane.

00:04:58.430 --> 00:05:00.880
Since I drew this out, what
is this right here?

00:05:00.880 --> 00:05:04.490
Well, once again, we have this
nonane backbone, one, two,

00:05:04.490 --> 00:05:06.980
three, four, five, six, seven,
eight, nine, and you want to

00:05:06.980 --> 00:05:10.420
number it so that this methyl
group is at the lowest number

00:05:10.420 --> 00:05:11.640
carbon as possible.

00:05:11.640 --> 00:05:13.320
So it's closer to this
end than that end.

00:05:13.320 --> 00:05:19.870
So it's one, two, three, four,
I'll do that in magenta, and

00:05:19.870 --> 00:05:23.930
you can keep numbering, five,
six, seven, eight, nine.

00:05:23.930 --> 00:05:28.780
So this one is going to be four,
and then you have your

00:05:28.780 --> 00:05:29.360
methyl group.

00:05:29.360 --> 00:05:31.570
It's just one carbon sitting
right there, so

00:05:31.570 --> 00:05:32.820
4-methylnonane.

00:05:42.560 --> 00:05:44.840
Now, let's go the other
way around.

00:05:44.840 --> 00:05:47.740
Let's say we start with the name
and we want to figure out

00:05:47.740 --> 00:05:48.490
its formula.

00:05:48.490 --> 00:05:52.310
And hopefully, this gives you
a good understanding how

00:05:52.310 --> 00:05:53.410
things will get more complex.

00:05:53.410 --> 00:05:55.020
And over the next few videos,
they'll get more and more

00:05:55.020 --> 00:05:56.550
complex, but you'll
see it's all at

00:05:56.550 --> 00:05:58.270
least reasonably logical.

00:05:58.270 --> 00:06:07.410
So let's say I were to give
you let me think of one.

00:06:07.410 --> 00:06:08.660
Let's say 2-propylheptane.

00:06:18.510 --> 00:06:20.780
So how do we take this apart?

00:06:20.780 --> 00:06:23.310
So the first thing you
see, it is an alkane.

00:06:23.310 --> 00:06:24.930
It's all going to
be single bonds.

00:06:24.930 --> 00:06:26.540
Heptane, this is kind
of the core.

00:06:26.540 --> 00:06:30.520
So the longest chain here, hept-
is the prefix for seven.

00:06:30.520 --> 00:06:32.530
So this tells us-- let
me make this clear.

00:06:32.530 --> 00:06:36.005
This tells us that we're dealing
with all single bonds.

00:06:39.790 --> 00:06:41.295
That's what the -ane tells us.

00:06:41.295 --> 00:06:43.700
If we had double bonds,
it would be -ene,

00:06:43.700 --> 00:06:44.970
triple bonds, -ine.

00:06:44.970 --> 00:06:46.680
We're going to see that in the
future, but let's stay

00:06:46.680 --> 00:06:48.220
simple right now.

00:06:48.220 --> 00:06:51.955
The hept- tells us we're dealing
with seven carbons.

00:06:54.570 --> 00:06:56.470
And then what does this
2-propyl mean?

00:06:56.470 --> 00:07:01.980
Well, propyl tells us-- what
was the prop- prefix?

00:07:01.980 --> 00:07:05.870
If methyl is one, ethyl is two,
propyl is three, butyl is

00:07:05.870 --> 00:07:12.180
four, so this is a three-carbon
group, and it's

00:07:12.180 --> 00:07:15.980
going to be attached to the
second carbon on the heptanes.

00:07:15.980 --> 00:07:18.520
Let me draw the heptane chain.

00:07:18.520 --> 00:07:23.420
So we have seven carbons, so
one, two, three, four, five,

00:07:23.420 --> 00:07:24.850
six, seven.

00:07:24.850 --> 00:07:29.490
And then on the second carbon--
just let me pick a

00:07:29.490 --> 00:07:30.220
good color here.

00:07:30.220 --> 00:07:33.580
On the second carbon, so if we
just number it one, two,

00:07:33.580 --> 00:07:38.170
three, four, five, six, seven,
on the second carbon right

00:07:38.170 --> 00:07:40.440
there, we have a propyl group.

00:07:40.440 --> 00:07:43.015
We have a propyl group, but it
is a three-carbon group.

00:07:43.015 --> 00:07:44.430
So let me draw the
propyl group.

00:07:44.430 --> 00:07:49.920
So we have a one, two, three,
three-carbon group, and it's

00:07:49.920 --> 00:07:53.170
attached at the second
carbon right there.

00:07:53.170 --> 00:07:56.330
So this molecule,
2-propylheptane

00:07:56.330 --> 00:07:58.240
looks just like this.

00:07:58.240 --> 00:08:00.610
Let's do another one, maybe one
that seems a little bit

00:08:00.610 --> 00:08:02.670
more difficult.

00:08:02.670 --> 00:08:08.850
Let's say that we have
6-butyltetradecane.

00:08:16.700 --> 00:08:18.880
And all of these might seem a
little daunting when you see

00:08:18.880 --> 00:08:21.450
it at first, but if you really
just break it up, it actually

00:08:21.450 --> 00:08:22.540
is pretty logical.

00:08:22.540 --> 00:08:24.945
So once again, we have the -ane
there, so it's all going

00:08:24.945 --> 00:08:27.110
to be single bonds.

00:08:27.110 --> 00:08:28.830
What is tetradec-?

00:08:28.830 --> 00:08:30.350
What is that prefix?

00:08:30.350 --> 00:08:33.240
Well, dec- is 10 and then
we have tetra-, that's 4

00:08:33.240 --> 00:08:34.620
plus 10, that's 14.

00:08:34.620 --> 00:08:36.510
Tetradec- is 14.

00:08:36.510 --> 00:08:37.850
So this is telling us
that we're dealing

00:08:37.850 --> 00:08:43.870
with a 14-carbon chain.

00:08:43.870 --> 00:08:45.240
Let's draw that.

00:08:45.240 --> 00:08:50.540
So one, two, three, four, five,
six, seven, eight, nine,

00:08:50.540 --> 00:08:54.380
ten, eleven, twelve, thirteen,
fourteen carbon chain.

00:08:54.380 --> 00:08:59.540
And if we number them, one, two,
three, four, five, six,

00:08:59.540 --> 00:09:04.910
seven, eight, nine, ten, eleven,
twelve, thirteen,

00:09:04.910 --> 00:09:10.530
fourteen, this tells us that
the sixth carbon will be

00:09:10.530 --> 00:09:12.360
butyl, will be a butyl group.

00:09:12.360 --> 00:09:14.410
So let's see, the sixth
carbon is right here.

00:09:14.410 --> 00:09:17.680
And butyl, and you should
memorize this, methyl is one,

00:09:17.680 --> 00:09:21.910
ethyl is two, propyl is three,
butyl is four, so we have a

00:09:21.910 --> 00:09:23.015
four-carbon group.

00:09:23.015 --> 00:09:25.290
So let me write this
in a pink maybe.

00:09:25.290 --> 00:09:33.140
So this is a four-carbon
group.

00:09:33.140 --> 00:09:35.650
When you call it an alkyl
group, it's attached to

00:09:35.650 --> 00:09:36.770
something else.

00:09:36.770 --> 00:09:39.870
So on the sixth carbon, we have
a four-carbon group, so

00:09:39.870 --> 00:09:41.990
one-- let me draw it this way.

00:09:41.990 --> 00:09:43.410
No, this is fine.

00:09:43.410 --> 00:09:49.540
Let's say we have one, two,
three, four, so that is our

00:09:49.540 --> 00:09:53.560
butyl right there, and it's
attached to the sixth carbon.

00:09:53.560 --> 00:09:55.700
It's attached to the sixth
carbon right there.

00:09:55.700 --> 00:09:57.900
So this is what our molecule
would look like.

00:09:57.900 --> 00:09:59.460
Now, let me ask you
a question.

00:09:59.460 --> 00:10:02.385
Would you ever see a
9-butyltetradecane?

00:10:04.990 --> 00:10:07.640
Would you ever see a
9-butyltetradecane?

00:10:07.640 --> 00:10:11.400
So something where-- so let
me redraw it like this.

00:10:11.400 --> 00:10:16.030
So a 9-butyltetradecane, so it
would have butyl over here.

00:10:16.030 --> 00:10:20.460
One, two, three, four, one,
two, three, four.

00:10:20.460 --> 00:10:23.880
Would you ever see a
9-butyltetradecane?

00:10:23.880 --> 00:10:25.060
Will you ever see
that written?

00:10:25.060 --> 00:10:27.510
You might say, oh, yeah,
Sal, you just drew it.

00:10:27.510 --> 00:10:29.320
And the reason why you won't
ever see that written is

00:10:29.320 --> 00:10:31.390
because there's a better
way to number it

00:10:31.390 --> 00:10:32.430
if it's like this.

00:10:32.430 --> 00:10:35.410
Instead of starting over here at
one at this end, you'd want

00:10:35.410 --> 00:10:36.810
to start one that end.

00:10:36.810 --> 00:10:39.810
So instead of saying
9-butyltetradecane, you should

00:10:39.810 --> 00:10:40.520
number it the other way.

00:10:40.520 --> 00:10:45.550
You should say this is one, two,
three, four, five, six.

00:10:45.550 --> 00:10:47.715
So this is actually also
6-butyltetradecane.

00:10:50.565 --> 00:10:51.580
Let me be sure I got that.

00:10:51.580 --> 00:10:54.320
One, two, three, four,
five, six.

00:10:54.320 --> 00:10:57.280
So instead of starting from the
left and making this the

00:10:57.280 --> 00:11:00.140
ninth carbon, you always want
to start numbering from the

00:11:00.140 --> 00:11:02.340
direction that has the lowest
number for the first group.


Naming simple alkanes

WEBVTT
Kind: captions
Language: en

00:00:00.560 --> 00:00:05.180
So when we name organic
molecules, the core of what we

00:00:05.180 --> 00:00:09.340
think about is just how many
carbons form the chain or form

00:00:09.340 --> 00:00:11.070
the ring that we're
looking at?

00:00:11.070 --> 00:00:13.630
So let's say I have something
like this.

00:00:13.630 --> 00:00:15.360
And hopefully, you're reasonably
familiar with what

00:00:15.360 --> 00:00:15.870
this represents.

00:00:15.870 --> 00:00:17.140
And I'll review it
a little bit.

00:00:17.140 --> 00:00:19.310
So let's say I have something
like this.

00:00:19.310 --> 00:00:21.390
Let's say that that
is the molecular

00:00:21.390 --> 00:00:22.660
structure right there.

00:00:22.660 --> 00:00:25.620
And so the first question you
should be asking is how many

00:00:25.620 --> 00:00:27.770
carbons are there?

00:00:27.770 --> 00:00:29.030
And some of you might
say, wait, how

00:00:29.030 --> 00:00:30.080
is that even a molecule?

00:00:30.080 --> 00:00:32.590
And just as a review, the
end point of every line

00:00:32.590 --> 00:00:34.120
represents a carbon.

00:00:34.120 --> 00:00:37.620
So that's a carbon, that's a
carbon, that's a carbon,

00:00:37.620 --> 00:00:39.860
that's a carbon, that's a
carbon, that's a carbon.

00:00:39.860 --> 00:00:46.950
And we have one, two, three,
four, five, six carbons and we

00:00:46.950 --> 00:00:49.210
have no double bonds.

00:00:49.210 --> 00:00:51.830
So if you have all of that
information you're ready to

00:00:51.830 --> 00:00:53.500
name this molecule.

00:00:53.500 --> 00:00:55.370
And before I actually name it,
let me just kind of give you

00:00:55.370 --> 00:00:57.290
all of the different prefixes.

00:00:57.290 --> 00:01:03.370
So if you have one carbon,
the prefix is meth-.

00:01:06.610 --> 00:01:10.400
If you have two carbons,
the prefix is eth-.

00:01:10.400 --> 00:01:12.930
And it's good to memorize
at least up to about 10.

00:01:12.930 --> 00:01:14.760
And actually it kind of
repeats after that.

00:01:14.760 --> 00:01:19.760
If you have three carbons,
the prefix is prop-.

00:01:19.760 --> 00:01:22.390
Prop- like propane, and you've
heard of ethane and methane.

00:01:22.390 --> 00:01:24.140
So you'll see all
of that soon.

00:01:24.140 --> 00:01:27.560
Four, you're talking
about but-.

00:01:27.560 --> 00:01:30.850
Five, and after five it kind
of becomes the traditional

00:01:30.850 --> 00:01:33.480
prefixes that we associate with
a lot of these numbers.

00:01:33.480 --> 00:01:36.770
So at five, it's pent-,
like pentagon.

00:01:36.770 --> 00:01:40.280
Six, it's hex- like hexagon.

00:01:40.280 --> 00:01:42.850
Seven is hept-.

00:01:42.850 --> 00:01:46.410
Eight is oct-, like octagon.

00:01:46.410 --> 00:01:49.000
Nine is non-.

00:01:49.000 --> 00:01:50.355
Ten is dec-.

00:01:53.150 --> 00:01:55.670
And then after that it kind of
starts to have a pattern here,

00:01:55.670 --> 00:01:58.260
and you're not going to really
deal with things much beyond

00:01:58.260 --> 00:02:00.620
the teens, but I'll just write
them down here just out of

00:02:00.620 --> 00:02:01.460
interest.

00:02:01.460 --> 00:02:05.850
Eleven is undec-.

00:02:05.850 --> 00:02:07.840
And for those of you who know
French, I'm not one of them,

00:02:07.840 --> 00:02:12.360
but I know that one in French
is une, or in Spanish, uno.

00:02:12.360 --> 00:02:15.400
So it's 1 and 10, 11.

00:02:15.400 --> 00:02:19.320
Twelve is dodec-.

00:02:19.320 --> 00:02:22.510
Do or dos, if you're speaking
Spanish, for 2.

00:02:22.510 --> 00:02:25.720
Dec for 10, 2 and
10, that's 12.

00:02:25.720 --> 00:02:27.980
Thirteen, you could imagine
what it's going to be.

00:02:27.980 --> 00:02:29.230
It is tridec-.

00:02:32.540 --> 00:02:34.750
Fourteen is tetradec-.

00:02:37.780 --> 00:02:40.330
A tetrapod is something
with four legs.

00:02:40.330 --> 00:02:42.520
And after that it, becomes
very systematic.

00:02:42.520 --> 00:02:44.560
At fifteen is pentadec-.

00:02:47.540 --> 00:02:50.730
Notice pent-, 5 and 10.

00:02:50.730 --> 00:02:52.870
Sixteen is hexadec-.

00:02:52.870 --> 00:02:54.690
Seventeen is heptadec-.

00:02:54.690 --> 00:02:56.390
So it just goes on
and on and on.

00:02:56.390 --> 00:02:58.220
I don't think I've got to go--
anyway, it's hexadec-,

00:02:58.220 --> 00:03:02.040
heptadec-, octodec- is
eighteen, nineteen is

00:03:02.040 --> 00:03:05.710
nonadec-, and then twenty
is actually iso-.

00:03:05.710 --> 00:03:07.140
But we won't even
go into that.

00:03:07.140 --> 00:03:08.570
This'll probably serve
our purposes.

00:03:08.570 --> 00:03:12.920
I mean, I could go up to
sixteen is hexadec-.

00:03:12.920 --> 00:03:15.690
So this is just how
many carbons are

00:03:15.690 --> 00:03:17.330
in our longest chain.

00:03:17.330 --> 00:03:19.460
What I drew here is
just one chain.

00:03:19.460 --> 00:03:21.430
So we could immediately--
let's try to name it.

00:03:21.430 --> 00:03:23.310
Well, how many carbons
do we have here?

00:03:23.310 --> 00:03:27.110
Well, we have one, two, three,
four, five, six carbons.

00:03:27.110 --> 00:03:30.260
So we'll be dealing with
hex- as a prefix.

00:03:30.260 --> 00:03:32.760
And then to get, I guess, the
post-fix on this prefix, or

00:03:32.760 --> 00:03:35.990
the kind of the root, you look
to see if there are any double

00:03:35.990 --> 00:03:36.940
bonds here.

00:03:36.940 --> 00:03:38.420
And there are no double
bonds here.

00:03:38.420 --> 00:03:41.470
And if we have no double bonds
in this carbon chain, we're

00:03:41.470 --> 00:03:43.560
dealing with an alkane.

00:03:43.560 --> 00:03:47.180
This is called an alkane, which
is a general term for

00:03:47.180 --> 00:03:50.805
all of the chains of carbons
that have no double bonds on

00:03:50.805 --> 00:03:53.390
them, or no triple bonds,
all single bonds.

00:03:53.390 --> 00:03:58.660
So in this situation, you take
hex- for six, so this is hex-.

00:03:58.660 --> 00:04:01.710
And then, because it's
an alkane, it gets

00:04:01.710 --> 00:04:03.170
the -ane from alkane.

00:04:03.170 --> 00:04:07.490
So this is hex-,
this is hexane.

00:04:07.490 --> 00:04:09.190
Let's do another one.

00:04:09.190 --> 00:04:12.120
Let's say I have this
thing right here.

00:04:12.120 --> 00:04:14.640
I'll draw-- let's make
it even longer.

00:04:14.640 --> 00:04:16.529
So let's say I have that
thing right there.

00:04:16.529 --> 00:04:18.160
So how many carbons
do we have?

00:04:18.160 --> 00:04:24.660
We have one, two, three, four,
five, six, seven carbons.

00:04:24.660 --> 00:04:27.850
They're all single bonds,
so it's an alkane.

00:04:27.850 --> 00:04:30.640
So this will be seven carbons.

00:04:30.640 --> 00:04:37.800
It is heptane because we
have all single bonds.

00:04:37.800 --> 00:04:41.900
Now, if things form a chain,
or if things form a ring, I

00:04:41.900 --> 00:04:45.850
should say, we put the prefix
cyclo- in front of it.

00:04:45.850 --> 00:04:48.930
So if I have-- let me show you
what I'm talking about.

00:04:48.930 --> 00:04:53.910
So if I just have five carbons,
one, two, three,

00:04:53.910 --> 00:04:56.480
four, so one, two, three,
four, five.

00:04:56.480 --> 00:04:57.520
OK, so that's five
right there.

00:04:57.520 --> 00:05:02.570
I have one, two, three, four,
five carbons in a chain.

00:05:02.570 --> 00:05:04.920
If I just have five carbons in
a chain like this, this would

00:05:04.920 --> 00:05:06.170
be pentane.

00:05:10.140 --> 00:05:12.260
But if I have five carbons
and they form a ring,

00:05:12.260 --> 00:05:13.090
so let me draw it.

00:05:13.090 --> 00:05:16.680
So it's one, two, three,
four, five carbons

00:05:16.680 --> 00:05:19.030
and it forms a ring.

00:05:19.030 --> 00:05:22.230
Let me make the drawing
a little bit better.

00:05:22.230 --> 00:05:23.890
So it's really, I'm just
drawing a pentagon.

00:05:23.890 --> 00:05:25.980
But notice, this has
five carbons on it.

00:05:25.980 --> 00:05:27.910
I can draw the carbons here.

00:05:27.910 --> 00:05:30.950
Carbon, carbon, carbon,
carbon, carbon.

00:05:30.950 --> 00:05:33.310
And just as a review, what you
don't see is the hydrogens

00:05:33.310 --> 00:05:33.900
they're bonded to.

00:05:33.900 --> 00:05:36.660
Each of these guys have two
bonds, so they must have two

00:05:36.660 --> 00:05:38.210
bonds with something else and
those are going to be with

00:05:38.210 --> 00:05:38.480
hydrogen,

00:05:38.480 --> 00:05:41.360
And I'lll draw it here just as
a bit of a review, but you

00:05:41.360 --> 00:05:45.160
notice very quickly, the drawing
gets extremely messy

00:05:45.160 --> 00:05:46.790
when you draw the
two hydrogens on

00:05:46.790 --> 00:05:48.100
each of these carbons.

00:05:48.100 --> 00:05:50.210
So it's a little bit
over-- maybe I

00:05:50.210 --> 00:05:52.260
shouldn't be doing that.

00:05:52.260 --> 00:05:52.800
But there you go.

00:05:52.800 --> 00:05:54.420
So it becomes very messy when
you draw the hydrogens, so

00:05:54.420 --> 00:05:56.130
it's better to just assume
that they're there.

00:05:56.130 --> 00:05:58.110
If we don't draw all four bonds
of the carbon, the other

00:05:58.110 --> 00:06:00.810
two bonds are going to
be with hydrogen.

00:06:00.810 --> 00:06:03.860
So here, you might say, OK, this
is an alkane, because I

00:06:03.860 --> 00:06:05.450
don't have any double
bonds here.

00:06:05.450 --> 00:06:07.380
All of these are single
bonds with the carbon.

00:06:07.380 --> 00:06:10.320
I have five carbons, so you
might say this is pentane, but

00:06:10.320 --> 00:06:11.380
you have to think about
one more thing.

00:06:11.380 --> 00:06:14.670
It's in a ring, so we add
the prefix cyclo- to it.

00:06:14.670 --> 00:06:18.010
So this is, because it's a ring,
we write cyclopentane.

00:06:23.150 --> 00:06:24.860
So let me just break
that apart.

00:06:24.860 --> 00:06:27.750
This tells us that we're
dealing with a ring.

00:06:27.750 --> 00:06:31.580
You see that this is
a ring right there.

00:06:31.580 --> 00:06:37.420
This tells us that we're dealing
with five carbons, and

00:06:37.420 --> 00:06:41.090
then this tells us right here,
the -ane part, that tells us

00:06:41.090 --> 00:06:43.880
that they are all
single bonds.

00:06:43.880 --> 00:06:47.180
All carbon-carbon
single bonds.

00:06:47.180 --> 00:06:48.940
No double or triple bonds.

00:06:48.940 --> 00:06:51.030
All single bonds.

00:06:51.030 --> 00:06:51.940
So let's go the other way.

00:06:51.940 --> 00:06:56.030
Let's start with the word and
let's see if we can figure out

00:06:56.030 --> 00:06:58.225
what the actual structure
would look like.

00:07:03.960 --> 00:07:05.810
Let's say I have cyclononane.

00:07:11.380 --> 00:07:13.040
So what is this telling me?

00:07:13.040 --> 00:07:15.120
This tells me I'm dealing
with a ring.

00:07:15.120 --> 00:07:16.570
That is a ring.

00:07:16.570 --> 00:07:18.170
It's going to have
a ring structure.

00:07:18.170 --> 00:07:22.520
It's going to have nine carbons,
nine C's, and then

00:07:22.520 --> 00:07:25.395
it's an alkane, so they're all
going to be single bonds.

00:07:34.560 --> 00:07:36.850
So if I want to draw it, I want
to draw nine carbons in a

00:07:36.850 --> 00:07:38.465
ring, it's not a trivial
thing to draw.

00:07:38.465 --> 00:07:46.190
I'll try my best, so let's see,
that's one, two, three,

00:07:46.190 --> 00:07:51.690
four, five, six, seven, eight.

00:07:54.620 --> 00:07:55.760
Let's see, let me draw it.

00:07:55.760 --> 00:07:58.470
I'll try a little a
better shot at it.

00:07:58.470 --> 00:08:06.900
So, let's see, you have one,
two, three, four, five, six,

00:08:06.900 --> 00:08:11.390
seven, eight, and then nine.

00:08:11.390 --> 00:08:13.910
And then you can connect the
last. So let me make sure that

00:08:13.910 --> 00:08:15.870
this is-- and obviously, I could
have drawn it better

00:08:15.870 --> 00:08:17.950
than that, but hopefully, you
can see all of the edges here.

00:08:17.950 --> 00:08:24.660
So I have one, two, three, four,
five, six, seven, eight,

00:08:24.660 --> 00:08:27.650
nine carbons.

00:08:27.650 --> 00:08:28.600
It's in a cycle.

00:08:28.600 --> 00:08:29.210
It's in a ring.

00:08:29.210 --> 00:08:30.460
I have nine carbons.

00:08:30.460 --> 00:08:34.919
They're all single bonded, so
this is cyclononane, although

00:08:34.919 --> 00:08:38.240
there's probably better ways to
draw that ring right there.

00:08:38.240 --> 00:08:47.050
So if someone were to tell you,
octane, and that word

00:08:47.050 --> 00:08:49.860
might feel familiar to you
from the gas station.

00:08:49.860 --> 00:08:53.310
They are literally talking about
the molecule octane.

00:08:53.310 --> 00:08:55.060
And now you know, or at least
you have a sense of what

00:08:55.060 --> 00:08:55.720
they're talking about.

00:08:55.720 --> 00:09:01.000
The oct- tells you that you
have eight carbons.

00:09:01.000 --> 00:09:03.680
There's no cyclo- in front of
it, so it's not a cyclo.

00:09:03.680 --> 00:09:05.180
It's just going to be a chain.

00:09:05.180 --> 00:09:08.680
And then the -ane part tells
you that they're all single

00:09:08.680 --> 00:09:11.580
bonds, so it's just going to be
eight carbons in a chain.

00:09:11.580 --> 00:09:16.580
One, two, three, four, five,
six, seven, eight.

00:09:16.580 --> 00:09:17.600
It's just going to
look like that.

00:09:17.600 --> 00:09:21.980
One, two, three, four, five,
six, seven, eight.

00:09:21.980 --> 00:09:24.290
In the next few videos, we'll
add more to these molecules

00:09:24.290 --> 00:09:26.990
and make the names even
more complex.


Representing structures of organic molecules

WEBVTT
Kind: captions
Language: en

00:00:00.740 --> 00:00:03.930
The one thing that probably
causes some of the most pain

00:00:03.930 --> 00:00:07.340
in chemistry, and in organic
chemistry, in particular, is

00:00:07.340 --> 00:00:10.990
just the notation and the
nomenclature or the naming

00:00:10.990 --> 00:00:12.000
that we use.

00:00:12.000 --> 00:00:14.380
And what I want to do here in
this video and really the next

00:00:14.380 --> 00:00:17.570
few videos is to just make sure
we have a firm grounding

00:00:17.570 --> 00:00:20.220
in the notation and in the
nomenclature or how we name

00:00:20.220 --> 00:00:21.450
things, and then everything
else will

00:00:21.450 --> 00:00:24.020
hopefully not be too difficult.

00:00:24.020 --> 00:00:26.430
So just to start off, and this
is really a little bit of

00:00:26.430 --> 00:00:29.810
review of regular chemistry,
if I just have a chain of

00:00:29.810 --> 00:00:31.730
carbons, and organic chemistry
is dealing

00:00:31.730 --> 00:00:34.040
with chains of carbons.

00:00:34.040 --> 00:00:36.540
Let me just draw a one-carbon
chain, so it's really kind of

00:00:36.540 --> 00:00:40.160
ridiculous to call it a chain,
but if we have one carbon over

00:00:40.160 --> 00:00:43.200
here and it has four valence
electrons, it

00:00:43.200 --> 00:00:44.790
wants to get to eight.

00:00:44.790 --> 00:00:46.480
That's the magic number
we learned in

00:00:46.480 --> 00:00:47.900
just regular chemistry.

00:00:47.900 --> 00:00:51.520
For all molecules, that's the
stable valence structure, I

00:00:51.520 --> 00:00:54.270
guess you could say it.

00:00:54.270 --> 00:00:56.560
A good partner to bond
with is hydrogen.

00:00:56.560 --> 00:01:03.490
So it has four valence electrons
and then hydrogen

00:01:03.490 --> 00:01:06.120
has one valence electron, so
they can each share an

00:01:06.120 --> 00:01:12.280
electron with each other
and then they

00:01:12.280 --> 00:01:13.550
both look pretty happy.

00:01:13.550 --> 00:01:15.970
I said eight's the magic number
for everybody except

00:01:15.970 --> 00:01:18.330
for hydrogen and helium.

00:01:18.330 --> 00:01:23.180
Both of them are happy because
they're only trying to fill

00:01:23.180 --> 00:01:24.820
their 1s orbital, so the
magic number for

00:01:24.820 --> 00:01:26.280
those two guys is two.

00:01:26.280 --> 00:01:28.110
So all of the hydrogens
now feel like

00:01:28.110 --> 00:01:29.370
they have two electrons.

00:01:29.370 --> 00:01:31.375
The carbon feels like
it has eight.

00:01:31.375 --> 00:01:33.520
Now, there's several
ways to write this.

00:01:33.520 --> 00:01:35.320
You could write it just like
this and you can see the

00:01:35.320 --> 00:01:39.510
electrons explicitly, or you
can draw little lines here.

00:01:39.510 --> 00:01:43.320
So I could also write this
exact molecule, which is

00:01:43.320 --> 00:01:45.740
methane, and we'll talk a little
bit more about why it's

00:01:45.740 --> 00:01:47.400
called methane later
in this video.

00:01:47.400 --> 00:01:51.200
I can write this exact structure
like this: a carbon

00:01:51.200 --> 00:01:52.745
bonded to four hydrogens.

00:01:56.410 --> 00:01:58.730
And the way that I've written
these bonds right here you

00:01:58.730 --> 00:02:01.810
could imagine that each of these
bonds consists of two

00:02:01.810 --> 00:02:05.330
electrons, one from the carbon
and one from the hydrogen.

00:02:05.330 --> 00:02:07.840
Now let's explore slightly
larger chains.

00:02:07.840 --> 00:02:09.919
So let's say I have a
two-carbon chain.

00:02:09.919 --> 00:02:11.600
Well, let me do a three-carbon
chain so it really

00:02:11.600 --> 00:02:12.790
looks like a chain.

00:02:12.790 --> 00:02:15.070
So if I were to draw everything
explicitly it might

00:02:15.070 --> 00:02:15.700
look like this.

00:02:15.700 --> 00:02:18.710
So I have a carbon.

00:02:18.710 --> 00:02:23.650
It has one, two, three,
four electrons.

00:02:23.650 --> 00:02:28.090
Maybe I have another carbon here
that has-- let me do the

00:02:28.090 --> 00:02:31.570
carbons in slightly different
shades of yellow.

00:02:31.570 --> 00:02:34.170
I have another carbon
here that has one,

00:02:34.170 --> 00:02:37.100
two, three, four electrons.

00:02:37.100 --> 00:02:39.110
And then let me do the other
carbon in that first yellow.

00:02:39.110 --> 00:02:40.500
And then I have another carbon
so we're going to have a

00:02:40.500 --> 00:02:41.850
three-carbon chain.

00:02:41.850 --> 00:02:47.040
It has one, two, three, four
valence electrons.

00:02:47.040 --> 00:02:49.130
Now, these other guys are
unpaired, and if you don't

00:02:49.130 --> 00:02:51.380
specify it, it's normally going
to be hydrogen, so let

00:02:51.380 --> 00:02:53.100
me draw some hydrogens
over here.

00:02:53.100 --> 00:02:56.980
So you're going to have a
hydrogen there, a hydrogen

00:02:56.980 --> 00:03:02.580
over there, a hydrogen over
here, a hydrogen over here, a

00:03:02.580 --> 00:03:06.740
hydrogen over there, a hydrogen
over here, almost

00:03:06.740 --> 00:03:09.490
done, a hydrogen there, and
then a hydrogen there.

00:03:09.490 --> 00:03:12.245
Now notice, in this molecular
structure that I've drawn, I

00:03:12.245 --> 00:03:13.850
have three carbons.

00:03:13.850 --> 00:03:16.120
They were each able to
form four bonds.

00:03:16.120 --> 00:03:19.080
This guy has bonds with three
hydrogens and another carbon.

00:03:19.080 --> 00:03:22.460
This guy has a bond with two
hydrogens and two carbons.

00:03:22.460 --> 00:03:25.280
This guy has a bond with three
hydrogens and then this carbon

00:03:25.280 --> 00:03:25.840
right here.

00:03:25.840 --> 00:03:28.750
And so this is a completely
valid molecular structure, but

00:03:28.750 --> 00:03:31.230
it was kind of a pain
to draw all of these

00:03:31.230 --> 00:03:32.730
valence electrons here.

00:03:32.730 --> 00:03:35.190
So what we typically would want
to do is, at least in

00:03:35.190 --> 00:03:36.680
this structure, and we're going
to see later in this

00:03:36.680 --> 00:03:39.010
video there's even simpler ways
to write it, so if we

00:03:39.010 --> 00:03:41.920
want at least do it with
these lines, we can

00:03:41.920 --> 00:03:43.680
draw it like this.

00:03:43.680 --> 00:03:47.400
So you have a carbon, carbon,
carbon, and then they are

00:03:47.400 --> 00:03:50.170
bonded to the hydrogens.

00:03:50.170 --> 00:03:52.560
So you'll almost never see it
written like this because this

00:03:52.560 --> 00:03:54.910
is just kind of crazy.

00:03:54.910 --> 00:03:58.000
Hyrdrogen, hydrogen-- at
least crazy to write.

00:03:58.000 --> 00:03:59.400
It takes forever.

00:03:59.400 --> 00:04:01.800
And it might be messy, like it
might not be clear where these

00:04:01.800 --> 00:04:02.870
electrons belong.

00:04:02.870 --> 00:04:04.640
I didn't write it as
clearly as I could.

00:04:04.640 --> 00:04:05.610
So they have two electrons
there.

00:04:05.610 --> 00:04:06.980
They share with these
two guys.

00:04:06.980 --> 00:04:08.305
Hopefully, that was
reasonably clear.

00:04:08.305 --> 00:04:10.340
But if we were to draw
it with the lines, it

00:04:10.340 --> 00:04:11.330
looks just like that.

00:04:11.330 --> 00:04:14.440
So it's a little bit neater,
faster to draw, same exact

00:04:14.440 --> 00:04:16.560
idea here and here.

00:04:16.560 --> 00:04:19.579
And in general, and we'll go
in more detail on it, this

00:04:19.579 --> 00:04:22.550
three-carbon chain, where
everything is a single bond,

00:04:22.550 --> 00:04:23.430
is propane.

00:04:23.430 --> 00:04:26.640
Let me write these words down
because it's helpful to get.

00:04:26.640 --> 00:04:29.080
This is methane.

00:04:29.080 --> 00:04:31.170
And you're going to see the
rhyme-- you're going to see

00:04:31.170 --> 00:04:34.420
the reason to this naming
soon enough.

00:04:34.420 --> 00:04:36.120
This is methane; this
is propane.

00:04:39.000 --> 00:04:42.910
And there's an even simpler
way to write propane.

00:04:42.910 --> 00:04:46.340
You could write it like this.

00:04:46.340 --> 00:04:48.360
Instead of explicitly drawing
these bonds, you could say

00:04:48.360 --> 00:04:52.010
that this part right here, you
could write that that part

00:04:52.010 --> 00:04:56.710
right there, that is CH3, so you
have a CH3, connected to

00:04:56.710 --> 00:05:06.350
a-- this is a CH2, that is CH2
which is then connected to

00:05:06.350 --> 00:05:08.100
another CH3.

00:05:08.100 --> 00:05:10.300
And the important thing is, no
matter what the notation, as

00:05:10.300 --> 00:05:13.010
long as you can figure out the
exact molecular structure, as

00:05:13.010 --> 00:05:15.770
long as you can-- so there's
this last CH3.

00:05:15.770 --> 00:05:18.850
Whether you have this, this,
or this, you know what the

00:05:18.850 --> 00:05:19.870
molecular structure is.

00:05:19.870 --> 00:05:22.780
You could draw any one of these
given any of the others.

00:05:22.780 --> 00:05:25.370
Now, there's an even simpler
way to write this.

00:05:25.370 --> 00:05:27.350
You could write it
just like this.

00:05:27.350 --> 00:05:29.030
Let me do it in a
different color.

00:05:29.030 --> 00:05:31.370
You literally could write it
so we have three carbons.

00:05:31.370 --> 00:05:34.550
So one, two, three.

00:05:34.550 --> 00:05:36.750
Now, this seems ridiculously
simple and you're like, how

00:05:36.750 --> 00:05:40.710
can this thing right here give
you the same information as

00:05:40.710 --> 00:05:43.180
all of these more complicated
ways to draw it?

00:05:43.180 --> 00:05:46.040
Well, in chemistry, and in
organic chemistry in

00:05:46.040 --> 00:05:50.670
particular, any of these-- let
me call it a line diagram or a

00:05:50.670 --> 00:05:51.870
line angle diagram.

00:05:51.870 --> 00:05:54.190
It's the simplest way and it's
actually probably the most

00:05:54.190 --> 00:05:57.350
useful way to show chains
of carbons or to

00:05:57.350 --> 00:05:58.860
show organic molecules.

00:05:58.860 --> 00:06:01.150
Once they start to get really,
really complicated, because

00:06:01.150 --> 00:06:04.050
then it's a pain to draw all of
the H's, but when you see

00:06:04.050 --> 00:06:07.470
something like this, you assume
that the end points of

00:06:07.470 --> 00:06:09.110
any lines have a carbon on it.

00:06:09.110 --> 00:06:11.970
So if you see something like
that, you assume that there's

00:06:11.970 --> 00:06:14.430
a carbon at that end point, a
carbon at that end point, and

00:06:14.430 --> 00:06:15.830
a carbon at that end point.

00:06:15.830 --> 00:06:18.790
And then you know that carbon
makes four bonds.

00:06:18.790 --> 00:06:20.880
There are no charges here.

00:06:20.880 --> 00:06:24.440
All the carbons are going to
make four bonds, and each of

00:06:24.440 --> 00:06:27.020
the carbons here, this carbon
has two bonds, so the other

00:06:27.020 --> 00:06:29.860
two bonds are implicitly going
to be with hydrogens.

00:06:29.860 --> 00:06:32.820
If they don't draw them, you
assume that they're going to

00:06:32.820 --> 00:06:33.460
be with hydrogens.

00:06:33.460 --> 00:06:36.170
This guy has one bond,
so the other three

00:06:36.170 --> 00:06:37.940
must be with hydrogen.

00:06:37.940 --> 00:06:42.200
This guy has one bond, so the
other three must be hydrogens.

00:06:42.200 --> 00:06:45.300
So just drawing that little line
angle thing right there,

00:06:45.300 --> 00:06:50.320
I actually did convey the exact
same information as this

00:06:50.320 --> 00:06:53.400
depiction, this depiction,
or this depiction.

00:06:53.400 --> 00:06:54.580
So you're going to see
a lot of this.

00:06:54.580 --> 00:06:56.050
This really simplifies things.

00:06:56.050 --> 00:06:57.930
And sometimes you see things
that are in between.

00:06:57.930 --> 00:07:00.820
You might see someone draw it
like this, where they'll write

00:07:00.820 --> 00:07:04.490
CH3, and then they'll
draw it like that.

00:07:04.490 --> 00:07:07.430
So that's kind of combining
this way of writing the

00:07:07.430 --> 00:07:11.270
molecule where you write the
CH3's for the end points, but

00:07:11.270 --> 00:07:13.660
then you implicitly have
the CH2 on the inside.

00:07:13.660 --> 00:07:15.590
You assume that this end point
right here is a C and it's

00:07:15.590 --> 00:07:17.850
bonded to two hydrogens.

00:07:17.850 --> 00:07:23.900
So these are all completely
valid ways of drawing the

00:07:23.900 --> 00:07:26.390
molecular structures of these
carbon chains or of these

00:07:26.390 --> 00:07:28.220
organic compounds.


Bill Gates talks about the Khan Academy at Aspen Ideas Festival 2010

WEBVTT
Kind: captions
Language: en

00:00:00.070 --> 00:00:01.945
MALE SPEAKER 1: What
could we do to improve K

00:00:01.945 --> 00:00:04.710
through 12 education
in America the best?

00:00:04.710 --> 00:00:07.920
BILL GATES: Well, today, if
you're motivated to learn,

00:00:07.920 --> 00:00:10.860
if you really, really,
really want to learn,

00:00:10.860 --> 00:00:14.530
this is an amazing
time for everyone.

00:00:14.530 --> 00:00:18.630
Because if you have
access to Wikipedia,

00:00:18.630 --> 00:00:22.120
if you have access to the
latest information on the web--

00:00:22.120 --> 00:00:25.060
there's a website that I've
just been using with my kids

00:00:25.060 --> 00:00:27.840
recently called Khan
Academy, K-H-A-N.

00:00:27.840 --> 00:00:31.300
Just one guy doing some
unbelievable 15-minute

00:00:31.300 --> 00:00:32.450
tutorials.

00:00:32.450 --> 00:00:36.820
There's great college lectures
out on academicearth.org.

00:00:36.820 --> 00:00:38.930
So if you're
motivated, now you can

00:00:38.930 --> 00:00:40.800
go to the very best lectures.

00:00:40.800 --> 00:00:42.540
MALE SPEAKER 1:
Speaking of allocation

00:00:42.540 --> 00:00:48.020
of society's resources, I think
you said, on your campus tour,

00:00:48.020 --> 00:00:52.670
at least at Harvard,
that the allocation of IQ

00:00:52.670 --> 00:00:57.690
points of our society to Wall
Street financial instruments

00:00:57.690 --> 00:01:00.780
was higher than it needed to
be, and that some of those IQ

00:01:00.780 --> 00:01:03.462
points should go
into other fields.

00:01:03.462 --> 00:01:04.170
BILL GATES: Yeah.

00:01:04.170 --> 00:01:08.240
Well if you say, OK, what
portion of society's IQ

00:01:08.240 --> 00:01:10.510
goes into studying
great teachers

00:01:10.510 --> 00:01:12.940
and documenting what
those great teachers do

00:01:12.940 --> 00:01:16.290
and how you would spread those
practices, what is effectively

00:01:16.290 --> 00:01:20.460
the R&D component in
education, it's about as close

00:01:20.460 --> 00:01:22.750
to nil as you can imagine.

00:01:22.750 --> 00:01:24.950
And yet, what's the
most important thing

00:01:24.950 --> 00:01:26.960
in terms of having
a society that

00:01:26.960 --> 00:01:30.260
has both got equality
of opportunity

00:01:30.260 --> 00:01:33.000
and is competitive
with other countries?

00:01:33.000 --> 00:01:34.400
It's education.

00:01:34.400 --> 00:01:38.260
And so it's this
mind-blowing misallocation.

00:01:38.260 --> 00:01:41.730
My favorite vignette is
that guy, Salman Khan.

00:01:41.730 --> 00:01:44.640
He was a hedge fund guy
making lots of money,

00:01:44.640 --> 00:01:46.760
and he quit to do these
little web videos.

00:01:46.760 --> 00:01:50.980
And so we have moved, I'd
say, about 160 IQ points

00:01:50.980 --> 00:01:56.980
from the hedge fund category
into the teaching many,

00:01:56.980 --> 00:01:59.210
many people in a
leveraged way category.

00:01:59.210 --> 00:02:03.170
So that was a good day,
the day that his wife

00:02:03.170 --> 00:02:05.450
let him quit his job.


Area of diagonal generated triangles of rectangle are equal

WEBVTT
Kind: captions
Language: en

00:00:00.780 --> 00:00:11.240
Let's say we've got a rectangle
and we have two diagonals

00:00:11.240 --> 00:00:15.140
across the rectangle-- that's
one of them, and then we have

00:00:15.140 --> 00:00:19.910
the other diagonal --and this
rectangle has a height of h--

00:00:19.910 --> 00:00:25.080
so that distance right there is
h --and it has a width of w.

00:00:29.200 --> 00:00:32.050
What we're going to show in
this video is that all of

00:00:32.050 --> 00:00:35.890
these four triangles
have the same area.

00:00:35.890 --> 00:00:38.595
Now right when you look at it,
it might be reasonably obvious

00:00:38.595 --> 00:00:47.060
that this bottom triangle will
have the same area as the top

00:00:47.060 --> 00:00:50.440
triangle, as this top kind
of upside down triangle.

00:00:50.440 --> 00:00:53.710
That these to have the same
area, that might be reasonably

00:00:53.710 --> 00:00:57.110
obvious. they have the same
dimension for their base, this

00:00:57.110 --> 00:00:59.860
width, and they have the same
height because this distance

00:00:59.860 --> 00:01:04.840
right here is exactly half of
the height of the rectangle.

00:01:04.840 --> 00:01:07.270
They are symmetric; they
are equal triangles.

00:01:07.270 --> 00:01:09.800
They have the same proportions.

00:01:09.800 --> 00:01:13.520
Now it's probably equally
obvious that this triangle on

00:01:13.520 --> 00:01:21.340
the left has the same area as
this triangle on the right.

00:01:21.340 --> 00:01:23.200
That's probably
equally obvious.

00:01:23.200 --> 00:01:26.960
What is not obvious is that
these orange triangles angles

00:01:26.960 --> 00:01:31.020
have the same area as these
green, blue triangles.

00:01:31.020 --> 00:01:32.320
And that's what we're
going to show right here.

00:01:32.320 --> 00:01:34.930
So all we have to do is really
calculate the areas of

00:01:34.930 --> 00:01:35.690
the different triangles.

00:01:35.690 --> 00:01:38.070
So let's do the orange
triangles first. and before

00:01:38.070 --> 00:01:40.310
doing that let's just
remind ourselves what the

00:01:40.310 --> 00:01:42.060
area of a triangle is.

00:01:42.060 --> 00:01:47.200
Area of a triangle is equal
to 1/2 times the base of

00:01:47.200 --> 00:01:50.030
the triangle times the
height of the triangle.

00:01:50.030 --> 00:01:52.260
That's just basic geometry.

00:01:52.260 --> 00:01:54.530
Not with that said, let's
figure out the area of

00:01:54.530 --> 00:01:55.660
the orange triangle.

00:01:58.250 --> 00:02:01.690
It's going to be 1/2
times the base.

00:02:01.690 --> 00:02:04.620
So the base of the orange
triangle is this distance

00:02:04.620 --> 00:02:07.400
right here: it is w.

00:02:07.400 --> 00:02:09.734
So 1/2 times w.

00:02:12.250 --> 00:02:14.800
I want to do that in a
different color; the

00:02:14.800 --> 00:02:17.770
color I wrote the w in.

00:02:17.770 --> 00:02:19.040
Now what's the height here?

00:02:21.820 --> 00:02:25.580
Well we already talked about
it: it's exactly half way up

00:02:25.580 --> 00:02:28.300
the height of the rectangle.

00:02:28.300 --> 00:02:32.815
So times 1/2 times the
height of the rectangle.

00:02:36.270 --> 00:02:37.700
So what's that going to be?

00:02:37.700 --> 00:02:43.295
You have 1/2 times 1/2 is 1/4
times width times height.

00:02:46.280 --> 00:02:49.820
So the area of that triangle
is 1/4 width height.

00:02:49.820 --> 00:02:50.810
So is that one.

00:02:50.810 --> 00:02:53.790
Same exact argument;
they have equal area.

00:02:53.790 --> 00:02:56.300
Now what's the area of
these green or these

00:02:56.300 --> 00:02:58.020
green/blue triangles?

00:02:58.020 --> 00:03:04.440
Well once again-- we'll do
this in a green color --area

00:03:04.440 --> 00:03:06.980
is equal to 1/2 base.

00:03:06.980 --> 00:03:08.680
So these guys are
turned on their side.

00:03:08.680 --> 00:03:13.550
The best base I can think of
is this distance right here.

00:03:13.550 --> 00:03:16.110
Or if you look at this triangle
it's this distance right here;

00:03:16.110 --> 00:03:19.900
it is the height of the
rectangle So now we're dealing

00:03:19.900 --> 00:03:25.080
with, the base in this case is
the height of the rectangle.

00:03:25.080 --> 00:03:26.890
Don't want you to
get too confused.

00:03:26.890 --> 00:03:31.210
The height is now
going to be what?

00:03:31.210 --> 00:03:32.950
So these triangles are turned
on the side, so what is

00:03:32.950 --> 00:03:36.170
this distance right here?

00:03:36.170 --> 00:03:39.640
Well it is exactly half
of the width, right?

00:03:39.640 --> 00:03:42.190
We're going exactly half of
this distance right here.

00:03:42.190 --> 00:03:44.740
This point right here is
exactly halfway between

00:03:44.740 --> 00:03:47.570
these two sides and halfway
between those two sides.

00:03:47.570 --> 00:03:50.880
So this distance right
here is 1/2 the width.

00:03:50.880 --> 00:03:55.400
Or the height of these sideways
triangles are 1/2 of the width.

00:03:59.410 --> 00:04:01.900
Little confusing: the base is
equal to the height of the

00:04:01.900 --> 00:04:05.960
rectangle, the height is equal
to 1/2 of the width. but if you

00:04:05.960 --> 00:04:09.870
do the math here, area is equal
to 1/2 times 1/2, which is

00:04:09.870 --> 00:04:12.450
1/4, height times width.

00:04:12.450 --> 00:04:16.560
Or you can just write that as
1/4 width times height, which

00:04:16.560 --> 00:04:18.040
is the exact same area.

00:04:18.040 --> 00:04:25.110
So the area here is 1/4 width
times height, which is the

00:04:25.110 --> 00:04:27.870
exact same area as each of
these orange triangles.

00:04:27.870 --> 00:04:31.570
And it makes sense because
each of them are exactly 1/4

00:04:31.570 --> 00:04:33.460
the area of the rectangle.

00:04:33.460 --> 00:04:35.540
Hopefully you enjoyed that.


Function inverse example 1

WEBVTT
Kind: captions
Language: en

00:00:00.830 --> 00:00:05.770
So we have f of x is equal to
negative x plus 4, and f of x

00:00:05.770 --> 00:00:08.460
is graphed right here on
our coordinate plane.

00:00:08.460 --> 00:00:11.690
Let's try to figure out
what the inverse of f is.

00:00:11.690 --> 00:00:14.770
And to figure out the inverse,
what I like to do is I set y, I

00:00:14.770 --> 00:00:18.590
set the variable y, equal to f
of x, or we could write that y

00:00:18.590 --> 00:00:22.130
is equal to negative x plus 4.

00:00:22.130 --> 00:00:25.040
Right now, we've solved
for y in terms of x.

00:00:25.040 --> 00:00:26.750
To solve for the inverse,
we do the opposite.

00:00:26.750 --> 00:00:29.600
We solve for x in terms of y.

00:00:29.600 --> 00:00:31.570
So let's subtract 4
from both sides.

00:00:31.570 --> 00:00:36.160
You get y minus 4 is
equal to negative x.

00:00:36.160 --> 00:00:39.000
And then to solve for x, we can
multiply both sides of this

00:00:39.000 --> 00:00:41.960
equation times negative 1.

00:00:41.960 --> 00:00:47.530
And so you get negative
y plus 4 is equal to x.

00:00:47.530 --> 00:00:50.390
Or just because we're always
used to writing the dependent

00:00:50.390 --> 00:00:52.770
variable on the left-hand side,
we could rewrite this as x is

00:00:52.770 --> 00:00:55.620
equal to negative y plus 4.

00:00:55.620 --> 00:00:58.300
Or another way to write it
is we could say that f

00:00:58.300 --> 00:01:06.840
inverse of y is equal
to negative y plus 4.

00:01:06.840 --> 00:01:09.470
So this is the inverse function
right here, and we've written

00:01:09.470 --> 00:01:13.360
it as a function of y, but we
can just rename the y as x

00:01:13.360 --> 00:01:14.950
so it's a function of x.

00:01:14.950 --> 00:01:16.080
So let's do that.

00:01:16.080 --> 00:01:23.120
So if we just rename this y as
x, we get f inverse of x is

00:01:23.120 --> 00:01:25.780
equal to the negative x plus 4.

00:01:25.780 --> 00:01:27.540
These two functions
are identical.

00:01:27.540 --> 00:01:30.730
Here, we just used y as the
independent variable, or

00:01:30.730 --> 00:01:31.760
as the input variable.

00:01:31.760 --> 00:01:34.660
Here we just use x, but they
are identical functions.

00:01:34.660 --> 00:01:37.840
Now, just out of interest,
let's graph the inverse

00:01:37.840 --> 00:01:40.220
function and see how it
might relate to this

00:01:40.220 --> 00:01:42.190
one right over here.

00:01:42.190 --> 00:01:44.250
So if you look at it,
it actually looks

00:01:44.250 --> 00:01:45.510
fairly identical.

00:01:45.510 --> 00:01:47.010
It's a negative x plus 4.

00:01:47.010 --> 00:01:48.395
It's the exact same function.

00:01:48.395 --> 00:01:51.630
So let's see, if we have-- the
y-intercept is 4, it's going

00:01:51.630 --> 00:01:52.950
to be the exact same thing.

00:01:52.950 --> 00:01:56.810
The function is
its own inverse.

00:01:56.810 --> 00:01:58.990
So if we were to graph it,
we would put it right

00:01:58.990 --> 00:01:59.930
on top of this.

00:02:02.560 --> 00:02:04.730
And so, there's a couple of
ways to think about it.

00:02:04.730 --> 00:02:07.530
In the first inverse function
video, I talked about how a

00:02:07.530 --> 00:02:10.390
function and their inverse--
they are the reflection

00:02:10.390 --> 00:02:12.030
over the line y equals x.

00:02:12.030 --> 00:02:14.220
So where's the line
y equals x here?

00:02:14.220 --> 00:02:16.445
Well, line y equals
x looks like this.

00:02:20.540 --> 00:02:25.710
And negative x plus 4 is
actually perpendicular to y is

00:02:25.710 --> 00:02:27.780
equal to x, so when you reflect
it, you're just kind of

00:02:27.780 --> 00:02:29.830
flipping it over, but it's
going to be the same line.

00:02:29.830 --> 00:02:32.670
It is its own reflection.

00:02:32.670 --> 00:02:34.470
Now, let's make sure that
that actually makes sense.

00:02:34.470 --> 00:02:38.700
When we're dealing with the
standard function right

00:02:38.700 --> 00:02:43.480
there, if you input a 2,
it gets mapped to a 2.

00:02:43.480 --> 00:02:48.750
If you input a 4, it
gets mapped to 0.

00:02:48.750 --> 00:02:50.370
What happens if you
go the other way?

00:02:50.370 --> 00:02:54.460
If you input a 2, well, 2
gets mapped to 2 either

00:02:54.460 --> 00:02:55.870
way, so that makes sense.

00:02:55.870 --> 00:02:59.180
For the regular function,
4 gets mapped to 0.

00:02:59.180 --> 00:03:02.320
For the inverse function,
0 gets mapped to 4.

00:03:02.320 --> 00:03:03.710
So it actually makes
complete sense.

00:03:03.710 --> 00:03:04.610
Let's think about
it another way.

00:03:04.610 --> 00:03:07.770
For the regular function-- let
me write it explicitly down.

00:03:07.770 --> 00:03:09.390
This might be obvious to
you, but just in case it's

00:03:09.390 --> 00:03:11.950
not, it might be helpful.

00:03:11.950 --> 00:03:14.440
Let's pick f of 5.

00:03:14.440 --> 00:03:18.020
f of 5 is equal to negative 1.

00:03:18.020 --> 00:03:23.900
Or we could say, the function f
maps us from 5 to negative 1.

00:03:23.900 --> 00:03:27.230
Now, what does f inverse do?

00:03:27.230 --> 00:03:31.190
What's f inverse of negative 1?

00:03:31.190 --> 00:03:33.325
f inverse of negative 1 is 5.

00:03:36.200 --> 00:03:41.000
Or we could say that f maps
us from negative 1 to 5.

00:03:41.000 --> 00:03:44.140
So once again, if you think
about kind of the sets, they're

00:03:44.140 --> 00:03:46.370
our domains and our ranges.

00:03:46.370 --> 00:03:49.070
So let's say that this is
the domain of f, this

00:03:49.070 --> 00:03:50.760
is the range of f.

00:03:50.760 --> 00:03:59.040
f will take us from
to negative 1.

00:03:59.040 --> 00:04:00.960
That's what the
function f does.

00:04:00.960 --> 00:04:04.920
And we see that f inverse takes
us back from negative 1 to 5.

00:04:04.920 --> 00:04:09.720
f inverse takes us back
from negative 1 to 5, just

00:04:09.720 --> 00:04:12.320
like it's supposed to do.

00:04:12.320 --> 00:04:15.200
Let's do one more of these.

00:04:15.200 --> 00:04:19.070
So here I have g of x is equal
to negative 2x minus 1.

00:04:19.070 --> 00:04:23.000
So just like the last problem,
I like to set y equal to this.

00:04:23.000 --> 00:04:25.340
So we say y is equal to g
of x, which is equal to

00:04:25.340 --> 00:04:27.740
negative 2x minus 1.

00:04:27.740 --> 00:04:29.950
Now we just solve for x.

00:04:29.950 --> 00:04:32.910
y plus 1 is equal
to negative 2x.

00:04:32.910 --> 00:04:34.920
Just added 1 to both sides.

00:04:34.920 --> 00:04:39.050
Now we can divide both sides of
this equation by negative 2,

00:04:39.050 --> 00:04:46.630
and so you get negative y over
2 minus 1/2 is equal to x, or

00:04:46.630 --> 00:04:52.420
we could write x is equal to
negative y over 2 minus 1/2, or

00:04:52.420 --> 00:04:56.260
we could write f inverse as a
function of y is equal to

00:04:56.260 --> 00:05:02.435
negative y over 2 minus 1/2, or
we can just rename y as x.

00:05:02.435 --> 00:05:08.270
And we could say that f inverse
of-- oh, let me careful here.

00:05:08.270 --> 00:05:09.250
That shouldn't be an f.

00:05:09.250 --> 00:05:11.300
The original function was
g , so let me be clear.

00:05:11.300 --> 00:05:21.850
That is g inverse of y is equal
to negative y over 2 minus 1/2

00:05:21.850 --> 00:05:24.340
because we started with a
g of x, not an f of x.

00:05:24.340 --> 00:05:26.120
Make sure we get our
notation right.

00:05:26.120 --> 00:05:31.010
Or we could just rename the y
and say g inverse of x is equal

00:05:31.010 --> 00:05:34.320
to negative x over 2 minus 1/2.

00:05:34.320 --> 00:05:35.140
Now, let's graph it.

00:05:35.140 --> 00:05:37.970
Its y-intercept
is negative 1/2.

00:05:37.970 --> 00:05:39.970
It's right over there.

00:05:39.970 --> 00:05:43.460
And it has a slope
of negative 1/2.

00:05:48.940 --> 00:05:52.760
Let's see, if we start at
negative 1/2, if we move over

00:05:52.760 --> 00:05:56.500
to 1 in the positive direction,
it will go down half.

00:05:56.500 --> 00:05:59.770
If we move over 1 again, it
will go down half again.

00:05:59.770 --> 00:06:01.650
If we move back-- so
it'll go like that.

00:06:01.650 --> 00:06:05.440
So the line, I'll try my
best to draw it, will

00:06:05.440 --> 00:06:07.830
look something like that.

00:06:07.830 --> 00:06:10.580
It'll just keep going, so it'll
look something like that, and

00:06:10.580 --> 00:06:13.170
it'll keep going in
both directions.

00:06:13.170 --> 00:06:15.400
And now let's see if this
really is a reflection over y

00:06:15.400 --> 00:06:21.910
equals x. y equals x looks
like that, and you can see

00:06:21.910 --> 00:06:22.750
they are a reflection.

00:06:22.750 --> 00:06:25.440
If you reflect this guy, if you
reflect this blue line, it

00:06:25.440 --> 00:06:27.220
becomes this orange line.

00:06:27.220 --> 00:06:30.885
But the general idea, you
literally just-- a function

00:06:30.885 --> 00:06:34.460
is originally expressed, is
solved for y in terms of x.

00:06:34.460 --> 00:06:35.530
You just do some algebra.

00:06:35.530 --> 00:06:38.750
Solve for x in terms of y, and
that's essentially your inverse

00:06:38.750 --> 00:06:41.120
function as a function of y,
but then you can rename

00:06:41.120 --> 00:06:43.700
it as a function of x.


Function inverses example 3

WEBVTT
Kind: captions
Language: en

00:00:00.950 --> 00:00:03.890
We have the function f of
x is equal to x minus

00:00:03.890 --> 00:00:06.230
1 squared minus 2.

00:00:06.230 --> 00:00:09.450
And they've constrained the
domain to x being less

00:00:09.450 --> 00:00:10.600
than or equal to 1.

00:00:10.600 --> 00:00:14.760
So we have the left half
of a parabola right here.

00:00:14.760 --> 00:00:17.410
They've constrained so that
it's not a full U parabola.

00:00:17.410 --> 00:00:20.470
And I'll let you think about
why that would make finding

00:00:20.470 --> 00:00:21.320
the inverse difficult.

00:00:21.320 --> 00:00:23.150
But let's try to find
the inverse here.

00:00:23.150 --> 00:00:25.500
And a good place to start
-- let's just set y

00:00:25.500 --> 00:00:26.830
being equal to f of x.

00:00:26.830 --> 00:00:29.200
You could say y is equal to
f of x or we could just

00:00:29.200 --> 00:00:35.430
write that y is equal to x
minus 1 squared minus 2.

00:00:35.430 --> 00:00:37.260
We know it's for

00:00:37.260 --> 00:00:39.960
x is less than or equal to 1.

00:00:39.960 --> 00:00:45.030
But right now we have y
solved for in terms of x.

00:00:45.030 --> 00:00:47.890
Or we've solved for y, to find
the inverse we're going to want

00:00:47.890 --> 00:00:51.980
to solve for x in terms of y.

00:00:51.980 --> 00:00:54.140
And we're going to
constrain y similarly.

00:00:54.140 --> 00:00:56.280
We could look at the graph and
we could say, well, in this

00:00:56.280 --> 00:01:00.000
graph right here, this is
defined for y being greater

00:01:00.000 --> 00:01:02.040
than or equal to negative 2.

00:01:02.040 --> 00:01:05.610
So we can maybe put in
parentheses y being greater

00:01:05.610 --> 00:01:07.880
than or equal to negative 2.

00:01:07.880 --> 00:01:10.480
Because this is going to be --
right now this is our range.

00:01:10.480 --> 00:01:12.580
But when we swap the x's
and y's, this is going

00:01:12.580 --> 00:01:13.350
to be our domain.

00:01:13.350 --> 00:01:16.070
So let's just keep that in
parentheses right there.

00:01:16.070 --> 00:01:18.150
So let's solve for x.

00:01:18.150 --> 00:01:20.030
So that's all you have to
do, to find the inverse.

00:01:20.030 --> 00:01:22.230
Solve for x and make sure
you keep track of the

00:01:22.230 --> 00:01:23.890
domains and the ranges.

00:01:23.890 --> 00:01:24.500
So, let's see.

00:01:24.500 --> 00:01:26.520
We could add 2 to both
sides of this equation.

00:01:26.520 --> 00:01:32.440
We get y plus 2 is equal
to x minus 1 squared.

00:01:32.440 --> 00:01:34.640
Minus 2, plus 2, so
those cancel out.

00:01:34.640 --> 00:01:37.020
And then, I'm just going to
switch to the y constraint.

00:01:37.020 --> 00:01:41.286
Because now it's not clear
what we're -- whether x is

00:01:41.286 --> 00:01:42.040
the domain or the range.

00:01:42.040 --> 00:01:43.790
But we know by the end
of this problem, y is

00:01:43.790 --> 00:01:44.670
going to be the domain.

00:01:44.670 --> 00:01:46.130
So let's just swap this here.

00:01:46.130 --> 00:01:50.090
So, 4 for y is greater than
or equal to negative 2.

00:01:50.090 --> 00:01:53.600
And we could also say, in
parentheses, x is less than 1.

00:01:53.600 --> 00:01:55.680
This is -- we haven't solved it
explicitly for either one, so

00:01:55.680 --> 00:01:58.050
we'll keep both of them
around right now.

00:01:58.050 --> 00:02:02.780
Now, to solve for x, you might
be tempted to just take the

00:02:02.780 --> 00:02:04.000
square root of both sides here.

00:02:04.000 --> 00:02:05.410
And you wouldn't be
completely wrong.

00:02:05.410 --> 00:02:07.850
But we have to be very,
very, very careful here.

00:02:07.850 --> 00:02:11.000
And this might not be something
that you've ever seen before.

00:02:11.000 --> 00:02:13.410
So this is an
interesting point here.

00:02:13.410 --> 00:02:19.340
We want the right side
to just be x minus 1.

00:02:19.340 --> 00:02:21.720
That's our goal here, in taking
the square root of both sides.

00:02:21.720 --> 00:02:25.270
We want to just have an
x minus 1 over there.

00:02:25.270 --> 00:02:31.090
Now, is x minus 1 a positive
or a negative number?

00:02:31.090 --> 00:02:35.350
Well, we've constrained our
x's to being less than 1.

00:02:35.350 --> 00:02:37.320
So we're dealing only in
a situation where x is

00:02:37.320 --> 00:02:38.610
less than or equal to 1.

00:02:38.610 --> 00:02:44.760
So if x is less than or equal
to 1, this is negative.

00:02:44.760 --> 00:02:48.000
This is negative.

00:02:48.000 --> 00:02:50.200
So we want to take the
negative square root.

00:02:50.200 --> 00:02:51.390
Let me just be very clear here.

00:02:51.390 --> 00:02:56.660
If I take negative 3 -- I take
negative 3 and I were to square

00:02:56.660 --> 00:02:58.780
it, that is equal to 9.

00:02:58.780 --> 00:03:02.830
Now, if we take the square root
of 9 -- if we take the square

00:03:02.830 --> 00:03:04.620
root -- let's say we take
the square root of both

00:03:04.620 --> 00:03:05.430
sides of this equation.

00:03:05.430 --> 00:03:08.280
And our goal is to get
back to negative 3.

00:03:08.280 --> 00:03:11.200
If we take the
positive square root.

00:03:11.200 --> 00:03:16.260
If we just take the principle
root of both sides of that, we

00:03:16.260 --> 00:03:18.410
would get 3 is equal to 3.

00:03:18.410 --> 00:03:19.740
But that's not our goal.

00:03:19.740 --> 00:03:21.660
We want to get back
to negative 3.

00:03:21.660 --> 00:03:28.050
So we want to take the negative
square root of our square.

00:03:28.050 --> 00:03:30.900
So because this expression is
negative and we want to get

00:03:30.900 --> 00:03:33.360
back to this expression, we
want to get back to this x

00:03:33.360 --> 00:03:36.420
minus 1, we need to take the
negative square root

00:03:36.420 --> 00:03:37.050
of both sides.

00:03:37.050 --> 00:03:39.500
You can always -- every
perfect square has a

00:03:39.500 --> 00:03:42.250
positive or a negative root.

00:03:42.250 --> 00:03:44.090
The principle root
is a positive root.

00:03:44.090 --> 00:03:45.870
But here we want to take the
negative root because this

00:03:45.870 --> 00:03:48.340
expression right here is
going to be negative.

00:03:48.340 --> 00:03:50.460
And that's what we
want to solve for.

00:03:50.460 --> 00:03:53.390
So let's take the negative
root of both sides.

00:03:53.390 --> 00:03:58.870
So you get the negative square
root of y plus 2 is equal to

00:03:58.870 --> 00:04:01.340
-- and I'll just write this
extra step here, just so you

00:04:01.340 --> 00:04:02.240
realize what we're doing.

00:04:02.240 --> 00:04:06.320
Is equal to the negative square
root, the negative square

00:04:06.320 --> 00:04:11.660
root of x minus 1 squared.

00:04:11.660 --> 00:04:15.310
For y is greater than or
equal to negative 2.

00:04:15.310 --> 00:04:17.410
And x is less than
or equal to 1.

00:04:17.410 --> 00:04:18.650
That's why the whole reason
we're going to take the

00:04:18.650 --> 00:04:20.450
negative square there.

00:04:20.450 --> 00:04:22.880
And then this expression
right here -- so let me

00:04:22.880 --> 00:04:24.220
just write the left again.

00:04:24.220 --> 00:04:29.470
Negative square root of y plus
2 is equal to the negative

00:04:29.470 --> 00:04:32.190
square root of x minus 1
squared is just going

00:04:32.190 --> 00:04:34.350
to be x minus 1.

00:04:34.350 --> 00:04:36.525
It's just going to
be x minus one.

00:04:36.525 --> 00:04:39.060
x minus 1 squared is
some positive quantity.

00:04:39.060 --> 00:04:41.730
The negative root is the
negative number that you

00:04:41.730 --> 00:04:43.570
have to square to get it.

00:04:43.570 --> 00:04:44.960
To get x minus 1 squared.

00:04:44.960 --> 00:04:46.470
So that just becomes x minus 1.

00:04:46.470 --> 00:04:48.370
Hopefully that doesn't
confuse you too much.

00:04:48.370 --> 00:04:50.170
We just want to get rid
of this squared sign.

00:04:50.170 --> 00:04:52.080
We want to make sure we
get the negative version.

00:04:52.080 --> 00:04:53.515
We don't want the positive
version, which would

00:04:53.515 --> 00:04:54.810
have been 1 minus x.

00:04:54.810 --> 00:04:56.020
Don't want to confuse you.

00:04:56.020 --> 00:04:58.350
So here, we now just
have to solve for x.

00:04:58.350 --> 00:04:58.910
Add 1.

00:04:58.910 --> 00:05:00.920
And let me write the 4.

00:05:00.920 --> 00:05:04.230
y is greater than or
equal to negative 2.

00:05:04.230 --> 00:05:06.020
Add 1 to both sides.

00:05:06.020 --> 00:05:13.470
You get negative square root of
y plus 2 plus 1 is equal to x

00:05:13.470 --> 00:05:17.320
for y is greater than or
equal to negative 2.

00:05:17.320 --> 00:05:21.120
Or, if we want to rewrite it,
we could say that x is equal to

00:05:21.120 --> 00:05:28.231
the negative square root of y
plus 2 plus 1 for y is greater

00:05:28.231 --> 00:05:29.120
than or equal to negative 2.

00:05:29.120 --> 00:05:31.840
Or if we want to write it in
terms, as an inverse function

00:05:31.840 --> 00:05:36.140
of y, we could say -- so we
could say that f inverse of y

00:05:36.140 --> 00:05:42.270
is equal to this, or f inverse
of y is equal to the negative

00:05:42.270 --> 00:05:48.500
square root of y plus 2 plus
1, for y is greater than

00:05:48.500 --> 00:05:49.830
or equal to negative 2.

00:05:49.830 --> 00:05:52.230
And now, if we wanted
this in terms of x.

00:05:52.230 --> 00:05:55.300
If we just want to rename
y as x we just replace

00:05:55.300 --> 00:05:56.980
the y's with x's.

00:05:56.980 --> 00:06:02.250
So we could write f
inverse of x -- I'm just

00:06:02.250 --> 00:06:03.390
renaming the y here.

00:06:03.390 --> 00:06:09.570
Is equal to the negative square
root of x plus 2 plus 1 for,

00:06:09.570 --> 00:06:12.580
I'm just renaming the y, for
x is greater than or

00:06:12.580 --> 00:06:13.930
equal to negative 2.

00:06:13.930 --> 00:06:16.150
And if we were to graph
this, let's see.

00:06:16.150 --> 00:06:18.670
If we started at x is equal
to negative 2, this is 0.

00:06:18.670 --> 00:06:22.560
So the point negative 2, 1 is
going to be on our graph.

00:06:22.560 --> 00:06:25.910
So negative 2, 1 is going
to be on our graph.

00:06:25.910 --> 00:06:30.590
Let's see, if we go to negative
1, negative 1 this will

00:06:30.590 --> 00:06:31.430
become a negative 1.

00:06:31.430 --> 00:06:34.650
Negative 1 is 0 on our graph.

00:06:34.650 --> 00:06:37.820
Negative 1, 0 is on our graph.

00:06:37.820 --> 00:06:38.500
And then, let's see.

00:06:38.500 --> 00:06:42.990
If we were to do, if we were
to put x is equal to 2 here.

00:06:42.990 --> 00:06:46.020
So x is equal to 2 is 4.

00:06:46.020 --> 00:06:48.550
4 square root,
principle root is 2.

00:06:48.550 --> 00:06:52.170
It becomes a negative 2 So
it becomes 2, negative 1.

00:06:52.170 --> 00:06:54.130
So that's on our
graph right there.

00:06:54.130 --> 00:06:55.850
So the graph is going
to look something like

00:06:55.850 --> 00:06:57.110
this, of f inverse.

00:06:57.110 --> 00:07:02.340
It's going to look something
like that right there.

00:07:02.340 --> 00:07:05.100
As you can see, it is a
reflection of our original

00:07:05.100 --> 00:07:08.030
f of x along the line
y is equal to x.

00:07:08.030 --> 00:07:13.070
Along the line y is equal to x.

00:07:13.070 --> 00:07:16.560
Because we've essentially just
swapped the x and the y.

00:07:16.560 --> 00:07:18.390
This is about as hard of
an inverse problem that

00:07:18.390 --> 00:07:19.540
I expect you to see.

00:07:19.540 --> 00:07:22.190
Especially in a precalculus
class because it really is

00:07:22.190 --> 00:07:24.790
tricky to realize that you
have to take the negative

00:07:24.790 --> 00:07:25.580
square root here.

00:07:25.580 --> 00:07:28.800
Because the way our domain was
constrained, this value right

00:07:28.800 --> 00:07:30.830
here is going to be negative.

00:07:30.830 --> 00:07:34.330
So to solve for it, you want to
have the negative square root.


Function inverses example 2

WEBVTT
Kind: captions
Language: en

00:00:00.620 --> 00:00:04.060
We've got the function f of x
is equal to x plus 2 squared

00:00:04.060 --> 00:00:07.350
plus 1, and we've constrained
our domain that x has to be

00:00:07.350 --> 00:00:09.380
greater than or equal
to negative 2.

00:00:09.380 --> 00:00:11.390
That's where we've
defined our function.

00:00:11.390 --> 00:00:13.180
And we want to
find its inverse.

00:00:13.180 --> 00:00:15.560
And I'll leave you to think
about why we had to constrain

00:00:15.560 --> 00:00:18.750
it to x being a greater than
or equal to negative 2.

00:00:18.750 --> 00:00:21.960
Wouldn't it have been possible
to find the inverse if we had

00:00:21.960 --> 00:00:24.390
just left it as the
full parabola?

00:00:24.390 --> 00:00:26.900
I'll leave you -- or maybe I'll
make a future video about that.

00:00:26.900 --> 00:00:29.450
But let's just figure
out the inverse here.

00:00:29.450 --> 00:00:31.520
So, like we've said in
the first video, in the

00:00:31.520 --> 00:00:34.930
introduction to inverses, we're
trying to find a mapping.

00:00:34.930 --> 00:00:39.120
Or, if we were to say that y --
if we were to say that y is

00:00:39.120 --> 00:00:42.350
equal to x plus 2
squared plus 1.

00:00:42.350 --> 00:00:45.480
This is the function you give
me an x and it maps to y.

00:00:45.480 --> 00:00:46.870
We want to go the other way.

00:00:46.870 --> 00:00:50.980
We want to take, I'll give you
a y and then map it to an x.

00:00:50.980 --> 00:00:53.430
So what we do is, we
essentially just solve

00:00:53.430 --> 00:00:55.920
for x in terms of y.

00:00:55.920 --> 00:00:57.610
So let's do that one
step at a time.

00:00:57.610 --> 00:01:00.600
So, the first thing to do, we
could subtract 1 from both

00:01:00.600 --> 00:01:02.610
sides of this equation.

00:01:02.610 --> 00:01:09.200
y minus 1 is equal to
x plus 2 squared.

00:01:09.200 --> 00:01:10.440
And now to solve here,
you might want to

00:01:10.440 --> 00:01:11.270
take the square root.

00:01:11.270 --> 00:01:13.180
And that actually will be
the correct thing to do.

00:01:13.180 --> 00:01:15.115
But it's very important to
think about whether you want

00:01:15.115 --> 00:01:17.680
to take the positive or
the negative square

00:01:17.680 --> 00:01:19.080
root at this step.

00:01:19.080 --> 00:01:22.200
So we've constrained our domain
to x is greater than or

00:01:22.200 --> 00:01:23.770
equal to negative 2.

00:01:23.770 --> 00:01:27.500
So this value right here, x
plus 2, if x is always greater

00:01:27.500 --> 00:01:29.870
than or equal to negative
2, x plus 2 will always be

00:01:29.870 --> 00:01:32.480
greater than or equal to 0.

00:01:32.480 --> 00:01:37.170
So this expression right here,
this right here is positive.

00:01:37.170 --> 00:01:38.160
This is positive.

00:01:38.160 --> 00:01:40.130
So we have a positive squared.

00:01:40.130 --> 00:01:44.840
So if we really want to get to
the x plus 2 in the appropriate

00:01:44.840 --> 00:01:48.370
domain, we want to take
the positive square root.

00:01:48.370 --> 00:01:50.530
And in the next video or the
video after that, we'll solve

00:01:50.530 --> 00:01:52.530
an example where you want to
take the negative square root.

00:01:52.530 --> 00:01:53.650
So we're going to take
theundefined positive square

00:01:53.650 --> 00:01:55.880
root, or just the principal
root, which is just the square

00:01:55.880 --> 00:01:57.910
root sign, of both sides.

00:01:57.910 --> 00:02:06.740
So you get the square root of y
minus 1 is equal to x plus 2.

00:02:06.740 --> 00:02:08.800
And one thing I should have
remembered to do is, from

00:02:08.800 --> 00:02:11.630
the beginning we had
a constraint on x.

00:02:11.630 --> 00:02:15.840
We had for x is greater than
or equal to negative 2.

00:02:15.840 --> 00:02:18.520
But what constraint
could we have on y?

00:02:18.520 --> 00:02:21.000
If you look at the graph right
here, x is greater than

00:02:21.000 --> 00:02:22.430
equal to negative 2.

00:02:22.430 --> 00:02:23.610
But what's why?

00:02:23.610 --> 00:02:27.880
What is the range of y-values
that we can get here?

00:02:27.880 --> 00:02:30.620
Well, if you just look at
the graph, y will always be

00:02:30.620 --> 00:02:32.060
greater than or equal to 1.

00:02:32.060 --> 00:02:34.670
And that just comes from the
fact that this term right

00:02:34.670 --> 00:02:40.400
here is always going to be
greater than or equal to 0.

00:02:40.400 --> 00:02:43.910
So the minimum value that the
function could take on is 1.

00:02:43.910 --> 00:02:45.620
So we could say for x is
greater than or equal to

00:02:45.620 --> 00:02:50.960
negative 2, and we could add
that y is always going to be

00:02:50.960 --> 00:02:53.790
greater than or equal to 1.

00:02:53.790 --> 00:02:55.710
y is always greater
than or equal to 1.

00:02:55.710 --> 00:02:58.130
The function is always
greater than or equal

00:02:58.130 --> 00:02:59.160
to that right there.

00:02:59.160 --> 00:03:00.530
To 1.

00:03:00.530 --> 00:03:02.730
And the reason why I want to
write it at the stage is

00:03:02.730 --> 00:03:04.830
because, you know, later
on, we're going to swap

00:03:04.830 --> 00:03:06.000
the the x's and y's.

00:03:06.000 --> 00:03:07.800
So let's just leave that there.

00:03:07.800 --> 00:03:10.130
So here we haven't explicitly
solved for x and y.

00:03:10.130 --> 00:03:14.700
But we can write for y is
greater than or equal to 1,

00:03:14.700 --> 00:03:18.300
this is going to be the domain
for our inverse, so to speak.

00:03:18.300 --> 00:03:22.040
And so here we can keep
it for y is greater

00:03:22.040 --> 00:03:22.980
than or equal to 1.

00:03:22.980 --> 00:03:24.790
This y constraint's
going to matter more.

00:03:24.790 --> 00:03:28.260
Because over here,
the domain is x.

00:03:28.260 --> 00:03:31.710
But for the inverse, the domain
is going to be the y-value.

00:03:31.710 --> 00:03:32.480
And then, let's see.

00:03:32.480 --> 00:03:35.520
We have the square root of y
minus 1 is equal to x plus 2.

00:03:35.520 --> 00:03:37.990
Now we can subtract
2 from both sides.

00:03:37.990 --> 00:03:44.570
We get the square root of y
minus 1 minus 2, is equal

00:03:44.570 --> 00:03:51.430
to x for y is greater
than or equal to 1.

00:03:51.430 --> 00:03:53.810
And so we've solved
for x in terms of y.

00:03:53.810 --> 00:03:56.960
Or, we could say, let me just
write it the other way.

00:03:56.960 --> 00:03:59.420
We could say, x is equal to,
I'm just swapping this. x is

00:03:59.420 --> 00:04:06.200
equal to the square root of y
minus one minus 2, for y is

00:04:06.200 --> 00:04:07.110
greater than or equal to one.

00:04:07.110 --> 00:04:08.550
So you see, now, the way
we've written it out.

00:04:08.550 --> 00:04:11.640
y is the input into the
function, which is going to be

00:04:11.640 --> 00:04:12.970
the inverse of that function.

00:04:12.970 --> 00:04:14.880
x the output. x is
now the range.

00:04:14.880 --> 00:04:19.440
So we could even rewrite
this as f inverse of y.

00:04:19.440 --> 00:04:25.170
That's what x is, is equal to
the square root of y minus 1

00:04:25.170 --> 00:04:30.110
minus 2, for y is greater
than or equal to 1.

00:04:30.110 --> 00:04:31.760
And this is the
inverse function.

00:04:31.760 --> 00:04:33.160
We could say this
is our answer.

00:04:33.160 --> 00:04:36.600
But many times, people want
the answer in terms of x.

00:04:36.600 --> 00:04:38.400
And we know we could
put anything in here.

00:04:38.400 --> 00:04:40.750
If we put an a here, we
take f inverse of a.

00:04:40.750 --> 00:04:43.750
It'll become the square root
of a minus 1 minus 2, 4.

00:04:43.750 --> 00:04:46.180
Well, assuming a is greater
than or equal to 1.

00:04:46.180 --> 00:04:47.530
But we could put an x in here.

00:04:47.530 --> 00:04:50.840
So we can just rename
the the y for x.

00:04:50.840 --> 00:04:52.950
So we could just do
a renaming here.

00:04:52.950 --> 00:04:55.630
So we can just rename y for x.

00:04:55.630 --> 00:04:59.340
And then we would get -- let
me scroll down a little bit.

00:04:59.340 --> 00:05:03.150
We would f inverse of x.

00:05:03.150 --> 00:05:03.870
I'll highlight it here.

00:05:03.870 --> 00:05:06.530
Just to show you, we're
renaming y with x.

00:05:06.530 --> 00:05:10.180
You could rename it with
anything really, is equal to

00:05:10.180 --> 00:05:13.680
the square root of x minus 1.

00:05:13.680 --> 00:05:16.980
Of x minus 1.

00:05:16.980 --> 00:05:23.380
Minus 2 for, we have to
rename this to, for x being

00:05:23.380 --> 00:05:25.840
greater than or equal to 1.

00:05:25.840 --> 00:05:30.300
And so we now have our inverse
function as a function of x.

00:05:30.300 --> 00:05:33.200
And if we were to graph it,
let's try our best to graph it.

00:05:33.200 --> 00:05:36.290
Maybe the easiest thing to do
is to draw some points here.

00:05:36.290 --> 00:05:38.700
So the smallest value
x can take on is 1.

00:05:38.700 --> 00:05:41.370
If you put a 1 here,
you get a 0 here.

00:05:41.370 --> 00:05:44.880
So the point 1, negative 2,
is on our inverse graph.

00:05:44.880 --> 00:05:49.000
So 1, negative 2
is right there.

00:05:49.000 --> 00:05:53.040
And then if we go to 2,
let's see, 2 minus 1 is 1.

00:05:53.040 --> 00:05:55.430
The principle root
of that is 1.

00:05:55.430 --> 00:05:56.100
Minus 2.

00:05:56.100 --> 00:05:58.710
So it's negative 1, so
the point, 2, negative

00:05:58.710 --> 00:06:00.960
1 is right there.

00:06:00.960 --> 00:06:01.890
And let's think about it.

00:06:01.890 --> 00:06:02.180
Let's see.

00:06:02.180 --> 00:06:04.410
If we did 5, I'm trying
take perfect squares.

00:06:04.410 --> 00:06:07.730
5 minus 1 is 4, minus 2.

00:06:07.730 --> 00:06:10.330
So the point 5, 2 is,
let me make sure.

00:06:10.330 --> 00:06:12.350
5 minus 1 is 4.

00:06:12.350 --> 00:06:13.220
Square root is 2.

00:06:13.220 --> 00:06:14.090
Minus 2 is 0.

00:06:14.090 --> 00:06:16.850
So the point 5, 0 is here.

00:06:16.850 --> 00:06:20.330
And so the inverse graph, it's
only defined for x greater

00:06:20.330 --> 00:06:22.200
than or equal to negative 1.

00:06:22.200 --> 00:06:25.040
So the inverse graph is going
to look something like this.

00:06:25.040 --> 00:06:29.260
It's going to look something
like, I started off

00:06:29.260 --> 00:06:30.200
well, and it got messy.

00:06:30.200 --> 00:06:32.760
So it's going to look
something like that.

00:06:32.760 --> 00:06:33.820
Just like that.

00:06:33.820 --> 00:06:36.520
And just like we saw, in the
first, the introduction to

00:06:36.520 --> 00:06:40.190
function inverses, these are
mirror images around

00:06:40.190 --> 00:06:41.510
the line y equals x.

00:06:41.510 --> 00:06:45.070
Let me graph y equals
x. y equals x.

00:06:45.070 --> 00:06:49.370
y equals x is that
line right there.

00:06:49.370 --> 00:06:52.060
Notice, they're mirror
images around that line.

00:06:52.060 --> 00:06:57.520
Over here, we map
the value 0 to 5.

00:06:57.520 --> 00:07:00.250
If x is 0, it gets mapped to 5.

00:07:00.250 --> 00:07:01.270
Here we go the other way.

00:07:01.270 --> 00:07:06.120
We're mapping 5 to -- we're
mapping 5 to the value 0.

00:07:06.120 --> 00:07:07.660
So that's why they're
mirror images.

00:07:07.660 --> 00:07:11.660
We've essentially
swapped the x and y.


Introduction to function inverses

WEBVTT
Kind: captions
Language: en

00:00:00.460 --> 00:00:03.040
Let's think about what
functions really do, and then

00:00:03.040 --> 00:00:06.590
we'll think about the idea of
an inverse of a function.

00:00:06.590 --> 00:00:08.450
So let's start with a pretty
straightforward function.

00:00:08.450 --> 00:00:13.510
Let's say f of x is
equal to 2x plus 4.

00:00:13.510 --> 00:00:20.340
And so if I take f of 2, f of 2
is going to be equal to 2 times

00:00:20.340 --> 00:00:24.560
2 plus 4, which is 4
plus 4, which is 8.

00:00:24.560 --> 00:00:32.470
I could take f of 3, which
is 2 times 3 plus 4,

00:00:32.470 --> 00:00:35.920
which is equal to 10.

00:00:35.920 --> 00:00:36.960
6 plus 4.

00:00:36.960 --> 00:00:38.280
So let's think about it
in a little bit more

00:00:38.280 --> 00:00:39.800
of an abstract sense.

00:00:39.800 --> 00:00:42.990
So there's a set of things that
I can input into this function.

00:00:42.990 --> 00:00:44.700
You might already be
familiar with that notion.

00:00:44.700 --> 00:00:46.050
It's the domain.

00:00:46.050 --> 00:00:48.760
The set of all of the things
that I can input into that

00:00:48.760 --> 00:00:52.760
function, that is the domain.

00:00:52.760 --> 00:00:57.290
And in that domain, 2 is
sitting there, you have 3 over

00:00:57.290 --> 00:00:59.700
there, pretty much you could
input any real number

00:00:59.700 --> 00:01:00.850
into this function.

00:01:00.850 --> 00:01:02.810
So this is going to be all
real, but we're making it a

00:01:02.810 --> 00:01:05.150
nice contained set here just
to help you visualize it.

00:01:05.150 --> 00:01:07.830
Now, when you apply the
function, let's think about

00:01:07.830 --> 00:01:10.000
it means to take f of 2.

00:01:10.000 --> 00:01:13.900
We're inputting a number, 2,
and then the function is

00:01:13.900 --> 00:01:15.790
outputting the number 8.

00:01:15.790 --> 00:01:19.270
It is mapping us from 2 to 8.

00:01:19.270 --> 00:01:22.490
So let's make another set here
of all of the possible values

00:01:22.490 --> 00:01:23.930
that my function can take on.

00:01:27.780 --> 00:01:30.300
And we can call that the range.

00:01:30.300 --> 00:01:33.320
There are more formal ways to
talk about this, and there's a

00:01:33.320 --> 00:01:36.640
much more rigorous discussion
of this later on, especially in

00:01:36.640 --> 00:01:39.060
the linear algebra playlist,
but this is all the different

00:01:39.060 --> 00:01:40.200
values I can take on.

00:01:40.200 --> 00:01:44.550
So if I take the number 2 from
our domain, I input it into the

00:01:44.550 --> 00:01:48.270
function, we're getting
mapped to the number 8.

00:01:48.270 --> 00:01:50.260
So let's let me draw that out.

00:01:50.260 --> 00:01:56.210
So we're going from 2 to
the number 8 right there.

00:01:56.210 --> 00:01:59.060
And it's being done
by the function.

00:01:59.060 --> 00:02:01.210
The function is
doing that mapping.

00:02:01.210 --> 00:02:05.190
That function is mapping
us from 2 to 8.

00:02:05.190 --> 00:02:09.180
This right here, that
is equal to f of 2.

00:02:09.180 --> 00:02:10.080
Same idea.

00:02:10.080 --> 00:02:18.810
You start with 3, 3 is being
mapped by the function to 10.

00:02:18.810 --> 00:02:20.480
It's creating an association.

00:02:20.480 --> 00:02:25.320
The function is mapping
us from 3 to 10.

00:02:25.320 --> 00:02:28.400
Now, this raises an
interesting question.

00:02:28.400 --> 00:02:32.840
Is there a way to get back from
8 to the 2, or is there a

00:02:32.840 --> 00:02:35.220
way to go back from
the 10 to the 3?

00:02:35.220 --> 00:02:37.340
Or is there some
other function?

00:02:37.340 --> 00:02:39.870
Is there some other function,
we can call that the inverse

00:02:39.870 --> 00:02:41.660
of f, that'll take us back?

00:02:41.660 --> 00:02:44.260
Is there some other
function that'll take

00:02:44.260 --> 00:02:46.200
us from 10 back to 3?

00:02:46.200 --> 00:02:48.970
We'll call that the inverse
of f, and we'll use that as

00:02:48.970 --> 00:02:52.790
notation, and it'll take
us back from 10 to 3.

00:02:52.790 --> 00:02:54.360
Is there a way to do that?

00:02:54.360 --> 00:02:58.070
Will that same inverse of f,
will it take us back from--

00:02:58.070 --> 00:03:02.290
if we apply 8 to it-- will
that take us back to 2?

00:03:07.230 --> 00:03:10.760
Now, all this seems very
abstract and difficult.

00:03:10.760 --> 00:03:13.990
What you'll find is it's
actually very easy to solve for

00:03:13.990 --> 00:03:17.030
this inverse of f, and I think
once we solve for it, it'll

00:03:17.030 --> 00:03:18.540
make it clear what
I'm talking about.

00:03:18.540 --> 00:03:20.820
That the function takes you
from 2 to 8, the inverse will

00:03:20.820 --> 00:03:23.360
take us back from 8 to 2.

00:03:23.360 --> 00:03:25.470
So to think about that, let's
just define-- let's just

00:03:25.470 --> 00:03:27.790
say y is equal to f of x.

00:03:30.620 --> 00:03:33.510
So y is equal to f of x,
is equal to 2x plus 4.

00:03:33.510 --> 00:03:39.110
So I can write just y is equal
to 2x plus 4, and this once

00:03:39.110 --> 00:03:40.420
again, this is our function.

00:03:40.420 --> 00:03:42.470
You give me an x,
it'll give me a y.

00:03:42.470 --> 00:03:43.990
But we want to go the
other way around.

00:03:43.990 --> 00:03:46.760
We want to give you
a y and get an x.

00:03:46.760 --> 00:03:50.490
So all we have to do is
solve for x in terms of y.

00:03:50.490 --> 00:03:51.160
So let's do that.

00:03:51.160 --> 00:03:54.460
If we subtract 4 from both
sides of this equation-- let me

00:03:54.460 --> 00:03:57.350
switch colors-- if we subtract
4 from both sides of this

00:03:57.350 --> 00:04:02.310
equation, we get y minus 4 is
equal to 2x, and then if we

00:04:02.310 --> 00:04:08.270
divide both sides of this
equation by 2, we get y over 2

00:04:08.270 --> 00:04:12.750
minus 2-- 4 divided by 2
is 2-- is equal to x.

00:04:12.750 --> 00:04:15.160
Or if we just want to write it
that way, we can just swap the

00:04:15.160 --> 00:04:20.100
sides, we get x is equal to
1/2y-- same thing as

00:04:20.100 --> 00:04:23.190
y over 2-- minus 2.

00:04:23.190 --> 00:04:26.270
So what we have here is
a function of y that

00:04:26.270 --> 00:04:28.750
gives us an x, which is
exactly what we wanted.

00:04:28.750 --> 00:04:33.510
We want a function of these
values that map back to an x.

00:04:33.510 --> 00:04:38.100
So we can call this-- we could
say that this is equal to--

00:04:38.100 --> 00:04:41.840
I'll do it in the same color--
this is equal to f inverse

00:04:41.840 --> 00:04:43.570
as a function of y.

00:04:43.570 --> 00:04:45.080
Or let me just write it
a little bit cleaner.

00:04:45.080 --> 00:04:49.050
We could say f inverse as a
function of y-- so we can have

00:04:49.050 --> 00:04:52.650
10 or 8-- so now the range is
now the domain for f inverse.

00:04:52.650 --> 00:04:58.120
f inverse as a function of y
is equal to 1/2y minus 2.

00:04:58.120 --> 00:05:00.570
So all we did is we started
with our original function, y

00:05:00.570 --> 00:05:05.300
is equal to 2x plus 4, we
solved for-- over here, we've

00:05:05.300 --> 00:05:07.870
solved for y in terms of x--
then we just do a little bit of

00:05:07.870 --> 00:05:12.250
algebra, solve for x in terms
of y, and we say that that is

00:05:12.250 --> 00:05:15.580
our inverse as a function of y.

00:05:15.580 --> 00:05:17.370
Which is right over here.

00:05:17.370 --> 00:05:19.570
And then, if we, you know, you
can say this is-- you could

00:05:19.570 --> 00:05:22.640
replace the y with an a, a b,
an x, whatever you want to do,

00:05:22.640 --> 00:05:26.240
so then we can just
rename the y as x.

00:05:26.240 --> 00:05:29.000
So if you put an x into this
function, you would get f

00:05:29.000 --> 00:05:34.660
inverse of x is equal
to 1/2x minus 2.

00:05:34.660 --> 00:05:38.340
So all you do, you solve for x,
and then you swap the y and the

00:05:38.340 --> 00:05:39.720
x, if you want to
do it that way.

00:05:39.720 --> 00:05:41.240
That's the easiest way
to think about it.

00:05:41.240 --> 00:05:44.110
And one thing I want to point
out is what happens when you

00:05:44.110 --> 00:05:46.255
graph the function
and the inverse.

00:05:46.255 --> 00:05:48.420
So let me just do a
little quick and dirty

00:05:48.420 --> 00:05:50.160
graph right here.

00:05:50.160 --> 00:05:52.190
And then I'll do a bunch of
examples of actually solving

00:05:52.190 --> 00:05:53.580
for inverses, but I really
just wanted to give

00:05:53.580 --> 00:05:55.030
you the general idea.

00:05:55.030 --> 00:05:57.010
Function takes you from the
domain to the range, the

00:05:57.010 --> 00:05:59.570
inverse will take you from that
point back to the original

00:05:59.570 --> 00:06:01.810
value, if it exists.

00:06:01.810 --> 00:06:04.500
So if I were to graph these--
just let me draw a little

00:06:04.500 --> 00:06:08.790
coordinate axis right here,
draw a little bit of a

00:06:08.790 --> 00:06:12.040
coordinate axis right there.

00:06:12.040 --> 00:06:16.630
This first function, 2x plus 4,
its y intercept is going to be

00:06:16.630 --> 00:06:24.520
1, 2, 3, 4, just like that, and
then its slope will

00:06:24.520 --> 00:06:25.610
look like this.

00:06:25.610 --> 00:06:31.470
It has a slope of 2, so it will
look something like-- its graph

00:06:31.470 --> 00:06:33.870
will look-- let me make it a
little bit neater than that--

00:06:33.870 --> 00:06:36.400
it'll look something like that.

00:06:36.400 --> 00:06:38.450
That's what that
function looks like.

00:06:38.450 --> 00:06:40.390
What does this
function look like?

00:06:40.390 --> 00:06:43.680
What does the inverse function
look like, as a function of x?

00:06:43.680 --> 00:06:46.980
Remember we solved for x,
and then we swapped the x

00:06:46.980 --> 00:06:47.900
and the y, essentially.

00:06:47.900 --> 00:06:51.570
We could say now that y is
equal to f inverse of x.

00:06:51.570 --> 00:06:56.400
So we have a y-intercept
of negative 2, 1, 2, and

00:06:56.400 --> 00:07:00.150
now the slope is 1/2.

00:07:00.150 --> 00:07:02.230
The slope looks like this.

00:07:02.230 --> 00:07:04.120
Let me see if I can draw it.

00:07:04.120 --> 00:07:08.510
The slope looks-- or the line
looks something like that.

00:07:08.510 --> 00:07:10.170
And what's the
relationship here?

00:07:10.170 --> 00:07:12.060
I mean, you know, these look
kind of related, it looks

00:07:12.060 --> 00:07:14.080
like they're reflected
about something.

00:07:14.080 --> 00:07:15.990
It'll be a little bit more
clear what they're reflected

00:07:15.990 --> 00:07:19.530
about if we draw the
line y is equal to x.

00:07:19.530 --> 00:07:22.250
So the line y equals
x looks like that.

00:07:22.250 --> 00:07:23.420
I'll do it as a dotted line.

00:07:29.740 --> 00:07:32.620
And you could see, you have
the function and its inverse,

00:07:32.620 --> 00:07:36.780
they're reflected about
the line y is equal to x.

00:07:36.780 --> 00:07:39.100
And hopefully, that
makes sense here.

00:07:39.100 --> 00:07:41.330
Because over here, on
this line, let's take

00:07:41.330 --> 00:07:42.530
an easy example.

00:07:42.530 --> 00:07:49.610
Our function, when you take
0-- so f of 0 is equal to 4.

00:07:49.610 --> 00:07:54.130
Our function is mapping 0 to 4.

00:07:54.130 --> 00:08:00.050
The inverse function, if
you take f inverse of 4, f

00:08:00.050 --> 00:08:02.920
inverse of 4 is equal to 0.

00:08:02.920 --> 00:08:08.120
Or the inverse function is
mapping us from 4 to 0.

00:08:08.120 --> 00:08:10.210
Which is exactly
what we expected.

00:08:10.210 --> 00:08:14.720
The function takes us from the
x to the y world, and then we

00:08:14.720 --> 00:08:17.240
swap it, we were swapping
the x and the y.

00:08:17.240 --> 00:08:18.060
We would take the inverse.

00:08:18.060 --> 00:08:21.055
And that's why it's reflected
around y equals x.

00:08:21.055 --> 00:08:24.030
So this example that I just
showed you right here, function

00:08:24.030 --> 00:08:30.070
takes you from 0 to 4-- maybe I
should do that in the function

00:08:30.070 --> 00:08:35.810
color-- so the function takes
you from 0 to 4, that's the

00:08:35.810 --> 00:08:39.680
function f of 0 is 4, you see
that right there, so it goes

00:08:39.680 --> 00:08:42.550
from 0 to 4, and then
the inverse takes us

00:08:42.550 --> 00:08:45.830
back from 4 to 0.

00:08:45.830 --> 00:08:49.380
So f inverse takes us
back from 4 to 0.

00:08:49.380 --> 00:08:50.430
You saw that right there.

00:08:50.430 --> 00:08:55.750
When you evaluate 4 here,
1/2 times 4 minus 2 is 0.

00:08:55.750 --> 00:08:57.730
The next couple of videos we'll
do a bunch of examples so you

00:08:57.730 --> 00:09:00.180
really understand how to solve
these and are able to do

00:09:00.180 --> 00:09:05.010
the exercises on our
application for this.