Addition of Rational  Numbers

WEBVTT
Kind: captions
Language: en

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Let's add some rational
numbers.

00:00:02.900 --> 00:00:05.350
And I'm using that word because
that's the word that

00:00:05.350 --> 00:00:08.640
this book uses, but in more
popular terminology we'll be

00:00:08.640 --> 00:00:10.480
adding fractions.

00:00:10.480 --> 00:00:14.100
So let's just go through all
of these, actually, just to

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see all of the examples.

00:00:15.080 --> 00:00:19.660
So first we're going to
have 3/7 plus 2/7.

00:00:19.660 --> 00:00:22.840
Our denominators are the same,
so we can just add the

00:00:22.840 --> 00:00:24.070
numerators.

00:00:24.070 --> 00:00:28.480
So our denominator is
7, 3 plus 2 is 5.

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That is a.

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Let me do every other.

00:00:31.960 --> 00:00:33.290
It would take forever
to do all of them.

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Not forever, but just more time
than I want to spend.

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So c is 5/16 plus 5/12.

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Our denominators are
not the same.

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We have to find a common
denominator, which has to be

00:00:47.700 --> 00:00:50.450
the least common-- it actually
could be any common multiple

00:00:50.450 --> 00:00:52.050
of these, but for simplicity
let's do the

00:00:52.050 --> 00:00:53.770
least common multiple.

00:00:53.770 --> 00:00:56.150
So what's the smallest number
that's a multiple

00:00:56.150 --> 00:00:58.215
of both 16 and 12?

00:00:58.215 --> 00:01:01.700
So let's see, 16 times 2
is 32, not there yet.

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Times 3, 48.

00:01:03.660 --> 00:01:04.599
That seems to work.

00:01:04.599 --> 00:01:06.990
12 goes into 48 four times.

00:01:06.990 --> 00:01:09.733
So let's use 48 as our
common denominator.

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So we had to multiply 16 times 3
to get to 48, so we're going

00:01:19.415 --> 00:01:23.890
to have to multiply
this 5 times 3.

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We're just multiplying the
numerator and the denominator

00:01:25.670 --> 00:01:28.090
by the same number, so we're
not really changing it.

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So 5 times 3 is 15.

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And then to get from this 12 to
this 48 right there, we had

00:01:36.850 --> 00:01:38.890
to multiply times 4.

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So then to get to 5 to this
numerator over here, we have

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to multiply times 4.

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5 times 4 is 20.

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Now we have the same
denominator.

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So this is going to be equal
to, our denominator is 48.

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And so we can add 15 plus
20, which is 35.

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And can we reduce this?

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Let's see, 5 does
not go into 48.

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7 does not go into 48.

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It looks like we're all done.

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Let's do e right there.

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8/25 plus 7 over 10.

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Once again, we don't have
a common denominator.

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But we can solve that.

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Let's make, let's see, 50 is the
smallest number that both

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of these go into.

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25 times 2, so that's 50.

00:02:32.340 --> 00:02:37.050
8 over 25, to go to 50
we multiply by 2.

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So the 8, we're going to
have to multiply by 2.

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So it's going to
be 16 over 50.

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And then the 7 over 10,
we're going to want

00:02:45.945 --> 00:02:47.930
to put it over 50.

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We multiply the 10 times
5, so we have to

00:02:51.750 --> 00:02:54.605
multiply the 7 times 5.

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So it's going to
be 35 over 50.

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Now that our denominators are
the same, we have it over 50.

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16 plus 35, what is that?

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10 plus 35 is 45,
plus 6 is 51.

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So it is 51 over 50.

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Problem g.

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Let me do it in a new color.

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Problem g.

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So here we have 7 over 15-- I'll
write the second one in a

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different color--
plus 2 over 9.

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Once again, the denominators
are different.

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Find a common denominator.

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What is the smallest number that
both 15 and 9 go into?

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Let's see, 15 times 2 is 30.

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Nope, not divisible by 9.

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15 times 3 is 45, that works.

00:03:47.670 --> 00:03:50.220
45 is divisible by 9.

00:03:50.220 --> 00:03:52.590
So we use 45.

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15 times 3 is 45, so
7 times 3 is 21.

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These two fractions
are equivalent.

00:04:02.850 --> 00:04:06.680
Plus we're going over 45.

00:04:06.680 --> 00:04:11.520
To get from 9 to 45, we have
to multiply times 5.

00:04:11.520 --> 00:04:14.420
So to get our numerator
over here, we have to

00:04:14.420 --> 00:04:15.980
multiply it times 5.

00:04:15.980 --> 00:04:18.420
So 2 times 5 is 10.

00:04:18.420 --> 00:04:22.422
2/9 is the same thing
as 10/45.

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So now we can add.

00:04:24.710 --> 00:04:27.130
We're adding fractions of 45.

00:04:27.130 --> 00:04:33.130
21 plus 10 is 31,
and we are done.

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Let's do one more problem down
here, a word problem.

00:04:36.900 --> 00:04:40.070
Nadia, Peter and Ian are pooling
their money to buy a

00:04:40.070 --> 00:04:41.640
gallon of ice cream.

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Nadia's the oldest and gets
the greatest allowance.

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She contributes 1/2 the cost.
So Nadia is contributing 1/2

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the cost. So that is
Nadia right there.

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Ian is next oldest and
contributes 1/3 of the cost.

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So Ian contributes 1/3.

00:05:02.280 --> 00:05:03.820
That is Ian.

00:05:03.820 --> 00:05:06.360
Peter, the youngest, gets the
smallest allowance and

00:05:06.360 --> 00:05:13.730
contributes 1/4 of the cost.
So Peter gives 1/4 of the

00:05:13.730 --> 00:05:17.560
cost. Peter contributes
1/4 of cost.

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They figure that this will
be enough money.

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When they get to the checkout,
they realize that they forgot

00:05:22.480 --> 00:05:24.000
about sales tax and
worry there will

00:05:24.000 --> 00:05:25.340
not be enough money.

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Amazingly, they have exactly
the right amount of money.

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What fraction of the cost of
ice cream was added as tax?

00:05:32.460 --> 00:05:35.640
Well, let's see, if we add 1/2
plus 1/3, plus 1/4 of the

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cost, let's see what we get.

00:05:37.640 --> 00:05:41.100
So we have to find a common
denominator, some number that

00:05:41.100 --> 00:05:44.250
is the least common multiple
of 2, 3, and 4.

00:05:44.250 --> 00:05:46.970
And let's see, 4, it would
have to be 12, right?

00:05:46.970 --> 00:05:49.150
12 is divisible by 2, it's
divisible by 3, and it's

00:05:49.150 --> 00:05:50.400
divisible by 4.

00:05:50.400 --> 00:05:56.480
So 1/2 is the same
thing as 6/12.

00:05:56.480 --> 00:05:58.750
2 times 6 is 12.

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1 times 6 is 6.

00:06:00.420 --> 00:06:01.240
These are equivalent.

00:06:01.240 --> 00:06:03.720
6 is 1/2 of 12.

00:06:03.720 --> 00:06:09.440
1/3, if we use 12 as a common
denominator, to go from 3 to

00:06:09.440 --> 00:06:11.570
12 you have to multiply by 4.

00:06:11.570 --> 00:06:14.190
So you take that 4 and
you multiply it by 1.

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4/12 is the same thing as 1/3.

00:06:17.620 --> 00:06:24.280
And then 1/4, if you use your
denominator 12, to go from 4

00:06:24.280 --> 00:06:27.410
to 12 you have to multiply by
3, so multiply the numerator

00:06:27.410 --> 00:06:30.080
by 3 as well, you get 3.

00:06:30.080 --> 00:06:31.360
So let's add these.

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So 6/12 plus 4/12, plus 3/12 is
going to be equal to-- our

00:06:36.660 --> 00:06:40.670
denominator's going to be 12--
it's going to be 6 plus 4,

00:06:40.670 --> 00:06:47.560
plus 3, which is equal to 6 plus
4 is 10, plus 3 is 13.

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So it's going to be
equal to 13/12.

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And this is as an improper
fraction.

00:06:53.000 --> 00:06:55.950
Or we could say that this is the
same thing, this is equal

00:06:55.950 --> 00:07:02.880
to 12/12 plus 1/12, or we could
say the same thing as

00:07:02.880 --> 00:07:04.420
12/12 is just 1, right?

00:07:04.420 --> 00:07:05.770
12 divided by 12 is 1.

00:07:05.770 --> 00:07:10.050
So this is 1 and 1/12.

00:07:10.050 --> 00:07:13.950
So when they pool their money,
they get 1 and 1/12 of the

00:07:13.950 --> 00:07:19.180
price of the ice cream that
they want to buy.

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So they say what fraction of
the cost of ice cream was

00:07:21.480 --> 00:07:22.310
added as tax?

00:07:22.310 --> 00:07:24.620
This is the exact amount that
they needed to pay.

00:07:24.620 --> 00:07:29.740
So clearly, 1 is the non-taxed
price of the ice cream, so

00:07:29.740 --> 00:07:32.760
this 1/12 was the amount
added as tax.

00:07:32.760 --> 00:07:35.740
So the answer to the question
is 1/12 of the price

00:07:35.740 --> 00:07:39.290
was added as tax.


Integers and Rational Numbers

WEBVTT
Kind: captions
Language: en

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Let's do some integer and
rational number problems.

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So this first one here, the tick
marks on the number line

00:00:08.340 --> 00:00:11.140
represent evenly spaced
integers.

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And just in case you forgot
what integers mean, that

00:00:13.540 --> 00:00:16.920
essentially is the counting
numbers, including 0 and the

00:00:16.920 --> 00:00:18.350
negative counting numbers.

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So 1, 2, 3, 4, those
are all integers.

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0 is an integer.

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Negative 1, negative
2, negative 3, and

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so on and so forth.

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Those are all integers.

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You might imagine the whole
numbers, those are also

00:00:29.710 --> 00:00:32.890
another way of saying
the integers.

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So they say find the values
of a, b, c, d and e.

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So these are evenly spaced, and
this is 0 right here, and

00:00:41.880 --> 00:00:44.070
this is 21.

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Let's see how many tick
marks it takes us to

00:00:45.570 --> 00:00:47.020
go from 0 to 21.

00:00:47.020 --> 00:00:55.080
We have one tick mark, 1, 2,
3, 4, 5, 6, 7 tick marks.

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So 7 tick marks take us to 21
and they're evenly spaced--

00:00:59.680 --> 00:01:02.750
they tell us that --evenly
spaced tick marks.

00:01:02.750 --> 00:01:06.230
So if 7 of these take us
to 21, there must be 3.

00:01:06.230 --> 00:01:07.580
Each must be a jump of 3.

00:01:07.580 --> 00:01:17.170
This must be 3, 6, 9, 12, 15,
18, and then bam, 21.

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And the way I figured that out,
I said 7 tick marks gets

00:01:19.320 --> 00:01:24.340
us to 21, 21 divided by
7 tick marks is 3.

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So each of those increment by 3,
so this goes to 0 and then

00:01:27.330 --> 00:01:29.360
this must be 3 to the
left of that.

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So it's negative
3 right there.

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So what are a, b, c and
d? a is negative 3.

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a is equal to negative 3.

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Right there, that's a 3.

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b is equal to 3 right there.

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c is equal to 9 right there.

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d is equal to 12, and
e is equal to 15.

00:01:52.940 --> 00:01:54.560
Not too bad.

00:01:54.560 --> 00:01:56.430
All right.

00:01:56.430 --> 00:02:00.640
Determine what fraction of the
whole each shaded region

00:02:00.640 --> 00:02:02.000
represents.

00:02:02.000 --> 00:02:03.020
All right.

00:02:03.020 --> 00:02:03.920
Well let's see.

00:02:03.920 --> 00:02:09.350
Here we have 3 sections
of this pie.

00:02:09.350 --> 00:02:12.875
So in our denominator, which is
the bottom of our fraction

00:02:12.875 --> 00:02:17.090
we'll write there's 3 possible
pieces, and only one of the

00:02:17.090 --> 00:02:19.500
possible 3 sections is shaded.

00:02:19.500 --> 00:02:21.910
So 1/3 is shaded.

00:02:21.910 --> 00:02:24.000
So this first one is 1/3.

00:02:24.000 --> 00:02:27.420
The second one is how many
sections do we have, and all

00:02:27.420 --> 00:02:32.950
the sections are equal sized,
so we have 1, 2, 3, 4, 5, 6,

00:02:32.950 --> 00:02:38.440
7, 8, 9, 10, 11, 12 sections.

00:02:38.440 --> 00:02:39.880
How many are shaded?

00:02:39.880 --> 00:02:43.720
1, 2, 3, 4, 5, 6, 7.

00:02:43.720 --> 00:02:46.140
So it is 7/12.

00:02:46.140 --> 00:02:48.740
That's what this shaded
region represents.

00:02:48.740 --> 00:02:52.030
And then here, let's see, how
many squares do we have?

00:02:52.030 --> 00:02:57.570
We have 1, 2, 3, 4, 5,
6, 7 along this side.

00:02:57.570 --> 00:03:02.450
And then we have 1, 2,
3, 4, 5, 6, 7, 8, 9,

00:03:02.450 --> 00:03:04.460
10 along that side.

00:03:04.460 --> 00:03:11.710
So we have a total of 70
squares, 7 times 10.

00:03:11.710 --> 00:03:14.220
And let's count how many
are shaded in.

00:03:14.220 --> 00:03:18.430
Well this is going to
be 7 right there.

00:03:18.430 --> 00:03:25.300
This is 7 right there, that
is 7 right there,

00:03:25.300 --> 00:03:28.360
that is 7 right there.

00:03:28.360 --> 00:03:29.190
So let me add this up.

00:03:29.190 --> 00:03:33.260
So so far I have 28,
that's 7 times 4.

00:03:33.260 --> 00:03:43.640
28 plus, let's see, I have
2-- this is 2, 2, 2,

00:03:43.640 --> 00:03:48.830
2, 2, 2, 2 and 2.

00:03:48.830 --> 00:03:49.760
So how many 2's are there.

00:03:49.760 --> 00:03:54.310
That's 1, 2, 3, 4,
5, 6, 7, 8 2's.

00:03:54.310 --> 00:03:57.510
So that 8 times 2, that's 16.

00:03:57.510 --> 00:03:59.850
So 28 plus 16.

00:03:59.850 --> 00:04:06.060
28 plus 10 is 38 plus
another 6 is 44.

00:04:06.060 --> 00:04:12.470
So we have 44 of the 70
squares filled in.

00:04:12.470 --> 00:04:16.750
And we can write this rational
number in simpler form.

00:04:16.750 --> 00:04:17.700
We can simplify it.

00:04:17.700 --> 00:04:21.440
Let's see, the numerator and
the denominator both are

00:04:21.440 --> 00:04:23.330
divisible by 2.

00:04:23.330 --> 00:04:28.980
So this is equal
to 22 over 35.

00:04:28.980 --> 00:04:30.320
And that's about as simple
as we can get.

00:04:30.320 --> 00:04:32.940
This is divisible by 2 and 11.

00:04:32.940 --> 00:04:35.660
This is not divisible
by either 2 or 11.

00:04:35.660 --> 00:04:36.220
So there you go.

00:04:36.220 --> 00:04:40.320
That's the fraction
in simplest form.

00:04:40.320 --> 00:04:42.040
Problem 3.

00:04:42.040 --> 00:04:45.540
Place the following sets of
rational numbers in order from

00:04:45.540 --> 00:04:48.560
least to greatest.

00:04:48.560 --> 00:04:49.530
So let's do part a.

00:04:49.530 --> 00:04:51.490
I might not do all of these
for the sake of time.

00:04:51.490 --> 00:04:56.050
1/2, 1/3 and 1/4.

00:04:56.050 --> 00:04:59.695
So if you think about a pie,
going on, if you take a pie

00:04:59.695 --> 00:05:03.790
and you divide it into just
2 pieces, 1/2 is half

00:05:03.790 --> 00:05:05.520
of that whole pie.

00:05:05.520 --> 00:05:08.990
That right there is 1/2.

00:05:08.990 --> 00:05:17.430
This right here, 1/4, 1/4 is--
you just take the same pie

00:05:17.430 --> 00:05:18.580
--1/4 is just that.

00:05:18.580 --> 00:05:26.870
So clearly 1/2 is greater than
1/4 or 1/4 is less than 1/2.

00:05:26.870 --> 00:05:27.920
I'll write it like that.

00:05:27.920 --> 00:05:29.790
And 1/3 is right in between.

00:05:29.790 --> 00:05:33.430
If you did 1/3-- it's maybe
harder to visualize here --but

00:05:33.430 --> 00:05:38.950
if this is 1/2, if we divided
this into thirds, 1/3

00:05:38.950 --> 00:05:39.850
is less than this.

00:05:39.850 --> 00:05:45.440
So if we divided this into
thirds, 3 equals sections, 1/3

00:05:45.440 --> 00:05:47.960
would be just like
that right here.

00:05:47.960 --> 00:05:50.190
1/3 is more than that.

00:05:50.190 --> 00:05:52.780
There's multiple ways we
could visualize this.

00:05:52.780 --> 00:05:57.890
So 1/4 is less than 1/3,
which is less than 1/2.

00:05:57.890 --> 00:06:01.000
In general, the larger
your denominator

00:06:01.000 --> 00:06:02.520
the smaller the fraction.

00:06:02.520 --> 00:06:05.680
I have only one 1 of 4, that's
less than 1 out of 3, and

00:06:05.680 --> 00:06:08.010
that's even less than 1 out of
2, which would be even less

00:06:08.010 --> 00:06:09.540
than 1 out of 1.

00:06:09.540 --> 00:06:10.860
So that's our first one.

00:06:10.860 --> 00:06:14.680
We put that in order:
1/4, 1/3, 1/2.

00:06:14.680 --> 00:06:17.950
Let me just do part to c, just
for the sake of time.

00:06:17.950 --> 00:06:28.150
So we have 39 over 60, we
have 49 over 80, and we

00:06:28.150 --> 00:06:31.350
have a 59 over 100.

00:06:31.350 --> 00:06:33.350
So the easiest way to do this,
if you had a calculator you

00:06:33.350 --> 00:06:35.800
would actually just divide each
of these numbers out and

00:06:35.800 --> 00:06:38.090
you would get a decimal, and
it's very easy to compare

00:06:38.090 --> 00:06:38.900
decimal numbers.

00:06:38.900 --> 00:06:42.140
But let's if we can do this a
little bit more artfully.

00:06:42.140 --> 00:06:44.230
So we can approximate it.

00:06:44.230 --> 00:06:46.570
We could say that this
is a little bit less.

00:06:46.570 --> 00:06:52.180
So 39 over 60, it's almost--
I'll make those squiggly equal

00:06:52.180 --> 00:06:56.340
signs for almost --it's
approximately 40 over 60,

00:06:56.340 --> 00:06:59.020
which is equal to 2/3.

00:06:59.020 --> 00:07:00.480
So this is almost 2/3.

00:07:00.480 --> 00:07:02.870
A little bit less than 2/3.

00:07:02.870 --> 00:07:10.420
This right here, 49 over 80,
it's almost 50 over 80, so

00:07:10.420 --> 00:07:13.390
it's almost 5/8.

00:07:13.390 --> 00:07:20.090
And then this right here, 59
over 80-- 59 over 100, sorry.

00:07:20.090 --> 00:07:27.910
It's almost equal to 60 over
100, which is equal to 3/5.

00:07:27.910 --> 00:07:30.480
And just to get a sense of, at
least now we might be able to

00:07:30.480 --> 00:07:33.860
do it in our head, 2/3,
that's approximately

00:07:33.860 --> 00:07:38.290
equal to 0.66 repeating.

00:07:38.290 --> 00:07:40.620
You could say 0.667, roughly
equal to that.

00:07:40.620 --> 00:07:42.480
That's the 2/3 right there.

00:07:42.480 --> 00:07:43.670
Let me do it in that
color so you know

00:07:43.670 --> 00:07:45.820
what I'm talking about.

00:07:45.820 --> 00:07:50.890
So this is approximately 0.667,
not exactly, we're

00:07:50.890 --> 00:07:52.330
estimating.

00:07:52.330 --> 00:07:57.460
This right here, 5/8-- So the
way I always think of eighths

00:07:57.460 --> 00:08:00.150
is 1/8 is .125.

00:08:00.150 --> 00:08:03.692
So 5/8-- Let me just divide
it right here, let

00:08:03.692 --> 00:08:05.220
me just do the work.

00:08:05.220 --> 00:08:10.030
So if we take 8 goes into 5, put
some decimals right here,

00:08:10.030 --> 00:08:15.210
8 goes into 5 zero times,
0 times 8 is 0.

00:08:15.210 --> 00:08:18.020
5 minus 0 is 5.

00:08:18.020 --> 00:08:19.710
Bring down this 0.

00:08:19.710 --> 00:08:22.710
8 goes into 50 six times.

00:08:22.710 --> 00:08:26.690
6 times 8 is 48.

00:08:26.690 --> 00:08:30.520
50 minus 48 is 2,
bring down a 0.

00:08:30.520 --> 00:08:34.679
8 goes into 20 two times.

00:08:34.679 --> 00:08:37.260
2 times 8 is 16.

00:08:37.260 --> 00:08:39.289
Subtract, you get a 4.

00:08:39.289 --> 00:08:40.580
Bring down this 0.

00:08:40.580 --> 00:08:41.740
40.

00:08:41.740 --> 00:08:43.630
8 goes into 40 five times.

00:08:43.630 --> 00:08:47.250
So this right here is 0.625.

00:08:47.250 --> 00:08:49.250
Not exactly, it's a little
bit less than this.

00:08:49.250 --> 00:08:51.300
This is a little bit
less than that.

00:08:51.300 --> 00:08:53.510
And 3/5 is pretty
straightforward.

00:08:53.510 --> 00:08:56.520
3/5 is the same thing
as 6/10, which is

00:08:56.520 --> 00:09:00.130
approximately equal to 0.6.

00:09:00.130 --> 00:09:02.590
So this is a little bit less
than 0.6, this is a little bit

00:09:02.590 --> 00:09:06.950
less than .625, this is a little
bit less than .667.

00:09:06.950 --> 00:09:09.240
So if you put them in order,
this is the smallest, then

00:09:09.240 --> 00:09:10.960
this one, then that one.

00:09:10.960 --> 00:09:12.000
So let me write it down.

00:09:12.000 --> 00:09:20.610
So the smallest is 59 over 100,
followed by 49 over 80,

00:09:20.610 --> 00:09:26.280
followed by the largest,
which is 39 over 60.

00:09:26.280 --> 00:09:27.790
Which might be a little
unintuitive because this

00:09:27.790 --> 00:09:29.970
actually has the smallest number
in the numerator, but

00:09:29.970 --> 00:09:31.620
it also has the smallest number
in the denominator.

00:09:31.620 --> 00:09:33.080
So you actually had
to work it out.

00:09:33.080 --> 00:09:36.560
They're actually reasonably
close to each other.

00:09:36.560 --> 00:09:39.150
Let's do problem number 5.

00:09:39.150 --> 00:09:41.960
Find the opposite of each
of the following.

00:09:41.960 --> 00:09:44.720
And when they say opposite,
I think they mean find the

00:09:44.720 --> 00:09:48.120
negative, or find the reflection
on the number line.

00:09:48.120 --> 00:09:50.450
So we essentially just have to
multiply all of these numbers

00:09:50.450 --> 00:09:52.150
by negative 1.

00:09:52.150 --> 00:09:59.440
So for part a, the opposite of
1.001 would be negative 1.001.

00:09:59.440 --> 00:10:05.040
For part b, the negative of 5
minus 11-- well we could write

00:10:05.040 --> 00:10:09.740
it --it's going to be negative
1 times-- Well,

00:10:09.740 --> 00:10:12.130
what's 5 minus 11?

00:10:12.130 --> 00:10:16.190
5 minus 11 is negative 6.

00:10:16.190 --> 00:10:18.160
This number right here
is negative 6.

00:10:18.160 --> 00:10:21.290
So the opposite of negative 6--
negative 1 times negative

00:10:21.290 --> 00:10:23.780
6 --is going to be equal to 6.

00:10:23.780 --> 00:10:25.210
Part c.

00:10:25.210 --> 00:10:26.250
x plus y.

00:10:26.250 --> 00:10:31.127
The opposite's going to be
negative 1 times x plus y, or

00:10:31.127 --> 00:10:33.540
we could have just written
negative times x plus y.

00:10:33.540 --> 00:10:35.510
You distribute the negative 1.

00:10:35.510 --> 00:10:39.440
This is equal to negative x,
and then negative 1 times

00:10:39.440 --> 00:10:42.000
plus y minus y.

00:10:42.000 --> 00:10:44.000
Finally, part d.

00:10:44.000 --> 00:10:47.660
Negative 1 times x minus y.

00:10:47.660 --> 00:10:48.880
I'm just taking the negative
of everything.

00:10:48.880 --> 00:10:51.340
So I'm taking the negative of
this, and so this is going to

00:10:51.340 --> 00:10:55.390
equal to negative 1 times
x, negative x.

00:10:55.390 --> 00:11:00.510
Then negative 1 times minus y or
negative y, that's plus y.

00:11:00.510 --> 00:11:03.630
Or we could write this
as y minus x.

00:11:03.630 --> 00:11:06.570
And that's actually a neat thing
to remember later on

00:11:06.570 --> 00:11:08.590
when you're manipulating
things algebraically.

00:11:08.590 --> 00:11:11.950
The negative of x minus
y is y minus x.

00:11:11.950 --> 00:11:13.000
We just did that right there.

00:11:13.000 --> 00:11:16.750
The negative of a minus
b is b minus a.

00:11:16.750 --> 00:11:19.280
You just swap the two when you
take the negative, and we just

00:11:19.280 --> 00:11:21.280
proved it for you right there.

00:11:21.280 --> 00:11:21.770
All right.

00:11:21.770 --> 00:11:23.550
Let's do a couple of these.

00:11:23.550 --> 00:11:27.300
Simplify the following absolute
value expression.

00:11:27.300 --> 00:11:29.360
Absolute value is essentially
telling you the

00:11:29.360 --> 00:11:31.330
distance from 0.

00:11:31.330 --> 00:11:37.010
So for example, if that's 0,
if I have the number 4.

00:11:37.010 --> 00:11:39.130
How far is 4 from 0?

00:11:39.130 --> 00:11:40.740
The absolute value of 4?

00:11:40.740 --> 00:11:42.790
Well it's 4 from 0.

00:11:42.790 --> 00:11:43.880
So you just say it's 4.

00:11:43.880 --> 00:11:45.210
So the absolute value
of positive

00:11:45.210 --> 00:11:46.910
numbers is just the number.

00:11:46.910 --> 00:11:49.100
Let's say I had negative 3.

00:11:49.100 --> 00:11:52.570
How far is negative 3 from 0?

00:11:52.570 --> 00:11:54.250
Well, I don't care whether it's
to the left of the right

00:11:54.250 --> 00:11:57.310
of 0, it's just 3
away from the 0.

00:11:57.310 --> 00:11:58.250
It's 3 to the left.

00:11:58.250 --> 00:11:59.830
So this is going to be 3.

00:11:59.830 --> 00:12:02.240
So one way to think about
absolute value is, whether

00:12:02.240 --> 00:12:04.610
it's positive or negative,
it becomes positive.

00:12:04.610 --> 00:12:07.040
It's that far from 0.

00:12:07.040 --> 00:12:08.330
So let's do these.

00:12:08.330 --> 00:12:10.940
So 11 minus the absolute
value of negative 4.

00:12:10.940 --> 00:12:14.960
This is equal to 11 minus-- the
absolute of negative 4 is

00:12:14.960 --> 00:12:19.660
just 4, which is equal to 7.

00:12:19.660 --> 00:12:23.030
I'll do every other one of these
problems. The absolute

00:12:23.030 --> 00:12:25.150
value of negative 5 minus 11.

00:12:25.150 --> 00:12:29.120
So that'll be equal to the
absolute value of negative 5

00:12:29.120 --> 00:12:32.310
minus 11 is negative 16.

00:12:32.310 --> 00:12:34.140
The absolute value
of negative 16.

00:12:34.140 --> 00:12:36.910
the absolute value of negative
16 is just 16.

00:12:36.910 --> 00:12:38.960
It's 16 to the left of 0.

00:12:38.960 --> 00:12:40.040
You could view it that
way, you could just

00:12:40.040 --> 00:12:41.310
get rid of its sign.

00:12:41.310 --> 00:12:44.280
And then finally here,
let's do part e.

00:12:44.280 --> 00:12:47.740
The negative of the absolute
value of negative 7 is equal

00:12:47.740 --> 00:12:50.650
to the negative of--
What's the absolute

00:12:50.650 --> 00:12:51.450
value of negative 7?

00:12:51.450 --> 00:12:52.780
It's just 7.

00:12:52.780 --> 00:12:54.490
So it's equal to negative 7.

00:12:54.490 --> 00:12:56.590
This right here is 7.

00:12:56.590 --> 00:12:59.380
Anyway, hopefully you
found that useful.


Word Problem Solving Strategies

WEBVTT
Kind: captions
Language: en

00:00:00.450 --> 00:00:03.050
Let's do some more word
problems. And this first one's

00:00:03.050 --> 00:00:04.660
a bit of a classic.

00:00:04.660 --> 00:00:10.020
Brit has $2.25 in nickels
and dimes.

00:00:10.020 --> 00:00:13.750
If she has 40 coins in total,
how many of each

00:00:13.750 --> 00:00:15.175
coin does she have?

00:00:15.175 --> 00:00:17.770
So let's define some
variables here.

00:00:17.770 --> 00:00:23.175
Let's let n equal the number
of nickels she has.

00:00:27.760 --> 00:00:37.810
And let d is equal to the
number of dimes she has.

00:00:37.810 --> 00:00:41.010
So let's see, the first thing we
know is that she has $2.25

00:00:41.010 --> 00:00:41.690
in nickels and dimes.

00:00:41.690 --> 00:00:43.760
So how much money
does she have?

00:00:43.760 --> 00:00:48.330
Well, if you take the number
of nickels, each nickel is

00:00:48.330 --> 00:00:49.630
worth $0.05.

00:00:49.630 --> 00:00:57.030
So the number of nickels times
0.05, that's how much money

00:00:57.030 --> 00:00:59.790
her nickels represent,
plus the amount of

00:00:59.790 --> 00:01:01.270
money her dimes represent.

00:01:01.270 --> 00:01:03.290
So she's got d dimes.

00:01:03.290 --> 00:01:08.940
They're each worth 0.1 of a
dollar, or .10, or 0.10 if we

00:01:08.940 --> 00:01:11.070
want to put the 0 in front
of the decimal.

00:01:11.070 --> 00:01:13.490
So this is the total amount
of money she has.

00:01:13.490 --> 00:01:18.390
If she has 5 nickels, you'd do
5 times $0.05, it'd be $0.25.

00:01:18.390 --> 00:01:21.370
If she had 4 dimes, that'd be
$0.40, and you would add it

00:01:21.370 --> 00:01:21.940
all together.

00:01:21.940 --> 00:01:24.550
Now, the problem tells us that
the total amount of money she

00:01:24.550 --> 00:01:30.400
has is $2.25.

00:01:30.400 --> 00:01:36.210
So this the first statement
right there can be translated

00:01:36.210 --> 00:01:38.180
into this equation right here.

00:01:38.180 --> 00:01:39.620
She has a certain number
of nickels and a

00:01:39.620 --> 00:01:40.930
certain number of dimes.

00:01:40.930 --> 00:01:44.760
You multiply $0.05 times the
nickels, plus $0.10 times the

00:01:44.760 --> 00:01:47.930
number of dimes, you
end up with 225.

00:01:47.930 --> 00:01:52.400
Now, this second statement,
she has 40 coins in total.

00:01:52.400 --> 00:01:54.790
And I'm assuming that she only
has nickels and dimes, that

00:01:54.790 --> 00:01:56.470
there aren't any other coins.

00:01:56.470 --> 00:01:59.380
So 40 coins in total, that
means that the number of

00:01:59.380 --> 00:02:02.820
nickels plus the number of dimes
has to be equal to 40.

00:02:02.820 --> 00:02:11.590
So the nickels plus the dimes
has got to be equal to 40.

00:02:11.590 --> 00:02:14.020
That's what that information
tells us.

00:02:14.020 --> 00:02:17.440
Now, we have two equations with
two unknowns, and there's

00:02:17.440 --> 00:02:18.530
two ways we can do this.

00:02:18.530 --> 00:02:21.050
We can do this as a system,
and we can subtract one

00:02:21.050 --> 00:02:22.400
equation from the other.

00:02:22.400 --> 00:02:23.490
Or we can do substitution.

00:02:23.490 --> 00:02:26.030
Let's do substitution.

00:02:26.030 --> 00:02:28.010
That's often a more intuitive
way to do it.

00:02:28.010 --> 00:02:30.650
So what I'm going to do is,
I'm going to solve for-- I

00:02:30.650 --> 00:02:33.800
could solve for either one-- but
I want to solve for-- I'll

00:02:33.800 --> 00:02:37.030
solve for d and then I'll
substitute whatever I get for

00:02:37.030 --> 00:02:38.670
the d in this equation
right here.

00:02:38.670 --> 00:02:42.070
So if we subtract n from both
sides of this equation-- so

00:02:42.070 --> 00:02:42.710
let me do that.

00:02:42.710 --> 00:02:45.760
Negative n plus this, minus n.

00:02:45.760 --> 00:02:48.350
Obviously, these cancel out and
you're just left with d is

00:02:48.350 --> 00:02:51.900
equal to 40 minus n.

00:02:51.900 --> 00:02:53.460
Now, we can take this.

00:02:53.460 --> 00:02:56.500
We just established using the
second equation, using this

00:02:56.500 --> 00:03:00.480
second statement, that d is
equal to 40 minus the number

00:03:00.480 --> 00:03:01.620
of nickels.

00:03:01.620 --> 00:03:05.790
And we can use that to
substitute back in for d.

00:03:05.790 --> 00:03:11.770
So this first equation will
now become 0.05 times our

00:03:11.770 --> 00:03:16.890
number of nickels, plus 0.--
I'll just write 0.1-- times

00:03:16.890 --> 00:03:18.790
our number of dimes.

00:03:18.790 --> 00:03:21.040
We already know that the
number of dimes is

00:03:21.040 --> 00:03:22.200
equal to 40 minus n.

00:03:22.200 --> 00:03:24.470
That's what that second
statement told us.

00:03:24.470 --> 00:03:29.820
So instead of d, I'm going to
substitute it with 40 minus n.

00:03:29.820 --> 00:03:32.670
40-- that's the number of dimes
we have. It's 40 minus

00:03:32.670 --> 00:03:36.750
the number of nickels, and
that is equal to $2.25.

00:03:36.750 --> 00:03:39.450
And now we can just try
to solve for n.

00:03:39.450 --> 00:03:47.720
So we get a 0.05n plus .1
times 40, that is 4.

00:03:47.720 --> 00:03:49.130
Let me show you that I'm
distributing it.

00:03:49.130 --> 00:03:57.970
So .1 times 40, that is 4 minus
0.1n-- I just distribute

00:03:57.970 --> 00:04:07.030
this times each of these terms--
is equal to $2.25.

00:04:07.030 --> 00:04:08.530
And then what can we do?

00:04:08.530 --> 00:04:16.199
Well, we can take .05 minus .1--
or your .10, depending on

00:04:16.199 --> 00:04:19.285
how you want to do it-- if you
take those two terms, you're

00:04:19.285 --> 00:04:28.870
actually going to get a negative
0.05n minus .10--

00:04:28.870 --> 00:04:30.870
this is twice as big as that,
so it's going to get a

00:04:30.870 --> 00:04:34.820
negative number-- plus 4--
so I still have that

00:04:34.820 --> 00:04:37.370
4-- is equal to $2.25.

00:04:37.370 --> 00:04:41.220
Now I can subtract 4 from both
sides of this equation.

00:04:41.220 --> 00:04:47.250
I get negative 0.05n is equal
to-- if you subtract 4 from

00:04:47.250 --> 00:04:50.840
the left, that 4's going to go
away-- you subtract 4 over

00:04:50.840 --> 00:04:59.020
here; 2.25 minus 4-- that's
negative 1.75.

00:04:59.020 --> 00:04:59.490
All right.

00:04:59.490 --> 00:05:02.020
Now we can multiply both sides
of this equation by negative

00:05:02.020 --> 00:05:04.960
1, just to make it a little bit
cleaner, so that becomes a

00:05:04.960 --> 00:05:07.280
plus and a plus.

00:05:07.280 --> 00:05:09.390
And then divide both
sides by .05.

00:05:09.390 --> 00:05:12.200
I could have just divided both
sides by negative .05 to begin

00:05:12.200 --> 00:05:14.160
with, but I like getting
everything positive.

00:05:14.160 --> 00:05:24.640
So if you divided by 0.05, you
get n is equal to $1.75

00:05:24.640 --> 00:05:28.176
divided by 0.05.

00:05:28.176 --> 00:05:30.235
So let's do this division.

00:05:30.235 --> 00:05:32.750
Let me do it in a
different color.

00:05:32.750 --> 00:05:40.880
0.05 divided into $1.75--
let's multiply both the

00:05:40.880 --> 00:05:43.680
divisor and the dividend
by 100.

00:05:43.680 --> 00:05:46.710
So we're going to go over two
decimal spaces, so this just

00:05:46.710 --> 00:05:48.080
becomes a 5.

00:05:48.080 --> 00:05:50.270
This becomes 175.

00:05:50.270 --> 00:05:53.450
This is the same thing
as 175 divided by 5.

00:05:53.450 --> 00:05:56.330
5 goes into 17 three times.

00:05:56.330 --> 00:05:59.160
3 times 5 is 15.

00:05:59.160 --> 00:06:02.480
17 minus 15 is 2.

00:06:02.480 --> 00:06:04.370
Bring down the 5.

00:06:04.370 --> 00:06:05.080
25.

00:06:05.080 --> 00:06:08.260
5 goes into 25 five times.

00:06:08.260 --> 00:06:10.400
5 times 5 is 25.

00:06:10.400 --> 00:06:11.740
You have no remainder.

00:06:11.740 --> 00:06:15.830
So this is equal to 35.

00:06:15.830 --> 00:06:19.170
So n is equal to 35, and now we
can solve for d. d is equal

00:06:19.170 --> 00:06:20.440
to 40 minus n.

00:06:20.440 --> 00:06:25.960
So d is equal to 40 minus
35, which is equal to 5.

00:06:25.960 --> 00:06:33.310
So we have 5 dimes
and 35 nickels.

00:06:33.310 --> 00:06:38.970
And just to verify that that
actually does make sense, 35

00:06:38.970 --> 00:06:40.600
nickels are going to
be worth how much?

00:06:40.600 --> 00:06:44.210
Well 35 times 5 cents is
going to get us to

00:06:44.210 --> 00:06:47.400
175 cents or $1.75.

00:06:47.400 --> 00:06:53.890
So this times 35, that is going
to be $1.75 in nickels.

00:06:53.890 --> 00:06:57.170
35 times 5 or times .05,
depending on how you

00:06:57.170 --> 00:06:58.060
want to view it.

00:06:58.060 --> 00:07:00.150
And then this is
going to be 5.

00:07:00.150 --> 00:07:03.740
So this right here is going
to be $0.50 in dimes.

00:07:03.740 --> 00:07:08.200
$1.75 plus $0.50,
that's $2.25.

00:07:08.200 --> 00:07:11.280
Let's do one more.

00:07:11.280 --> 00:07:15.480
A pattern of squares is
out together-- oh.

00:07:15.480 --> 00:07:18.620
A pattern of squares is
out together as shown.

00:07:18.620 --> 00:07:19.870
I guess they're hanging out.

00:07:19.870 --> 00:07:22.340
How many squares are in
the 12th diagram?

00:07:22.340 --> 00:07:23.150
So let's see what's happening.

00:07:23.150 --> 00:07:29.280
So this is the first, that is
the second, that is the third.

00:07:29.280 --> 00:07:33.830
Let's see here, each side, this
is a 1 by 1 square, and

00:07:33.830 --> 00:07:37.720
then here it's 2 by 2, but then
they take out this block

00:07:37.720 --> 00:07:39.140
right over there.

00:07:39.140 --> 00:07:40.170
So let's see.

00:07:40.170 --> 00:07:44.020
So one way you could think about
it is this is 2 by 2,

00:07:44.020 --> 00:07:51.250
and this is a 3 by 3, but they
took out the 2 by 2.

00:07:51.250 --> 00:07:57.120
And so the next one is going to
be a 4 by 4 where they took

00:07:57.120 --> 00:08:01.140
out the 3 by 3.

00:08:01.140 --> 00:08:05.000
So I guess one way that we can
describe the number of

00:08:05.000 --> 00:08:07.826
squares-- actually, the easiest
way, you don't have to

00:08:07.826 --> 00:08:14.780
think about that is, look, I
have 3 here, and then I have 3

00:08:14.780 --> 00:08:16.340
minus 1 over here.

00:08:16.340 --> 00:08:18.150
Let me do that in a
different color.

00:08:18.150 --> 00:08:20.570
3 minus 1 over here.

00:08:20.570 --> 00:08:25.220
Here I have 2, and here
I have 2 minus 1.

00:08:25.220 --> 00:08:27.740
And then here I have 1, and then
here I have 1 minus 1.

00:08:27.740 --> 00:08:28.840
I have nothing.

00:08:28.840 --> 00:08:33.799
So in general, the number of
squares for any n is going to

00:08:33.799 --> 00:08:43.169
be n plus, then we
have n minus 1.

00:08:43.169 --> 00:08:48.910
Plus n minus 1, which is the
same thing as 2n minus 1.

00:08:48.910 --> 00:08:50.300
And it works, right.

00:08:50.300 --> 00:08:51.290
2n minus 1.

00:08:51.290 --> 00:08:53.550
2 times 3 minus 1 is 5.

00:08:53.550 --> 00:08:55.890
1, 2, 3, 4, 5.

00:08:55.890 --> 00:08:57.350
So works for these three.

00:08:57.350 --> 00:09:01.520
So the 12th square, so when n is
equal to 12, it's going to

00:09:01.520 --> 00:09:06.460
be 2 times 12 minus 1, which is
equal to 24 minus 1, which

00:09:06.460 --> 00:09:12.710
is equal to 23 squares
in the 12th diagram.

00:09:12.710 --> 00:09:15.450
Actually, I said lets do only
one, let's do another one.

00:09:15.450 --> 00:09:17.610
We're in the zone.

00:09:17.610 --> 00:09:21.470
Grace starts biking at
12 miles per hour.

00:09:21.470 --> 00:09:26.220
1 hour later, Dan starts biking
at 15 miles per hour.

00:09:26.220 --> 00:09:29.900
Following the same route, how
long will it take him to catch

00:09:29.900 --> 00:09:31.430
up with Grace?

00:09:31.430 --> 00:09:39.940
So let t equal the time
Grace is biking.

00:09:39.940 --> 00:09:41.510
Actually, let me
do it this way.

00:09:41.510 --> 00:09:48.180
Let me do it the time
Dan is biking.

00:09:48.180 --> 00:09:51.010
Then how long will
Grace be biking?

00:09:51.010 --> 00:09:53.430
Well, he starts an hour-- let's
see, did I say, yeah--

00:09:53.430 --> 00:09:55.420
an hour later, he starts.

00:09:55.420 --> 00:09:58.380
So whatever number this is,
Grace is going to be biking

00:09:58.380 --> 00:09:59.390
for 1 more hour.

00:09:59.390 --> 00:10:07.370
So t plus 1 is equal to
time Grace is biking.

00:10:07.370 --> 00:10:09.025
When this is 0 this
is going to be 1.

00:10:09.025 --> 00:10:13.480
She will have already been
biking for 1 hour.

00:10:13.480 --> 00:10:17.240
And we just have to remember
that distance is equal to rate

00:10:17.240 --> 00:10:18.990
times time.

00:10:18.990 --> 00:10:28.450
So Dan's distance is going to
be equal to 15 miles an hour

00:10:28.450 --> 00:10:30.580
times the time in hours.

00:10:30.580 --> 00:10:32.250
This is Dan's distance.

00:10:32.250 --> 00:10:36.630
And then Grace's distance--
maybe I'll call that g, or

00:10:36.630 --> 00:10:42.420
let's say D for Grace, that's
her distance as well-- is

00:10:42.420 --> 00:10:45.500
going to be equal to
12 miles an hour.

00:10:45.500 --> 00:10:48.730
And her time in terms of t
is going to be t plus 1.

00:10:48.730 --> 00:10:50.280
She's going to be always
be traveling for 1

00:10:50.280 --> 00:10:52.090
more hour than Dan.

00:10:52.090 --> 00:10:54.350
12 times t plus 1.

00:10:54.350 --> 00:10:57.290
And the point at which they
catch up, that's the point at

00:10:57.290 --> 00:10:59.380
which these two things are
going to be equal to.

00:10:59.380 --> 00:11:01.330
That Dan's distance
is going to be

00:11:01.330 --> 00:11:02.760
equal to Grace's distance.

00:11:02.760 --> 00:11:07.260
So we just have to set these two
to be equal to each other.

00:11:07.260 --> 00:11:13.750
So you get 15t is going to be
equal to-- that is how far Dan

00:11:13.750 --> 00:11:17.910
has been riding his bike-- and
then it's going to be equal to

00:11:17.910 --> 00:11:20.080
12 times t plus 1.

00:11:20.080 --> 00:11:22.020
That is how far Grace
has ridden her bike.

00:11:22.020 --> 00:11:25.270
And remember, her time is always
going to be 1 hour more

00:11:25.270 --> 00:11:26.290
than Dan's.

00:11:26.290 --> 00:11:28.720
So let's just solve for time.

00:11:28.720 --> 00:11:36.350
So we get 15t is equal
to 12t plus 1.

00:11:36.350 --> 00:11:39.010
12t plus 1.

00:11:39.010 --> 00:11:43.810
And then if we subtract 12--
I'm sorry, 12t plus 12-- I

00:11:43.810 --> 00:11:45.190
don't want to make
a mistake there.

00:11:45.190 --> 00:11:47.840
If we subtract 12t from
both sides, you get

00:11:47.840 --> 00:11:52.940
3t is equal to 12.

00:11:52.940 --> 00:11:56.940
Divide both sides by 3-- let me
scroll down a little bit--

00:11:56.940 --> 00:11:59.990
you get t is equal to 4.

00:11:59.990 --> 00:12:05.290
So after 4 hours, after Dan has
been riding for 4 hours,

00:12:05.290 --> 00:12:07.400
he will catch up with Grace.

00:12:07.400 --> 00:12:12.240
This is Dan, and if Dan's ridden
for 4 hours, Grace has

00:12:12.240 --> 00:12:14.370
been riding for 6 hours.

00:12:14.370 --> 00:12:17.060
6 hours is how far Grace
has ridden for.

00:12:17.060 --> 00:12:23.030
And Dan after 4 hours,
4 times 15, he would

00:12:23.030 --> 00:12:25.860
have gone 60 miles.

00:12:25.860 --> 00:12:30.220
And Grace after-- sorry, there's
1 more, this isn't a

00:12:30.220 --> 00:12:32.300
six-- she goes 1 more
hour than Dan.

00:12:32.300 --> 00:12:35.000
So it's 5, she's traveled
for 5 hours.

00:12:35.000 --> 00:12:39.730
So Grace would have gone 5 hours
at 12 miles an hour.

00:12:39.730 --> 00:12:46.100
And that, once again,
is also 60 miles.

00:12:46.100 --> 00:12:50.480
So they meet at exactly
the 60 mile mark.


Word Problem Solving Plan 1

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:04.280
Let's do some word problems.
It says this year

00:00:04.280 --> 00:00:06.790
you got a 5% raise.

00:00:06.790 --> 00:00:10.830
If your new salary is $45,000,
what was your

00:00:10.830 --> 00:00:12.960
salary before the raise?

00:00:12.960 --> 00:00:15.790
So let's say my old
salary is x.

00:00:15.790 --> 00:00:20.670
x is equal to old salary.

00:00:20.670 --> 00:00:24.030
And I got a 5% raise
off of that salary.

00:00:24.030 --> 00:00:32.549
So x plus 5% of x, plus 5%
times x-- that's my raise

00:00:32.549 --> 00:00:37.880
right there-- is going to be
equal to my new salary.

00:00:37.880 --> 00:00:39.390
Well my new salary, they
tell us what it is.

00:00:39.390 --> 00:00:44.680
It's $45,000.

00:00:44.680 --> 00:00:46.850
And let's see if we can
algebraically manipulate this

00:00:46.850 --> 00:00:47.500
a little bit.

00:00:47.500 --> 00:00:53.050
5% is the same thing
as 0.05, 5%.

00:00:53.050 --> 00:00:55.960
Per cent literally
means per 100.

00:00:55.960 --> 00:00:58.580
So this is the same thing
as 5 over 100.

00:00:58.580 --> 00:01:05.770
This is 5%, per 100, which is
the same thing as 0.05.

00:01:05.770 --> 00:01:09.960
So here we could say this is
the same thing is 1x plus--

00:01:09.960 --> 00:01:17.630
let me write that x a little bit
nicer-- 1x plus 0.05x is

00:01:17.630 --> 00:01:21.230
equal to 45,000.

00:01:21.230 --> 00:01:25.670
1 of something plus 0.05
of that same something.

00:01:25.670 --> 00:01:27.550
You can add the coefficients.

00:01:27.550 --> 00:01:28.530
So you add those two.

00:01:28.530 --> 00:01:36.200
You're going to get 1.05x
is equal to 45,000.

00:01:36.200 --> 00:01:45.720
And then to solve for x, you
divide both sides by 1.05.

00:01:45.720 --> 00:01:47.280
These will just cancel out.

00:01:47.280 --> 00:01:54.195
And we're going to get x is
equal to 45,000 over 1.05.

00:01:54.195 --> 00:01:56.850
And let's figure out
what that is.

00:01:56.850 --> 00:02:04.820
So 45-- one, two, three--
$45,000 divided by 1.05 is

00:02:04.820 --> 00:02:09.240
equal to $42,857.

00:02:09.240 --> 00:02:18.160
I'll say that again, $42,857.

00:02:18.160 --> 00:02:21.540
This is how much your annual
income was last year.

00:02:21.540 --> 00:02:23.160
Let's make sure that
that makes sense.

00:02:23.160 --> 00:02:25.240
Let's see if that works out.

00:02:25.240 --> 00:02:28.370
So if we check it, let's
say we start with this.

00:02:28.370 --> 00:02:30.300
Let's take 5% of that.

00:02:30.300 --> 00:02:34.700
So times 0.05 is equal to-- so
that's the amount of raise you

00:02:34.700 --> 00:02:40.350
had-- plus your old salary,
plus 42,857.

00:02:40.350 --> 00:02:42.060
And, of course, we round
it off a little bit,

00:02:42.060 --> 00:02:43.500
so it won't be exact.

00:02:43.500 --> 00:02:47.020
And it equals almost
exactly $45,000.

00:02:47.020 --> 00:02:49.280
We're like 14 cents
below that.

00:02:49.280 --> 00:02:50.740
So $45,000.

00:02:50.740 --> 00:02:54.090
So this answer checks out.

00:02:54.090 --> 00:02:56.100
Let's do this next one.

00:02:56.100 --> 00:03:02.930
It costs $250-- let me write
this down-- it costs $250 to

00:03:02.930 --> 00:03:06.570
carpet a room that
is 14 by 18.

00:03:06.570 --> 00:03:11.860
How much does it cost to carpet
a room that is 9 by 10?

00:03:11.860 --> 00:03:13.700
And the units here are feet.

00:03:13.700 --> 00:03:16.910
Well in case you have never
carpeted a room, you normally

00:03:16.910 --> 00:03:19.910
carpet a room by the amount of
area the room is, or the

00:03:19.910 --> 00:03:21.920
square footage of the room.

00:03:21.920 --> 00:03:25.000
So what we could do is use this
information up here to

00:03:25.000 --> 00:03:27.475
figure out how much it costs
per square foot.

00:03:33.360 --> 00:03:36.220
So in this situation we had
14 by 18, so how many

00:03:36.220 --> 00:03:37.760
square feet is that?

00:03:37.760 --> 00:03:38.920
So I'll multiply it out.

00:03:38.920 --> 00:03:41.790
18 times 14.

00:03:41.790 --> 00:03:44.580
4 times 8 is 32.

00:03:44.580 --> 00:03:48.840
1 times 4 is 4, plus 3 is 7.

00:03:48.840 --> 00:03:50.220
You have a 0.

00:03:50.220 --> 00:03:51.810
1 times 8 is 8.

00:03:51.810 --> 00:03:53.800
1 times 1 is 1.

00:03:53.800 --> 00:03:55.170
You have a 2.

00:03:55.170 --> 00:03:58.470
You have a 15.

00:03:58.470 --> 00:04:00.500
And then you have a 2.

00:04:00.500 --> 00:04:05.430
So it's 252 square feet.

00:04:05.430 --> 00:04:15.990
So 14 times 18, this
is 252 square feet.

00:04:15.990 --> 00:04:17.260
Which is kind of a
strange number.

00:04:17.260 --> 00:04:22.180
It's almost exactly $1.00 per
square foot, but not exactly

00:04:22.180 --> 00:04:23.800
equal to $1.00 per
square foot.

00:04:23.800 --> 00:04:26.240
So if you want to know per
square foot, they charged us

00:04:26.240 --> 00:04:33.830
$250 for 252 square feet.

00:04:33.830 --> 00:04:35.150
So it's going to be a
little bit less than

00:04:35.150 --> 00:04:36.450
$1.00 per square foot.

00:04:36.450 --> 00:04:38.550
Let me confirm my math.

00:04:38.550 --> 00:04:44.040
So you have 18 times
14 is equal to 252.

00:04:44.040 --> 00:04:52.530
And then so they're charging us
$250 for 252 square feet.

00:04:52.530 --> 00:04:57.820
So it's like 99.2 cents
per square foot.

00:04:57.820 --> 00:05:00.960
So if we write this as a
function, they're going to

00:05:00.960 --> 00:05:05.720
charge us as a function of,
let's say the square footage,

00:05:05.720 --> 00:05:08.760
or let me say as a function of
the area where a is the area

00:05:08.760 --> 00:05:09.800
of our floor.

00:05:09.800 --> 00:05:11.960
They're going to charge us--
what was that number I just

00:05:11.960 --> 00:05:20.620
had-- 99.2 cents, 0.992 times
our area in square footage.

00:05:20.620 --> 00:05:22.980
This is an expression
of how much they're

00:05:22.980 --> 00:05:23.510
going to charge us.

00:05:23.510 --> 00:05:25.010
So here, what's our area?

00:05:25.010 --> 00:05:26.460
We have 9 by 10.

00:05:26.460 --> 00:05:31.550
So this is our area is equal
to 90 square feet.

00:05:31.550 --> 00:05:36.620
So the cost of 90 square feet
of carpeting for 90 square

00:05:36.620 --> 00:05:40.170
feet is going to be a little
bit less than $90.00.

00:05:40.170 --> 00:05:42.100
We know this is a little
bit less than

00:05:42.100 --> 00:05:43.630
$1.00 per square foot.

00:05:43.630 --> 00:05:44.440
We can actually do it.

00:05:44.440 --> 00:05:49.580
It's 0.992 times 90.

00:05:49.580 --> 00:05:52.830
So let's multiply
this times 90.

00:05:52.830 --> 00:06:01.410
It's equal to like $89.29.

00:06:01.410 --> 00:06:03.830
If I were to have written this
problem, I would have made

00:06:03.830 --> 00:06:07.780
this number 250 so that it would
have been exactly $1.00

00:06:07.780 --> 00:06:09.840
per square foot, and so we
wouldn't have had to break out

00:06:09.840 --> 00:06:10.560
the calculator.

00:06:10.560 --> 00:06:13.160
But anyway, I didn't
write this problem.

00:06:13.160 --> 00:06:16.590
Problem 6.

00:06:16.590 --> 00:06:21.150
It costs $12.00 to get into the
San Diego County Fair, and

00:06:21.150 --> 00:06:23.340
$1.50 per ride.

00:06:23.340 --> 00:06:27.530
If Rena spent $24.00 in total,
how may rides did she go on?

00:06:27.530 --> 00:06:31.510
So no matter what, it's going
to cost you $12.00,

00:06:31.510 --> 00:06:33.390
plus $1.50 per ride.

00:06:33.390 --> 00:06:37.950
Plus $1.50 times the number
of rides you go on.

00:06:37.950 --> 00:06:41.960
This is how much it's going to
cost for you to spend your day

00:06:41.960 --> 00:06:43.780
at the San Diego County Fair.

00:06:43.780 --> 00:06:45.370
I could write this
as a function.

00:06:45.370 --> 00:06:49.680
I could write cost as a function
of rides is $12.00

00:06:49.680 --> 00:06:51.805
plus $1.50 times the
number of rides.

00:06:54.510 --> 00:06:59.560
So we know that Rena spent
$24.00 in total.

00:06:59.560 --> 00:07:01.710
So what was the number
of rides she went on?

00:07:01.710 --> 00:07:02.820
Well we could solve for this.

00:07:02.820 --> 00:07:04.260
We could just look at
this part of the

00:07:04.260 --> 00:07:06.580
equation right there.

00:07:06.580 --> 00:07:08.680
So what do we get?

00:07:08.680 --> 00:07:11.750
We get 12 plus-- I'll
just write this as--

00:07:11.750 --> 00:07:14.990
1.5r is equal to 24.

00:07:14.990 --> 00:07:18.370
Let's subtract 12 from both
sides of the equation.

00:07:18.370 --> 00:07:21.500
So that becomes-- so I'll write
it here-- negative 12

00:07:21.500 --> 00:07:23.470
plus negative 12.

00:07:23.470 --> 00:07:25.290
Subtract the 12 from
both sides.

00:07:25.290 --> 00:07:28.770
You get 1.5r is equal to 12.

00:07:28.770 --> 00:07:30.670
And divide both sides by 1.5.

00:07:33.670 --> 00:07:43.300
You get r is equal to-- what's
12 divided by 1.5-- that is 8.

00:07:43.300 --> 00:07:45.660
1.5 times 8 is 12.

00:07:45.660 --> 00:07:46.780
1-1/2 times 8.

00:07:46.780 --> 00:07:50.380
One 8 is 8, and 1/2 of
an 8 is 4, that's 12.

00:07:50.380 --> 00:07:52.650
So she went on 8 rides.

00:07:52.650 --> 00:07:54.060
And let's verify.

00:07:54.060 --> 00:07:57.430
Let's verify by looking at
what's the cost of 8 rides?

00:07:57.430 --> 00:08:02.550
It's going to be $12.00 plus
1.5 times the 8 rides.

00:08:02.550 --> 00:08:04.850
So that's going to be
$12.00 plus, well

00:08:04.850 --> 00:08:08.200
1.5 times 8 is $12.00.

00:08:08.200 --> 00:08:11.230
So this is going to be
equal to $24.00.

00:08:11.230 --> 00:08:15.100
So we have verified
our problem.

00:08:15.100 --> 00:08:15.830
All right.

00:08:15.830 --> 00:08:17.220
Let's do one more.

00:08:17.220 --> 00:08:19.900
The sum of angles in a triangle
is 180 degrees.

00:08:19.900 --> 00:08:21.530
That's good to know
in general.

00:08:21.530 --> 00:08:25.040
You'll see that many
times in geometry.

00:08:25.040 --> 00:08:25.980
Let me write this down.

00:08:25.980 --> 00:08:28.520
So let me draw some random
triangle right there.

00:08:28.520 --> 00:08:31.250
So that is my triangle.

00:08:31.250 --> 00:08:33.684
If the second angle is twice the
size of the first angle.

00:08:33.684 --> 00:08:35.570
So let me make my first angle.

00:08:35.570 --> 00:08:37.350
Let's say that the angle is x.

00:08:37.350 --> 00:08:39.140
That's my first angle.

00:08:39.140 --> 00:08:41.450
The second angle is twice the
size of the first angle.

00:08:41.450 --> 00:08:43.730
So let's say this is the second
angle, that's 2x.

00:08:43.730 --> 00:08:46.540
And the third angle is
three times the size

00:08:46.540 --> 00:08:47.420
of the first angle.

00:08:47.420 --> 00:08:51.950
So the third angle is three
times this one, so that's 3x.

00:08:51.950 --> 00:08:55.910
What are the measures of the
angles in the triangle?

00:08:55.910 --> 00:08:57.480
Well they tell us that
they all have to

00:08:57.480 --> 00:08:59.790
add up to 180 degrees.

00:08:59.790 --> 00:09:06.030
So we get x plus 2x
plus 3x has to be

00:09:06.030 --> 00:09:08.650
equal to 180 degrees.

00:09:08.650 --> 00:09:10.450
And I could even write
this as 1x.

00:09:10.450 --> 00:09:11.990
Just remember that there's
a coefficient there.

00:09:11.990 --> 00:09:15.180
So 1 of something plus 2 of
something plus 3 of something.

00:09:15.180 --> 00:09:16.710
That's going to be 6
of the somethings.

00:09:16.710 --> 00:09:17.830
1 plus 2 plus 3.

00:09:17.830 --> 00:09:23.140
So 6x is equal to 180 degrees.

00:09:23.140 --> 00:09:25.320
Divide both sides by 6.

00:09:28.540 --> 00:09:34.110
And you get x is equal to
180 divided by 6 is 30.

00:09:34.110 --> 00:09:36.450
So this angle right
here is 30.

00:09:36.450 --> 00:09:38.410
This angle right here is
two times that, right?

00:09:38.410 --> 00:09:39.050
It's 2x.

00:09:39.050 --> 00:09:42.200
So this angle right
here is 60.

00:09:42.200 --> 00:09:44.280
And this angle right here
is three times that.

00:09:44.280 --> 00:09:46.500
That's equal to 90.

00:09:46.500 --> 00:09:49.370
So let's see if this meets
our requirements.

00:09:49.370 --> 00:09:52.950
This is a 30, 60, 90.

00:09:52.950 --> 00:09:54.360
They definitely add up to 180.

00:09:54.360 --> 00:09:58.140
30 plus 60 is 90,
plus 90 is 180.

00:09:58.140 --> 00:10:00.360
And then if we look at the
requirements, the second angle

00:10:00.360 --> 00:10:04.200
is twice the size of the first.
Right, 60 is twice 30.

00:10:04.200 --> 00:10:07.070
And the third angle is three
times the size of the first.

00:10:07.070 --> 00:10:08.980
90 is 3 times 30.

00:10:08.980 --> 00:10:10.230
So it all worked out.


Functions as graphs

WEBVTT
Kind: captions
Language: en

00:00:00.630 --> 00:00:04.320
Let's do some problems with
plotting points and graphical

00:00:04.320 --> 00:00:06.110
representations of functions.

00:00:06.110 --> 00:00:09.070
So I didn't write the problem
here, but what they ask us to

00:00:09.070 --> 00:00:11.590
do here is figure out
the coordinates of

00:00:11.590 --> 00:00:13.800
each of these points.

00:00:13.800 --> 00:00:15.200
So let's just do that.

00:00:15.200 --> 00:00:19.350
So first we have this
point a right there.

00:00:19.350 --> 00:00:23.240
So I'll just write a is
at the coordinate.

00:00:23.240 --> 00:00:27.090
So we write the x-coordinate
first. So its x-coordinate is

00:00:27.090 --> 00:00:31.780
how far to the left or the right
it is of the origin.

00:00:31.780 --> 00:00:37.830
And it is 1, 2, 3, 4, 5, 6 to
the left, or it is negative 6

00:00:37.830 --> 00:00:41.230
of the origin, or it's at the
coordinate negative 6.

00:00:41.230 --> 00:00:44.420
So negative 6, that's
right there.

00:00:44.420 --> 00:00:47.950
And then its y-coordinate, which
is how high it is, that

00:00:47.950 --> 00:00:49.330
is right there.

00:00:49.330 --> 00:00:50.850
That's 4.

00:00:50.850 --> 00:00:53.790
Negative 6, 4.

00:00:53.790 --> 00:00:57.060
Let's do b right there.

00:00:57.060 --> 00:01:00.890
b is at the coordinate-- let's
see, its x-coordinate, and

00:01:00.890 --> 00:01:04.580
just drop down there, is 7.

00:01:04.580 --> 00:01:09.950
And its y-coordinate, how
high it is, is 6.

00:01:09.950 --> 00:01:10.730
All right.

00:01:10.730 --> 00:01:11.980
Let's do c.

00:01:14.830 --> 00:01:18.410
The x-coordinate, and you can
read that, it's negative 8.

00:01:18.410 --> 00:01:21.110
It's 8 to the left
of the origin.

00:01:21.110 --> 00:01:22.550
Negative 8.

00:01:22.550 --> 00:01:26.000
And its y-coordinate, it's
2 below the origin.

00:01:26.000 --> 00:01:31.120
So its y-coordinate right
there is negative 2.

00:01:31.120 --> 00:01:34.200
This is, I think,
not too painful.

00:01:34.200 --> 00:01:37.590
Part d, or coordinate
d, or point d.

00:01:37.590 --> 00:01:42.880
Point d, its y-coordinate,
it's at 4.

00:01:42.880 --> 00:01:46.990
And then its-- I'm sorry, its
x-coordinate is at 4, and then

00:01:46.990 --> 00:01:50.860
its y-coordinate, how far down
it is, or in this vertical

00:01:50.860 --> 00:01:53.730
axis, it is-- this looks
like negative 7.

00:01:56.450 --> 00:02:00.025
And then, finally,
we are on e.

00:02:00.025 --> 00:02:02.720
I'm picking out a nice
color for e.

00:02:02.720 --> 00:02:05.420
e right there.

00:02:05.420 --> 00:02:07.460
Its x-coordinate is 5.

00:02:07.460 --> 00:02:10.340
It's right on the x-axis
at x is equal to 5.

00:02:10.340 --> 00:02:13.350
And its y-coordinate-- well,
it isn't above or below the

00:02:13.350 --> 00:02:16.960
origin or the x-axis, so
its y-coordinate is 0.

00:02:16.960 --> 00:02:19.930
You could draw a line and
go straight there to 0.

00:02:19.930 --> 00:02:21.120
So there we go.

00:02:21.120 --> 00:02:22.940
We figured out all of
those coordinates.

00:02:22.940 --> 00:02:27.220
Now this problem 5 here-- let me
scroll over a little bit--

00:02:27.220 --> 00:02:32.150
they say determine whether each
relation is a function.

00:02:32.150 --> 00:02:35.990
So here the trick is to realize
that a relation is not

00:02:35.990 --> 00:02:40.040
a function if they define two
values for a given x.

00:02:40.040 --> 00:02:41.660
Let me give an example here.

00:02:41.660 --> 00:02:51.680
So if I wrote that f of x is
equal to 5 if x is equal to 1,

00:02:51.680 --> 00:02:55.820
or it's equal to 6 if
x is equal to 1.

00:02:55.820 --> 00:02:57.070
This makes no sense.

00:03:00.310 --> 00:03:01.710
Why doesn't it make any sense?

00:03:01.710 --> 00:03:03.810
Because if I put a 1 in there,
I don't know what

00:03:03.810 --> 00:03:04.510
I'm going to produce?

00:03:04.510 --> 00:03:06.000
Am I going to produce a 5?

00:03:06.000 --> 00:03:07.460
Am I going to produce a 6?

00:03:07.460 --> 00:03:09.230
This is a badly structured
function.

00:03:09.230 --> 00:03:11.720
This is not a function.

00:03:11.720 --> 00:03:15.360
So if there's any situation for
the same input value, they

00:03:15.360 --> 00:03:18.880
define two different output
values, we're not going to

00:03:18.880 --> 00:03:20.030
have a function.

00:03:20.030 --> 00:03:22.860
So let's see if they
do that here.

00:03:22.860 --> 00:03:26.120
So in this first part, part a,
they say, if you could imagine

00:03:26.120 --> 00:03:28.160
if x is 1 then y is 7.

00:03:28.160 --> 00:03:31.010
If x is 2, y is 7 as well.

00:03:31.010 --> 00:03:32.130
That is OK.

00:03:32.130 --> 00:03:35.500
You could have two x values
getting the same y value.

00:03:35.500 --> 00:03:44.020
For example, that'd be like
saying f of x is equal to 7 if

00:03:44.020 --> 00:03:46.740
x is equal to 1 or 2.

00:03:46.740 --> 00:03:48.540
This is completely fine.

00:03:48.540 --> 00:03:52.020
For two different x values you
can get the same output value,

00:03:52.020 --> 00:03:56.160
but you can't have two different
x values giving the

00:03:56.160 --> 00:04:02.120
same-- sorry-- you can't have
the same x value producing two

00:04:02.120 --> 00:04:03.060
different outputs.

00:04:03.060 --> 00:04:05.870
Because then you don't know,
hey, if you went f of 1, you

00:04:05.870 --> 00:04:08.100
don't know what f of 1 is.

00:04:08.100 --> 00:04:10.100
f of 1, is it a 5, is it 6?

00:04:10.100 --> 00:04:10.900
You don't know.

00:04:10.900 --> 00:04:12.780
Here you know what f
of 1 is, it's 7.

00:04:12.780 --> 00:04:15.440
Here you know what f
of 2 is, it's 7.

00:04:15.440 --> 00:04:17.050
So, so far so good.

00:04:17.050 --> 00:04:18.850
So when you have 2,
you have a 7.

00:04:18.850 --> 00:04:21.570
When your input is
3, you get an 8.

00:04:21.570 --> 00:04:23.810
When your input is
4, you get an 8.

00:04:23.810 --> 00:04:27.840
So, for example, our function
definition, so it's 8 if x is

00:04:27.840 --> 00:04:30.550
equal to 3 or 4.

00:04:30.550 --> 00:04:31.960
And then 5.

00:04:31.960 --> 00:04:37.770
And then our function is equal
to 9 if x is equal to 5.

00:04:37.770 --> 00:04:40.010
So part a, this is our function
definition, right

00:04:40.010 --> 00:04:42.250
here, for part a, which is a
completely legitimate function

00:04:42.250 --> 00:04:42.680
definition.

00:04:42.680 --> 00:04:46.080
You give me any value 1, 2, 3,
4, or 5, which would be the

00:04:46.080 --> 00:04:51.740
domain in this situation, and I
will tell you what the value

00:04:51.740 --> 00:04:54.000
of that function is at
any of those points.

00:04:54.000 --> 00:04:56.230
And the range would
be 7, 8, or 9.

00:04:56.230 --> 00:04:59.660
So part a is definitely
a function.

00:04:59.660 --> 00:05:01.110
Now part b.

00:05:01.110 --> 00:05:02.940
Let's see, if x is 1, y is 1.

00:05:02.940 --> 00:05:06.140
But then they say, if x
is 1, y is negative 1.

00:05:06.140 --> 00:05:07.560
Well, that makes no sense.

00:05:07.560 --> 00:05:09.760
They're doing this right here.

00:05:09.760 --> 00:05:12.202
They're trying to make a
function, where they say this

00:05:12.202 --> 00:05:16.790
function is going to be equal
to 1 if x is equal to 1, but

00:05:16.790 --> 00:05:21.180
then it's going to be equal to
negative 1 if x is equal to 1.

00:05:21.180 --> 00:05:24.370
So if I were to take f of
1, I don't know what

00:05:24.370 --> 00:05:24.840
it's going to be.

00:05:24.840 --> 00:05:25.700
Is it 1 or negative 1?

00:05:25.700 --> 00:05:27.910
I don't know, do I take this
1 or do I take that 1?

00:05:27.910 --> 00:05:29.870
So this is not a function.

00:05:29.870 --> 00:05:32.850
So part b is not a function.

00:05:32.850 --> 00:05:36.270
That relation is
not a function.

00:05:36.270 --> 00:05:38.060
All right.

00:05:38.060 --> 00:05:40.450
Let's do a couple more.

00:05:40.450 --> 00:05:42.520
Problem 6.

00:05:42.520 --> 00:05:45.185
Write the function rule
for each graph.

00:05:45.185 --> 00:05:48.450
So we have this little
v looking thing.

00:05:48.450 --> 00:05:50.910
So we could write it
a couple of ways.

00:05:50.910 --> 00:05:52.425
Let's call it f of x.

00:05:52.425 --> 00:05:55.120
And you could call it g of x, or
h of x, but if you haven't

00:05:55.120 --> 00:05:57.440
used your f yet, people
tend to use f of x.

00:05:57.440 --> 00:05:58.940
So this is x.

00:05:58.940 --> 00:06:01.980
So let's see, it looks like
it's one line when x is

00:06:01.980 --> 00:06:04.320
greater than the 0, and
another line when x

00:06:04.320 --> 00:06:06.320
is less than 0.

00:06:06.320 --> 00:06:10.960
So it's one thing when x is,
let's say, greater than or

00:06:10.960 --> 00:06:11.990
equal to 0.

00:06:11.990 --> 00:06:14.470
And another thing when
x is less than 0.

00:06:14.470 --> 00:06:16.490
And I'm going to merge
the two in a second.

00:06:16.490 --> 00:06:18.520
So what does the line
look like here?

00:06:18.520 --> 00:06:21.280
When x is 0, y is 0.

00:06:21.280 --> 00:06:23.860
When x is 2, y is 1.

00:06:23.860 --> 00:06:26.690
When x is 4, y is 2.

00:06:26.690 --> 00:06:28.370
It looks like no matter
what x is, y is going

00:06:28.370 --> 00:06:29.630
to be 1/2 of that.

00:06:29.630 --> 00:06:32.630
When x is 6, y is 3.

00:06:32.630 --> 00:06:38.440
So it's equal to 1/2x, when
x is greater than 0.

00:06:38.440 --> 00:06:41.070
And then when x is less
than 0, when x is

00:06:41.070 --> 00:06:42.930
negative 2, y is 1.

00:06:42.930 --> 00:06:45.620
When x is negative 4, y is 2.

00:06:45.620 --> 00:06:48.750
So here it looks like it's
the negative 1/2 of it.

00:06:48.750 --> 00:06:52.600
Negative 1/2 times negative
4 is positive 2.

00:06:52.600 --> 00:06:56.790
So it's negative 1/2 times
x, when x is less than 0.

00:06:56.790 --> 00:06:58.870
So this is a completely
legitimate answer.

00:06:58.870 --> 00:07:01.660
But if we wanted to make it a
little bit simpler, or clean

00:07:01.660 --> 00:07:05.670
it up a little bit, we could
write this function definition

00:07:05.670 --> 00:07:09.640
as f of x being equal to--
instead of dividing it between

00:07:09.640 --> 00:07:12.340
greater than or less than 0,
let's just take the absolute

00:07:12.340 --> 00:07:16.890
value of x and then multiply
that times 1/2.

00:07:16.890 --> 00:07:18.920
Because here, that obviously
works for positive values,

00:07:18.920 --> 00:07:21.430
because the absolute value
of x will be equal to x.

00:07:21.430 --> 00:07:24.402
But then for negative values,
the absolute value of x is

00:07:24.402 --> 00:07:26.170
equal to negative x.

00:07:26.170 --> 00:07:28.150
So you take negative 2 here.

00:07:28.150 --> 00:07:30.910
Negative 2, take the absolute
value, you get 2

00:07:30.910 --> 00:07:33.570
times 1/2 is 1.

00:07:33.570 --> 00:07:35.890
So either of these would
be legitimate function

00:07:35.890 --> 00:07:37.630
definitions.

00:07:37.630 --> 00:07:38.870
Problem 10.

00:07:38.870 --> 00:07:42.270
Use the vertical line test-- let
me switch colors for this

00:07:42.270 --> 00:07:45.500
one-- use the vertical line test
to determine whether each

00:07:45.500 --> 00:07:46.570
relation is a function.

00:07:46.570 --> 00:07:49.530
Now, the vertical line test is
just a visual way of doing

00:07:49.530 --> 00:07:52.980
exactly what we did in this
problem over here.

00:07:52.980 --> 00:07:57.370
Something is only a function
if for a given x value, you

00:07:57.370 --> 00:08:00.670
only have one y value.

00:08:00.670 --> 00:08:03.670
So for example, when they say a
vertical line, that means at

00:08:03.670 --> 00:08:05.970
any point I should be able to
draw a vertical line on this

00:08:05.970 --> 00:08:09.240
function and only intersect
it once.

00:08:09.240 --> 00:08:13.420
Because a vertical line says
when x is equal to 3, there's

00:08:13.420 --> 00:08:16.400
only one value that I can
get on our function.

00:08:16.400 --> 00:08:17.810
That's what I'm doing with
a vertical line.

00:08:17.810 --> 00:08:22.510
When x is equal to 0, there's
only one point on the function

00:08:22.510 --> 00:08:26.290
that is mapped from
x is equal to 0.

00:08:26.290 --> 00:08:29.150
So this one right here
is a function.

00:08:29.150 --> 00:08:31.950
Any vertical line you draw
will only intersect the

00:08:31.950 --> 00:08:33.559
function once.

00:08:33.559 --> 00:08:37.039
This one, very clearly, you
draw any vertical line.

00:08:37.039 --> 00:08:38.970
You draw any vertical line
here and you're going to

00:08:38.970 --> 00:08:41.700
intersect the graph twice.

00:08:41.700 --> 00:08:44.110
So this is saying, this vertical
line that I just

00:08:44.110 --> 00:08:49.590
drew, this is essentially saying
f of 2 could be this

00:08:49.590 --> 00:08:50.400
point over here.

00:08:50.400 --> 00:08:54.060
This looks like, I don't know,
maybe it's equal to, it could

00:08:54.060 --> 00:08:58.810
be equal to 1.9 or it
could be-- what is

00:08:58.810 --> 00:09:00.790
this?-- negative 1.9.

00:09:00.790 --> 00:09:02.620
This is not a proper function
definition.

00:09:02.620 --> 00:09:06.010
I don't know whether f of 2 is
this value or that value.

00:09:06.010 --> 00:09:10.300
I don't know whether f of 0 is
that value or that value.

00:09:10.300 --> 00:09:12.510
And you could keep going across
the whole number line.

00:09:12.510 --> 00:09:14.950
You don't know whether f of 5
is this value or this value.

00:09:14.950 --> 00:09:16.870
So this is not a function.

00:09:16.870 --> 00:09:18.990
This relation is
not a function.

00:09:25.280 --> 00:09:27.620
And it's the same logic as we
did before, but we're saying

00:09:27.620 --> 00:09:28.290
the vertical line.

00:09:28.290 --> 00:09:30.350
You could draw a vertical line
and you can intersect the

00:09:30.350 --> 00:09:34.110
relation or the graph twice,
so it's not a function.


Domain and range of a function

WEBVTT
Kind: captions
Language: en

00:00:00.590 --> 00:00:04.300
Let's do some example problems
dealing with functions and

00:00:04.300 --> 00:00:05.990
their domains and ranges.

00:00:05.990 --> 00:00:10.330
Just as a review, a function
is just an operator-- let's

00:00:10.330 --> 00:00:12.860
say this function is f; that
tends to be the most typical

00:00:12.860 --> 00:00:16.309
letter for functions-- that
operates on some input, in

00:00:16.309 --> 00:00:21.140
this case, the input is x, and
it produces some output y.

00:00:21.140 --> 00:00:24.950
Or you could view it as you take
some input x, put it into

00:00:24.950 --> 00:00:28.740
your function box f, and it's
going to operate on it and

00:00:28.740 --> 00:00:30.980
produce some y.

00:00:30.980 --> 00:00:36.390
And the set of values that x
can take on is the domain.

00:00:36.390 --> 00:00:39.320
The set of values that f can
legitimately operate on,

00:00:39.320 --> 00:00:40.390
that's the domain.

00:00:40.390 --> 00:00:43.110
And the set of values
y that f can

00:00:43.110 --> 00:00:47.140
produce, that is the range.

00:00:47.140 --> 00:00:50.080
Now, with that in mind, let's
figure out, one, the function

00:00:50.080 --> 00:00:53.440
definitions for each of these
problems here, these example

00:00:53.440 --> 00:00:58.250
problems, and then figure out
the domains and the ranges.

00:00:58.250 --> 00:01:02.140
So here we have Dustin charges
$10.00 per hour-- so let me

00:01:02.140 --> 00:01:05.870
write that down-- $10.00 per
hour for mowing lawns.

00:01:05.870 --> 00:01:08.900
So how much does he charge
as a function of hours?

00:01:08.900 --> 00:01:12.020
So let's say Dustin charges, so
the function, I'll call it

00:01:12.020 --> 00:01:13.670
d for Dustin.

00:01:13.670 --> 00:01:16.460
Dustin charges as a
function of hours.

00:01:16.460 --> 00:01:17.385
It depends on hours.

00:01:17.385 --> 00:01:20.440
The input into our box
is going to be

00:01:20.440 --> 00:01:21.950
hours that he worked.

00:01:21.950 --> 00:01:26.290
It's going to be equal to the
number of hours times 10.

00:01:26.290 --> 00:01:31.020
It's going to be 10 times-- I
won't write the times there.

00:01:31.020 --> 00:01:33.640
Maybe I can just write 10h.

00:01:33.640 --> 00:01:37.490
And, of course, he can't work
negative hours, so we could

00:01:37.490 --> 00:01:41.210
write h is going to be greater
than or equal to 0.

00:01:41.210 --> 00:01:42.890
That's our function
definition.

00:01:42.890 --> 00:01:45.740
If you say, hey, how much is he
going to charge for working

00:01:45.740 --> 00:01:46.750
half an hour?

00:01:46.750 --> 00:01:51.160
You put 1/2 in here, which
essentially substitutes this h

00:01:51.160 --> 00:01:52.120
with a 1/2.

00:01:52.120 --> 00:01:53.240
You do 10 times 1/2.

00:01:53.240 --> 00:01:55.970
Let me write that down
on the side here.

00:01:55.970 --> 00:01:58.780
So Dustin's going to charge
for 1/2 hour.

00:01:58.780 --> 00:02:01.470
He's going to charge 10--
wherever you see the h,

00:02:01.470 --> 00:02:07.050
replace it with 1/2-- times 1/2,
which is equal to $5.00.

00:02:07.050 --> 00:02:08.210
That's our function
definition.

00:02:08.210 --> 00:02:11.660
I was just showing you an
example of applying it.

00:02:11.660 --> 00:02:14.023
Now, let's figure out the
domain and the range.

00:02:17.980 --> 00:02:20.480
So I almost explicitly say
it here, you can't

00:02:20.480 --> 00:02:21.500
work negative hours.

00:02:21.500 --> 00:02:25.900
This number, this function,
is not defined if h

00:02:25.900 --> 00:02:28.380
is less than 0.

00:02:28.380 --> 00:02:29.630
So we could say non-negative.

00:02:33.790 --> 00:02:38.580
And we could say real numbers,
which it could be everything

00:02:38.580 --> 00:02:41.700
including pi and e
and all of that.

00:02:41.700 --> 00:02:43.320
But you can't legitimately
charge

00:02:43.320 --> 00:02:45.760
someone 10 times pi dollars.

00:02:45.760 --> 00:02:49.760
That's not a legitimate amount
that one can charge someone in

00:02:49.760 --> 00:02:51.440
the real world, because at the
end of the day, you have to

00:02:51.440 --> 00:02:53.050
round to the penny.

00:02:53.050 --> 00:02:54.370
You can't actually
charge them that.

00:02:54.370 --> 00:02:56.040
So any number that he
charges is going to

00:02:56.040 --> 00:02:58.010
be a rational number.

00:02:58.010 --> 00:02:59.880
It can be expressed
as a fraction.

00:02:59.880 --> 00:03:02.680
So we can kind of take out
numbers like e and pi.

00:03:02.680 --> 00:03:04.440
So if we want to be really cute
about it, we would say

00:03:04.440 --> 00:03:05.860
non-negative.

00:03:05.860 --> 00:03:11.210
rational numbers.

00:03:11.210 --> 00:03:14.160
And these are just numbers that
can be expressed as a

00:03:14.160 --> 00:03:16.010
fraction, which are most
numbers, just not these

00:03:16.010 --> 00:03:23.040
numbers that just keep not
repeating and all of that.

00:03:23.040 --> 00:03:26.700
So non-negative rational
numbers is what he

00:03:26.700 --> 00:03:28.890
can input in here.

00:03:28.890 --> 00:03:36.060
And then the range, which is
the valid-- once again,

00:03:36.060 --> 00:03:38.750
whatever number you put in for
h, remember, you can only put

00:03:38.750 --> 00:03:42.350
in non-negative values for h,
non-negative rational values

00:03:42.350 --> 00:03:46.310
for h, so whatever numbers you
put in for h, you're going to

00:03:46.310 --> 00:03:50.530
get positive values for how much
he charges for 10h, for

00:03:50.530 --> 00:03:51.880
the value of the function.

00:03:51.880 --> 00:03:58.090
So once again it's going to be
non-negative, and if you put a

00:03:58.090 --> 00:04:00.700
rational number here and it's
being expressed as a fraction

00:04:00.700 --> 00:04:02.130
and you multiply it by 10,
it's still going to be a

00:04:02.130 --> 00:04:03.380
rational number.

00:04:08.030 --> 00:04:09.840
And if you want to be really
cute, you could say any number

00:04:09.840 --> 00:04:11.860
that could be expressed
as a dollar sign.

00:04:11.860 --> 00:04:13.730
But anyway, I don't want to get
too cute on this problem.

00:04:13.730 --> 00:04:14.960
I think you get the idea.

00:04:14.960 --> 00:04:15.660
Let's do more

00:04:15.660 --> 00:04:17.649
problems. All right.

00:04:17.649 --> 00:04:21.100
Here we have Maria doing
some tutoring.

00:04:21.100 --> 00:04:25.410
Maria charges $25.00 per hour
for tutoring math with a

00:04:25.410 --> 00:04:27.740
minimum charge of $15.00.

00:04:27.740 --> 00:04:32.010
So Maria charges as a
function of hours.

00:04:32.010 --> 00:04:36.710
So she's going to charge
$15.00 if you don't get

00:04:36.710 --> 00:04:37.650
enough hours in.

00:04:37.650 --> 00:04:41.510
So at $25.00 per hour, in order
to make $15.00, you've

00:04:41.510 --> 00:04:45.980
got to work 15/25, so
it's 3/5 of an hour.

00:04:45.980 --> 00:04:53.100
So if her hours are less than
3/5 and greater than or equal

00:04:53.100 --> 00:04:57.270
to 0, she's going to
charge $15.00.

00:04:57.270 --> 00:05:00.490
Because if she only worked 1/5
of an hour, the bill would

00:05:00.490 --> 00:05:02.150
have been $5.00, but
she says she has a

00:05:02.150 --> 00:05:04.120
minimum charge of $15.00.

00:05:04.120 --> 00:05:08.390
So she's going to charge $15.00
up until 3/5 of an

00:05:08.390 --> 00:05:10.110
hour, or 36 minutes.

00:05:10.110 --> 00:05:15.365
And then after that, for h
greater than 3/5, she's going

00:05:15.365 --> 00:05:17.400
to charge $25.00 an hour.

00:05:17.400 --> 00:05:21.310
She's going to charge
25 times h.

00:05:21.310 --> 00:05:24.990
So that's her function
definition right there.

00:05:24.990 --> 00:05:28.890
Now, what is the domain,
the domain for

00:05:28.890 --> 00:05:32.050
Maria's billing function?

00:05:32.050 --> 00:05:34.940
Well, once again, this
is only defined for

00:05:34.940 --> 00:05:36.190
non-negative numbers.

00:05:40.470 --> 00:05:43.030
And by the same logic we used
here, we got a little cute.

00:05:43.030 --> 00:05:47.860
We said, oh, she really can't
charge e hours or pi hours.

00:05:47.860 --> 00:05:49.590
That's not realistic
for her to measure.

00:05:49.590 --> 00:05:51.900
Everything's going to be
expressible as a fraction.

00:05:51.900 --> 00:05:53.930
For example, even 5,
if you say 5 hours,

00:05:53.930 --> 00:05:54.820
that's still a fraction.

00:05:54.820 --> 00:05:56.790
That's 5 over 1.

00:05:56.790 --> 00:06:01.840
So let's say non-negative
rational numbers.

00:06:01.840 --> 00:06:05.260
And what's the range of her
charging function or her

00:06:05.260 --> 00:06:07.010
billing function?

00:06:07.010 --> 00:06:10.660
Well, it has to be
at least $15.00.

00:06:10.660 --> 00:06:13.005
No matter what, she's going to
charge $15.00, so it's going

00:06:13.005 --> 00:06:15.430
to be $15.00 or more.

00:06:15.430 --> 00:06:22.920
So it's going to be rational
numbers greater

00:06:22.920 --> 00:06:30.420
than or equal to $15.00.

00:06:30.420 --> 00:06:33.200
There's no situation in which
she will charge $14.00.

00:06:33.200 --> 00:06:34.230
There's no situation
in which she would

00:06:34.230 --> 00:06:35.960
charge negative $1.00.

00:06:35.960 --> 00:06:40.380
Everything's going to be greater
than or equal to 15.

00:06:40.380 --> 00:06:42.950
So now we have these more
abstract function definitions,

00:06:42.950 --> 00:06:45.260
so now can stick really
any number in it.

00:06:45.260 --> 00:06:45.930
Well, we're not going
to deal with

00:06:45.930 --> 00:06:48.010
complexes, any real number.

00:06:48.010 --> 00:06:49.610
But let's see if they limit
it a little bit.

00:06:49.610 --> 00:06:50.453
So we have f of x.

00:06:50.453 --> 00:06:56.400
Let me rewrite it. f of x is
equal to 15x minus 12.

00:06:56.400 --> 00:06:58.870
So what's the domain here?

00:06:58.870 --> 00:07:03.420
What's our domain?

00:07:03.420 --> 00:07:06.016
Well, I can stick any real
number x in here and I'm just

00:07:06.016 --> 00:07:08.230
going to multiply it by 15
and then subtract 12.

00:07:08.230 --> 00:07:09.280
I could put pi there.

00:07:09.280 --> 00:07:10.780
I could put the square
of 2 there.

00:07:10.780 --> 00:07:12.450
I could put e there.

00:07:12.450 --> 00:07:13.900
I could do all sorts
of crazy things.

00:07:13.900 --> 00:07:15.935
So I'm going to say
all real numbers.

00:07:21.080 --> 00:07:22.880
Any real number I put in for
x, I'm going to get a

00:07:22.880 --> 00:07:24.540
legitimate output.

00:07:24.540 --> 00:07:26.705
And then what is my range?

00:07:29.460 --> 00:07:33.730
Well, once again, this can take
on any value out there.

00:07:33.730 --> 00:07:35.410
I can get to any negative
value if I

00:07:35.410 --> 00:07:36.990
make x negative enough.

00:07:36.990 --> 00:07:38.970
I'm going to subtract 12 from
it, but I could get to any

00:07:38.970 --> 00:07:39.660
negative value.

00:07:39.660 --> 00:07:40.920
I could get to zero.

00:07:40.920 --> 00:07:42.880
I could get to any
positive value.

00:07:42.880 --> 00:07:49.096
So this is also going to
be all real numbers.

00:07:51.840 --> 00:07:52.900
Now we have this function.

00:07:52.900 --> 00:07:56.560
It says f of x is equal to--
they forgot the equal sign--

00:07:56.560 --> 00:07:59.970
2x squared plus 5.

00:07:59.970 --> 00:08:03.660
So what is the domain?

00:08:03.660 --> 00:08:07.020
What are all of the valid values
for x that I could

00:08:07.020 --> 00:08:07.800
stick here?

00:08:07.800 --> 00:08:08.700
Well, I could stick
anything here.

00:08:08.700 --> 00:08:09.550
I could put e there.

00:08:09.550 --> 00:08:11.030
2 times e squared plus 5.

00:08:11.030 --> 00:08:12.810
I could put a square
root of 2 there.

00:08:12.810 --> 00:08:14.220
I could put a negative
number there.

00:08:14.220 --> 00:08:16.910
So I could really take
on any real number.

00:08:16.910 --> 00:08:26.260
All real numbers, positive or
negative or otherwise, or even

00:08:26.260 --> 00:08:27.420
non-rational.

00:08:27.420 --> 00:08:30.590
And then what is
the range here?

00:08:30.590 --> 00:08:32.260
And this is interesting.

00:08:32.260 --> 00:08:35.520
Because no matter what number
I put here, this value right

00:08:35.520 --> 00:08:38.559
here is going to be
greater than zero.

00:08:38.559 --> 00:08:41.045
This is going to be, right
here, non-negative.

00:08:44.110 --> 00:08:45.490
Even if you put a really
negative number here, you're

00:08:45.490 --> 00:08:47.040
going to square it, so it's
going to become a really

00:08:47.040 --> 00:08:48.640
positive number when
you square it.

00:08:48.640 --> 00:08:52.460
So this expression right here
is going to be non-negative.

00:08:52.460 --> 00:08:55.890
So if you take a non-negative
number and you add it to 5,

00:08:55.890 --> 00:08:58.120
you're always going to get
something greater than or

00:08:58.120 --> 00:09:00.140
equal to 5.

00:09:00.140 --> 00:09:02.250
Worst case, this is a zero,
and then you get a 5.

00:09:02.250 --> 00:09:04.690
But if this is either slightly
less than zero or slightly

00:09:04.690 --> 00:09:07.750
more than zero, you're going to
get values greater than 5.

00:09:07.750 --> 00:09:16.740
So the range is all real
numbers greater

00:09:16.740 --> 00:09:19.450
than or equal to 5.

00:09:19.450 --> 00:09:21.980
Your f of x could never take
on the value zero.

00:09:21.980 --> 00:09:25.080
There's no way to get zero.

00:09:25.080 --> 00:09:26.330
Problem 5.

00:09:29.690 --> 00:09:34.900
They have f of x is
equal to 1 over x.

00:09:34.900 --> 00:09:38.550
So, once again, our domain.

00:09:38.550 --> 00:09:40.550
Well, we could put any
value in for x.

00:09:40.550 --> 00:09:41.960
Any real number will
work here.

00:09:41.960 --> 00:09:43.480
We're just going to take
the inverse of it.

00:09:43.480 --> 00:09:48.830
So this is going to be all
reals, all real numbers.

00:09:48.830 --> 00:09:51.380
And then what is the range?

00:09:51.380 --> 00:09:53.450
This is interesting as well.

00:09:53.450 --> 00:09:58.640
Because no matter what I put
here, I can put anything here.

00:09:58.640 --> 00:10:00.550
Oh, actually, let me be
very, very careful.

00:10:00.550 --> 00:10:03.670
This is not defined for
x is equal to 0.

00:10:03.670 --> 00:10:08.980
All real that are not
equal to zero.

00:10:08.980 --> 00:10:11.970
If I put zero here, I'm
going to get 1/0.

00:10:11.970 --> 00:10:13.260
It will be undefined.

00:10:13.260 --> 00:10:14.020
Let me write that down.

00:10:14.020 --> 00:10:18.690
I almost made a mistake.
f of 0 is undefined.

00:10:18.690 --> 00:10:20.380
And that literally means
we don't know what

00:10:20.380 --> 00:10:21.150
to do with the zero.

00:10:21.150 --> 00:10:24.680
We have not defined a way
to operate on a zero.

00:10:24.680 --> 00:10:26.400
So f of 0 is undefined.

00:10:26.400 --> 00:10:31.020
So the domain is all reals
except for zero.

00:10:31.020 --> 00:10:33.120
And now what's the range?

00:10:33.120 --> 00:10:37.210
Well, once again, we can take on
any value here except zero.

00:10:37.210 --> 00:10:38.465
We could try to approach zero.

00:10:38.465 --> 00:10:41.220
If x got really, really, really,
really, really large,

00:10:41.220 --> 00:10:43.500
then this will approach zero.

00:10:43.500 --> 00:10:46.890
If x got really, really
negative, it'll approach zero.

00:10:46.890 --> 00:10:51.130
But in no circumstance will we
actually ever get to zero.

00:10:51.130 --> 00:10:58.210
So once again, all reals that
are not equal to zero.

00:10:58.210 --> 00:11:01.830
All real numbers except
for zero.

00:11:01.830 --> 00:11:02.370
All right.

00:11:02.370 --> 00:11:03.080
Problem 6.

00:11:03.080 --> 00:11:06.300
What is the range of the
function y is equal to x

00:11:06.300 --> 00:11:10.100
squared minus 5 when the
domain is-- so they're

00:11:10.100 --> 00:11:11.300
defining the domain.

00:11:11.300 --> 00:11:12.110
They're restricting it.

00:11:12.110 --> 00:11:16.470
They're saying the domain is
just the numbers minus 2,

00:11:16.470 --> 00:11:20.410
minus 1, 0, 1, and 2.

00:11:20.410 --> 00:11:24.640
So this function definition
really should say f of x is

00:11:24.640 --> 00:11:32.660
equal to x squared minus 5 for x
is equal to-- we could write

00:11:32.660 --> 00:11:39.180
any member for x is a member
of the set minus 2,

00:11:39.180 --> 00:11:43.070
minus 1, 0, 1, 2.

00:11:43.070 --> 00:11:45.420
It's not defined for x being
anything else to that.

00:11:45.420 --> 00:11:46.860
You don't know what
to do with it.

00:11:46.860 --> 00:11:49.600
You can only apply it if
x is one of these.

00:11:49.600 --> 00:11:50.590
So what's the range?

00:11:50.590 --> 00:11:54.035
Well, the range is all of the
functions, all the values that

00:11:54.035 --> 00:11:56.200
f of x can take on.

00:11:56.200 --> 00:12:03.960
So the range is going to be the
set of values that we get

00:12:03.960 --> 00:12:06.900
when we put in all of
these different x's.

00:12:06.900 --> 00:12:09.280
So let's try this first
one: negative 2.

00:12:09.280 --> 00:12:12.040
Negative 2 squared minus 5.

00:12:12.040 --> 00:12:13.830
That's 4 minus 5.

00:12:13.830 --> 00:12:15.615
That's negative 1.

00:12:15.615 --> 00:12:17.220
Let's do negative 1.

00:12:17.220 --> 00:12:19.910
Negative 1 squared minus 5.

00:12:19.910 --> 00:12:22.000
That is 1 minus 5.

00:12:22.000 --> 00:12:24.050
That is negative 4.

00:12:24.050 --> 00:12:25.140
You have 0.

00:12:25.140 --> 00:12:26.450
0 squared.

00:12:26.450 --> 00:12:30.750
So f of 0, you have 0 squared
minus 5, that's negative 5.

00:12:30.750 --> 00:12:31.560
And you have 1.

00:12:31.560 --> 00:12:34.940
1 squared minus 5, that's
negative 4, so it's

00:12:34.940 --> 00:12:36.180
already in our range.

00:12:36.180 --> 00:12:37.630
Then you have 2.

00:12:37.630 --> 00:12:41.110
2 squared minus 5, that's
negative 1,

00:12:41.110 --> 00:12:42.020
already in our range.

00:12:42.020 --> 00:12:46.190
So that's all of the values that
this function can take

00:12:46.190 --> 00:12:47.670
on, given that we've
restricted it

00:12:47.670 --> 00:12:49.380
to only these inputs.