Finding trig functions of special angles example

WEBVTT
Kind: captions
Language: en

00:00:00.282 --> 00:00:01.900
Voiceover:The first thing
I want to do in this video

00:00:01.900 --> 00:00:04.797
is given the information
given about this triangle.

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Figuring out the unknowns,

00:00:06.481 --> 00:00:09.353
figuring out the length here this value a,

00:00:09.353 --> 00:00:12.089
figuring out this length in orange, b,

00:00:12.089 --> 00:00:14.491
figuring out the measure of this angle

00:00:14.491 --> 00:00:16.859
which we'll call alpha in
figuring out in radians

00:00:16.859 --> 00:00:18.578
and figuring out the
measure of this angle,

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right over here in blue
that we could call beta.

00:00:20.540 --> 00:00:22.369
I encourage you to pause the video now

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and given the information that we have

00:00:24.598 --> 00:00:26.326
try to figure out these unknowns.

00:00:27.787 --> 00:00:29.694
Assuming you've given it a try,

00:00:29.694 --> 00:00:32.582
so let's see what we can determine.

00:00:32.582 --> 00:00:34.876
We know two of the side lengths

00:00:34.876 --> 00:00:38.162
and they're the same, they
both have side length two

00:00:38.162 --> 00:00:41.660
and they've given us one of
the angles right over here,

00:00:41.660 --> 00:00:44.819
this yellow angle has a measure
of pi over three radians.

00:00:44.819 --> 00:00:46.772
One thing that jumps out of me

00:00:46.772 --> 00:00:48.996
is that this is an isosceles triangle.

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An isosceles triangle has two sides

00:00:52.799 --> 00:00:55.220
that have the same length

00:00:55.220 --> 00:00:58.457
so this side is congruent to
that side right over there.

00:00:58.457 --> 00:01:00.321
That means that the base angles,

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the corresponding base angles

00:01:02.055 --> 00:01:03.315
are also going to be congruent

00:01:03.315 --> 00:01:04.116
and that's just because

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this is a property of isosceles triangles.

00:01:05.707 --> 00:01:07.238
These two are congruent

00:01:07.238 --> 00:01:09.099
so the two base angles
are going to be congruent.

00:01:09.099 --> 00:01:13.157
That means that beta is also equal to

00:01:13.157 --> 00:01:15.981
pi over three radians.

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This is also beta is also equal to

00:01:19.374 --> 00:01:23.404
pi over three radians.

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Now let's think about ...

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Now what else we can
attempt to figure out.

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Well if we know beta is pi over three

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and we know this angle in yellow

00:01:33.213 --> 00:01:34.556
is pi over three radians,

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we can figure out what
this angle over here

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in white is

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and I don't know, let's call that angle,

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the measure of that angle let's call it x.

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If we're thinking in terms of radians

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x plus, this angle plus
pi over three radians,

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plus pi over three radians,

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this is the sum of the
interior pi over three.

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This is the sum of the interior angles

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of this larger triangle right over here,

00:02:01.717 --> 00:02:04.104
they're going to sum up to pi radians.

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If we're thinking in degrees
it would be 180 degrees

00:02:06.433 --> 00:02:09.276
so those are going to
sum up to pi radians.

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What's x going to be

00:02:11.228 --> 00:02:13.040
and this one you can
almost look straight at it.

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You say, this is going to be pi over three

00:02:14.777 --> 00:02:16.705
but if you want to solve a
little bit more systematically

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you can subtract these two from both sides

00:02:19.133 --> 00:02:25.086
and you get x is equal to,

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I'll rewrite this orange
pi is three pi over three,

00:02:30.813 --> 00:02:35.906
this is the same thing as
pi, minus pi over three,

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minus pi over three,

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which of course three pi minus pi minus pi

00:02:42.129 --> 00:02:46.294
is going to be pi over three.

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These angles measure is
also pi over three radians.

00:02:50.712 --> 00:02:53.132
Now that's interesting
because these shows us

00:02:53.132 --> 00:02:55.066
that this is actually
an equilateral triangle.

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All of the sides, when all of the sides

00:02:57.939 --> 00:03:00.871
have the same angle measure that means …

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when all the angles have
the same angle measure

00:03:04.696 --> 00:03:06.792
that means all the sides
are going to be congruent.

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If this side is two, this side is two

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then that means that this
entire side right over here,

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this entire side right
over here is equal to two.

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Now what else can we
attempt to figure out.

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Well let's see

00:03:24.907 --> 00:03:27.012
we know this is pi over three radians.

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We know that this is pi,

00:03:31.842 --> 00:03:32.775
we know beta is pi over three,

00:03:32.775 --> 00:03:34.071
we know that this is …

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If we're thinking degrees 90 degrees

00:03:36.022 --> 00:03:37.716
which is the same thing as pi over two

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so if we know these two
angles of a triangle

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we can figure out what alpha is.

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We know that alpha, let me write it here.

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Alpha plus pi over three,

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plus this right angle which
in radians is pi over two.

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We'll do in white.

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Plus pi over two, that's
this angle right over here,

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plus pi over two are going to equal …

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or the sum of the interior
angles of a triangle ...

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Remember we're looking at
this triangle right over here

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that's going to be equal to

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if we thought in degrees it
would be 180 degrees in radians

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it would be equal to pi.

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We get alpha,

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alpha is equal to pi which
is the same thing ...

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Let's see I'll find a
common denominator of six.

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We can write that as six pi over six,

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minus pi over three which
is the same thing as

00:04:31.869 --> 00:04:34.567
two pi over six.

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I'm just subtracting this from both sides

00:04:36.186 --> 00:04:37.528
and I wrote it like that.

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Minus pi over two which
is the same thing as

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three pi over six

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and that's going to be equal to

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six minus two minus three is one.

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That's going to be equal to pi over six.

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Alpha is pie over six

00:04:53.892 --> 00:04:57.203
so let me write that alpha is
equal to pi over six radians.

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This over here is pi over six radians.

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Now what's interesting about that

00:05:04.323 --> 00:05:07.582
is that shows that alpha is exactly half

00:05:07.582 --> 00:05:11.451
of this pi over three radian,
white angle over here.

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That has measure angle at alpha

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and this one also has alpha,

00:05:15.153 --> 00:05:16.916
a measure of alpha radians.

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Pi over six plus pi over
six is pi over three.

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Now what's interesting about that is

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that gives us confidence
that this thing over here

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is going to be bisecting
this side over here

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that this length

00:05:34.549 --> 00:05:37.047
is equivalent to this
length right over here.

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We are bisecting this angle

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and so when we get to the other side

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and this is an isosceles triangle,

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we could actually now prove

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that these are congruent triangles

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that you have a side, an angle and a side,

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side angle side.

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This side corresponds to that side

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so these two sides need to be congruent.

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Now because these two sides are congruent

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we know this has length b

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then this has also length b

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so we know that b plus b is equal to two,

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or we could just say b is equal to one.

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Now we could use that
information to solve for a

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using the Pythagorean Theorem.

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This is a right triangle
so we could say a squared

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plus b squared which is one squared

00:06:27.782 --> 00:06:32.395
is equal to two squared,

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or we could say, let's say a squared

00:06:35.989 --> 00:06:39.557
is going to be equal to
two squared minus one

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so it's going to be equal to three.

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I just subtracted one from both sides

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and so we would have a

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is equal to the square root of three.

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If you remember your
special right triangles

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from geometry class your
30, 60, 90 triangles

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you'll recognize that this is one of them

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but instead of calling
it a 30, 60, 90 triangle

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we could call it a pi
over six, pi over three,

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pi over two radian triangle

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but it really is in degrees 30, 60, 90

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and we see it has a properties
of the 30, 60, 90 triangle

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where the shorter side

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or the side opposite the
pi over six radian angle

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has half the length of the hypotenuse

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and then the longer non hypotenuse side

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is square root of three
times the shorter side,

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the side opposite the 60 degree angle

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or the pi over three radian angle

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has the length of square root of three

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times the shortest side.

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Just like that we were able to figure out

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all of the missing information in this.

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Now given that, what I want to do

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is figure out what the
sine, cosine, and tangent

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of alpha actually are

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and I could have done it for alpha

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or I could have done it for beta

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but I'll just do it for alpha

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and actually I'll leave
it as an exercise to you

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to do it for beta.

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Let's figure out

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so the sine of alpha

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and I encourage you to pause the video

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and try to think of what this is.

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Well the sine of alpha is the same as

00:08:07.619 --> 00:08:09.254
sine of pi over six,

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and remember Sohcahtoa definition.

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Let me do Toa in a different color.

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Sohcahtoa.

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Sine is opposite over hypotenuse

00:08:23.696 --> 00:08:25.593
so for this angle right over here

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the opposite side is one,

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so it's equal to one over the hypotenuse,

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one over two.

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Sine of pi over six is one half.

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Now what is cosine of alpha

00:08:37.614 --> 00:08:41.979
or what is cosine of pi over six?

00:08:41.979 --> 00:08:44.460
Well cosine is adjacent over hypotenuse

00:08:44.460 --> 00:08:47.531
so the adjacent side to this angle,

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that's not the hypotenuse,
it's square root of three

00:08:49.761 --> 00:08:52.347
and the hypotenuse once again is two.

00:08:52.347 --> 00:08:54.775
The cosine of pi over six

00:08:54.775 --> 00:08:57.349
is square root of three over two

00:08:57.349 --> 00:09:05.637
and then finally for the
tangent of pi over six,

00:09:08.590 --> 00:09:10.856
we could just take it
to sine over the cosine

00:09:10.856 --> 00:09:13.187
or we could say it's the
opposite over the adjacent.

00:09:13.187 --> 00:09:16.479
The opposite is one,

00:09:16.479 --> 00:09:18.514
the opposite angle to alpha is one

00:09:18.514 --> 00:09:20.612
and the adjacent is square root of three

00:09:20.612 --> 00:09:22.580
so it's one over the square root of three

00:09:22.580 --> 00:09:25.426
and if you don't like irrational
numbers in the denominator

00:09:25.426 --> 00:09:27.179
we can multiply it by
the square root of three

00:09:27.179 --> 00:09:28.411
over the square of three

00:09:28.411 --> 00:09:35.692
to get square root of three over three

00:09:35.692 --> 00:09:37.385
and we are done.

00:09:37.385 --> 00:09:39.925
We have figured out a
lot about this triangle.


Solving triangle in unit circle

WEBVTT
Kind: captions
Language: en

00:00:01.002 --> 00:00:03.460
Voiceover:So we have depicted
here is a unit circle

00:00:03.460 --> 00:00:09.300
centered at point A and the
point B lies on the circle.

00:00:09.300 --> 00:00:11.172
And then we dropped a perpendicular from

00:00:11.172 --> 00:00:16.047
point B to point D, point D
lies on the positive X-axis.

00:00:16.047 --> 00:00:19.578
So they form this triangle
ABD, and they tell us at angle

00:00:19.578 --> 00:00:25.091
BAD, angle bad, it has a
measure of pi over four radians.

00:00:25.091 --> 00:00:28.501
What I want to do, in this
video, is use our knowledge

00:00:28.501 --> 00:00:32.918
of trigonometry and use
our knowledge of triangles

00:00:32.918 --> 00:00:35.158
in order to figure out several things.

00:00:35.158 --> 00:00:36.995
So the first thing we want
to figure out is what's the

00:00:36.995 --> 00:00:53.453
radian measure of angle ABD?

00:00:53.453 --> 00:00:55.855
Actually let's just do that
first and then I'll talk about

00:00:55.855 --> 00:00:59.300
the other things. that
we need to think about.

00:00:59.300 --> 00:01:01.119
So I assumed you've paused the video

00:01:01.119 --> 00:01:03.628
and tried to do this on your own.

00:01:03.628 --> 00:01:05.708
So let's think about what ABD would be.

00:01:05.708 --> 00:01:07.597
We know that two of the
angles of this triangle,

00:01:07.597 --> 00:01:09.602
so if you know two of the
angles of this triangle,

00:01:09.602 --> 00:01:11.403
you should be able to
figure out the third.

00:01:11.403 --> 00:01:13.077
Now what might be a little
bit unfamiliar, we're use

00:01:13.077 --> 00:01:15.278
to saying that the sum
of the interior angles

00:01:15.278 --> 00:01:18.368
of a triangle add up
to 180 degrees, but now

00:01:18.368 --> 00:01:20.249
we're thinking in terms of radians.

00:01:20.249 --> 00:01:22.437
So we could say that the sum of the angles

00:01:22.437 --> 00:01:25.568
of a triangle add up to,
instead of saying 180

00:01:25.568 --> 00:01:29.143
degrees, 180 degrees is the
same thing as pi radians.

00:01:29.143 --> 00:01:33.738
So this angle plus that angle
are going to add up to pi.

00:01:33.738 --> 00:01:36.373
So lets just say that
this right over here,

00:01:36.373 --> 00:01:41.637
lets just say measure
of angle ABD in radians,

00:01:41.637 --> 00:01:48.176
plus pi over four, plus,
this is a right angle.

00:01:48.176 --> 00:01:50.292
What would that be in radians?

00:01:50.292 --> 00:01:52.813
Well a right angle in radians, a 90 degree

00:01:52.813 --> 00:01:55.627
angle in radians, is pi over two radians.

00:01:55.627 --> 00:01:58.103
So plus pi over two.

00:01:58.103 --> 00:02:00.432
When you take the sum of
them, the interior angles

00:02:00.432 --> 00:02:03.123
of this triangle, they're
going to add up to pi radians.

00:02:03.123 --> 00:02:06.087
Which is of course the
same thing as 180 degrees.

00:02:06.087 --> 00:02:09.259
And now we can solve for
the measure of angle ABD.

00:02:09.259 --> 00:02:16.820
Measure of angle ABD is equal to pi minus

00:02:16.820 --> 00:02:20.907
pi over two, minus pi over four.

00:02:20.907 --> 00:02:23.623
I just subtracted these
two from both the sides.

00:02:23.623 --> 00:02:25.527
So this is going to be
equal to, you can put

00:02:25.527 --> 00:02:28.872
a common denominator of
four, then this is four pi

00:02:28.872 --> 00:02:33.531
over four, this is minus two pi over four,

00:02:33.531 --> 00:02:36.889
and this, of course,
is minus pi over four.

00:02:36.889 --> 00:02:41.189
So four minus two minus one
is going to get us to one.

00:02:41.189 --> 00:02:44.643
So this is going to
get us to pi over four.

00:02:44.643 --> 00:02:47.192
So the measure of angle
ABD is actually the same

00:02:47.192 --> 00:02:52.505
as the measure of angle
BAD, it is pi over four.

00:02:52.505 --> 00:02:54.986
So that angle right over
there is pi over four.

00:02:54.986 --> 00:02:57.874
Now what does that help us with?

00:02:57.874 --> 00:03:00.963
So if we know that this is pi over four

00:03:00.963 --> 00:03:03.105
and that is pi over four
radians, and once again

00:03:03.105 --> 00:03:04.541
we know that this is a unit circle.

00:03:04.541 --> 00:03:08.185
So we know the length
of segment AB, which is

00:03:08.185 --> 00:03:10.993
a radius of the circle or is
the radius of the circle is

00:03:10.993 --> 00:03:14.180
length one, what else do we
know about this triangle?

00:03:14.180 --> 00:03:17.118
Can we figure out the
lengths of segment AD

00:03:17.118 --> 00:03:20.280
and the length of segment DB?

00:03:20.280 --> 00:03:23.231
Well sure, because we have two base angles

00:03:23.231 --> 00:03:25.498
that have the same measure,
that means that the

00:03:25.498 --> 00:03:28.494
corresponding sides are also
going to have the same measure.

00:03:28.494 --> 00:03:30.419
That means that this side is going

00:03:30.419 --> 00:03:32.111
to be congruent to that side.

00:03:32.111 --> 00:03:34.182
I can reorient it in a way that might make

00:03:34.182 --> 00:03:37.104
it a little easier to realize.

00:03:37.104 --> 00:03:41.070
If we were to flip it over,
maybe not completely fit it over

00:03:41.070 --> 00:03:43.653
but if we were to make it look like this.

00:03:43.653 --> 00:03:46.119
So the triangle, we
could make it look like,

00:03:46.119 --> 00:03:50.572
a little bit more like this.

00:03:50.572 --> 00:03:52.993
Actually I want to make it
look like a right angle though.

00:03:52.993 --> 00:03:57.591
So my triangle, let me make
it look like, there you go.

00:03:57.591 --> 00:04:07.031
So if this is D, this is B, this
is A, this is our right angle.

00:04:07.031 --> 00:04:09.702
Now this is pi over four radians and

00:04:09.702 --> 00:04:11.854
this is also pi over four radians.

00:04:11.854 --> 00:04:13.482
When your two base angles are the same,

00:04:13.482 --> 00:04:15.625
you know you're dealing
with an isosceles triangle.

00:04:15.625 --> 00:04:20.025
Because they're not all the
same, it's not equilateral.

00:04:20.025 --> 00:04:22.454
If all the angles were the
same this would be equilateral.

00:04:22.454 --> 00:04:25.883
This is an isosceles
non-equilateral triangle.

00:04:25.883 --> 00:04:28.036
So if your base angles
are the same, then you

00:04:28.036 --> 00:04:30.563
also know that the corresponding
sides are going to be the same.

00:04:30.563 --> 00:04:35.448
These two sides are the same,
this is an isosceles triangle.

00:04:35.448 --> 00:04:36.978
So how does that help us figure out

00:04:36.978 --> 00:04:39.100
the lengths of the sides?

00:04:39.100 --> 00:04:43.089
Well if you say that
this side has length X,

00:04:43.089 --> 00:04:45.021
that means that this side has length X.

00:04:45.021 --> 00:04:47.965
If this side has length X,
then this side has length X.

00:04:47.965 --> 00:04:50.070
and now we can use the
Pythagorean Theorem.

00:04:50.070 --> 00:04:56.092
We can say that this X
squared plus this X squared

00:04:56.092 --> 00:05:00.103
is equal to the hypotenuse
square is equal to one square.

00:05:00.103 --> 00:05:02.901
Or we could write that
two X squared is equal

00:05:02.901 --> 00:05:06.716
to one or that X squared
is equal to one over

00:05:06.716 --> 00:05:11.714
two or taking the principal
root of both sides,

00:05:11.714 --> 00:05:15.528
we get X is equal to one
over the square root of two.

00:05:15.528 --> 00:05:19.814
And a lot of folks don't like
having a radical denominator,

00:05:19.814 --> 00:05:22.572
they don't like having a rational
number in the denominator.

00:05:22.572 --> 00:05:25.595
So we could rationalize the denominator by

00:05:25.595 --> 00:05:28.107
multiplying by the square root of two over

00:05:28.107 --> 00:05:30.525
the square root of two,
which would be, let's see

00:05:30.525 --> 00:05:32.137
the numerator will have
the square root of two,

00:05:32.137 --> 00:05:34.432
and the denominator
we're just going to have

00:05:34.432 --> 00:05:37.448
square root of two times
square root of two is just two.

00:05:37.448 --> 00:05:39.389
So we've already been able to figure out

00:05:39.389 --> 00:05:40.262
several interesting things

00:05:40.262 --> 00:05:42.394
We've been able to figure
the angle of ABD in radians.

00:05:42.394 --> 00:05:44.781
We're able to figure out
lengths of segment AD

00:05:44.781 --> 00:05:47.437
and the length of segment BD.

00:05:47.437 --> 00:05:49.392
Now what I want to do
is figure out what are

00:05:49.392 --> 00:05:53.485
the sine, cosine, and tangent
of pi over four radians,

00:05:53.485 --> 00:05:55.734
given all of the work we have done?

00:05:55.734 --> 00:05:58.573
So let's first think
about, what is the sine?

00:05:58.573 --> 00:06:01.368
We'll do this in the color orange.

00:06:01.368 --> 00:06:02.511
Given all the work we have done,

00:06:02.511 --> 00:06:04.688
what is the sine of pi over four radians?

00:06:04.688 --> 00:06:06.588
And I encourage you once
again, pause the video,

00:06:06.588 --> 00:06:08.154
think about the unit circle definition of

00:06:08.154 --> 00:06:12.949
trig functions, and
think about what this is.

00:06:12.949 --> 00:06:15.270
Well the unit circle
definition of trig functions

00:06:15.270 --> 00:06:17.349
in this angle, this pi over
four radians, is forming

00:06:17.349 --> 00:06:19.508
an angle with the positive X axis.

00:06:19.508 --> 00:06:22.316
And where it's terminal ray
intersects the unit circle,

00:06:22.316 --> 00:06:24.882
the X and Y coordinates of this point

00:06:24.882 --> 00:06:28.704
are what specify the cosine and sine.

00:06:28.704 --> 00:06:31.604
So the coordinates of
this point are going to be

00:06:31.604 --> 00:06:37.315
the cosine of pi over four
radians is the X coordinate

00:06:37.315 --> 00:06:40.134
and the sine of pi over
four radians is the

00:06:40.134 --> 00:06:42.877
Y coordinates, sine of pi over four.

00:06:42.877 --> 00:06:44.847
So what's the Y coordinate going to be

00:06:44.847 --> 00:06:46.989
if we want pi over four?

00:06:46.989 --> 00:06:49.554
Well it's going to be this
length, right over here.

00:06:49.554 --> 00:06:51.283
Which is the same thing as this length,

00:06:51.283 --> 00:06:53.691
which is the length of X,
which is the square root

00:06:53.691 --> 00:06:56.149
of two over two.

00:06:56.149 --> 00:06:58.284
Now what is the cosine of pi over four?

00:06:58.284 --> 00:07:00.245
Once again I encourage you to pause

00:07:00.245 --> 00:07:01.873
the video to think about it.

00:07:01.873 --> 00:07:03.043
What's this X coordinate?

00:07:03.043 --> 00:07:06.814
The X coordinate, is this
distance right over here,

00:07:06.814 --> 00:07:08.821
which is once again going to be X,

00:07:08.821 --> 00:07:11.879
which is square root of two over two.

00:07:11.879 --> 00:07:18.220
Now, what is the tangent of
pi over four going to be?

00:07:18.220 --> 00:07:21.074
Well the tangent of pi over
four is sine of pi over four

00:07:21.074 --> 00:07:25.346
over cosine of pi over four.

00:07:25.346 --> 00:07:27.714
Now both of these are
the exact same thing.

00:07:27.714 --> 00:07:28.947
They're both square root of two over two.

00:07:28.947 --> 00:07:30.258
So you're going to have
square root of two over two,

00:07:30.258 --> 00:07:34.273
divided by two over two, well
that's just going to give you one.

00:07:34.273 --> 00:07:35.994
And that also makes
since, because remember

00:07:35.994 --> 00:07:39.411
the tangent of this angle
is the slope of this line.

00:07:39.411 --> 00:07:41.822
And we see the slope,
for every X we move in

00:07:41.822 --> 00:07:44.575
the horizontal direction we move X up.

00:07:44.575 --> 00:07:48.247
So our change in Y over
change in X is essentially

00:07:48.247 --> 00:07:51.927
X over X, which is equal to one.


Introduction to function composition

WEBVTT
Kind: captions
Language: en

00:00:00.707 --> 00:00:02.195
Voiceover:So we have three different

00:00:02.195 --> 00:00:03.346
function definitions here.

00:00:03.346 --> 00:00:06.579
This is F of X in blue, here
we map between different

00:00:06.579 --> 00:00:10.653
values of T and what G of T would be.

00:00:10.653 --> 00:00:13.996
So you could use this as
a definition of G of T.

00:00:13.996 --> 00:00:18.363
And here we map from X to H of X.

00:00:18.363 --> 00:00:20.025
So for example, when X is equal

00:00:20.025 --> 00:00:22.636
to three, H of X is equal to zero.

00:00:22.636 --> 00:00:26.204
When X is equal to one,
H of X is equal to two.

00:00:26.204 --> 00:00:27.424
And actually let me number

00:00:27.424 --> 00:00:30.459
this one, two, three, just like that.

00:00:30.459 --> 00:00:33.562
Now what I want to do in
this video is introduce you

00:00:33.562 --> 00:00:37.991
to the idea of composing functions.

00:00:37.991 --> 00:00:40.455
Now what does it mean
to compose functions?

00:00:40.455 --> 00:00:43.190
Well that means to build
up a function by composing

00:00:43.190 --> 00:00:46.693
one function of other functions or I guess

00:00:46.693 --> 00:00:48.102
you could think of nesting them.

00:00:48.102 --> 00:00:49.446
What do I mean by that?

00:00:49.446 --> 00:00:53.641
Well, let's think about
what it means to evaluate

00:00:53.687 --> 00:00:57.138
F of, not X, but we're
going to evaluate F of,

00:00:57.138 --> 00:01:00.066
actually let's just start
with a little warm-up.

00:01:00.066 --> 00:01:07.443
Let's evaluate F of G of two.

00:01:07.443 --> 00:01:09.075
Now what do you think this is going to be

00:01:09.075 --> 00:01:10.514
and I encourage you to pause this video

00:01:10.514 --> 00:01:13.074
and think about it on your own.

00:01:13.074 --> 00:01:15.106
Well it seems kind of daunting at first,

00:01:15.106 --> 00:01:17.186
if you're not very
familiar with the notation,

00:01:17.186 --> 00:01:19.154
but we just have to
remember what a function is.

00:01:19.154 --> 00:01:20.402
A function is just a mapping from

00:01:20.402 --> 00:01:22.355
one set of numbers to another.

00:01:22.355 --> 00:01:24.739
So for example, when
we're saying G of two,

00:01:24.739 --> 00:01:27.971
that means take the number two, input it

00:01:27.971 --> 00:01:31.509
into the function G and
then you're going to get

00:01:31.509 --> 00:01:36.454
an output which we are
going to call G of two.

00:01:36.454 --> 00:01:38.772
Now we're going to use
that output, G of two,

00:01:38.772 --> 00:01:41.507
and then input it into the function F.

00:01:41.507 --> 00:01:47.759
So we're going to input
it into the function F,

00:01:47.759 --> 00:01:49.054
and what we're going to get is F

00:01:49.054 --> 00:01:55.881
of the thing that we
inputted, F of G of two.

00:01:55.881 --> 00:01:57.392
So let's just take it step by step.

00:01:57.392 --> 00:01:59.856
What is G of two?

00:01:59.856 --> 00:02:08.746
Well when T is equal to two,
G of two is negative three.

00:02:09.176 --> 00:02:15.135
So when I put negative three
into F, what am I going to get?

00:02:15.135 --> 00:02:22.430
Well, I'm going to get negative
three squared minus one,

00:02:22.430 --> 00:02:27.118
which is nine minus one which
is going to be equal to eight.

00:02:27.118 --> 00:02:29.343
So this right over here is equal to eight.

00:02:29.343 --> 00:02:33.983
F of G of two is going
to be equal to eight.

00:02:33.983 --> 00:02:37.326
Now, what would, using
this same exact logic,

00:02:37.326 --> 00:02:46.869
what would F of H of two be?

00:02:46.869 --> 00:02:48.362
And once again, I encourage
you to pause the video

00:02:48.362 --> 00:02:50.730
and think about it on your own.

00:02:50.730 --> 00:02:53.818
Well let's think about
it this way, instead of

00:02:53.818 --> 00:02:57.257
doing it using this little
diagram, here everywhere

00:02:57.257 --> 00:03:01.610
you see the input is X,
whatever the input is

00:03:01.610 --> 00:03:03.355
you square it and minus one.

00:03:03.355 --> 00:03:06.682
Here the input is H of
two, and so we're going to

00:03:06.682 --> 00:03:10.234
take the input, which is H
of two, and we're going to

00:03:10.234 --> 00:03:17.065
square it and we're going to subtract one.

00:03:17.065 --> 00:03:21.241
So F of H of two is H of
two squared minus one.

00:03:21.241 --> 00:03:23.450
Now what is H of two?

00:03:23.450 --> 00:03:26.698
When X is equal to two, H of two is one.

00:03:26.698 --> 00:03:32.297
So H of two is one, so since
H of two is equal to one,

00:03:32.297 --> 00:03:35.561
this simplifies two one squared minus one,

00:03:35.561 --> 00:03:37.513
well that's just going to be one minus one

00:03:37.513 --> 00:03:42.522
which is equal to zero.

00:03:42.522 --> 00:03:44.698
We could have done it
with the diagram way,

00:03:44.698 --> 00:03:46.826
we could have said, hey
we're going to input

00:03:46.826 --> 00:03:54.585
two into H, if you input
two into H you get one,

00:03:54.585 --> 00:03:57.321
so that is H of two right over here.

00:03:57.321 --> 00:04:00.762
So that is H of two, and then
we're going to input that

00:04:00.762 --> 00:04:09.770
into F, which is going
to give us F of one.

00:04:09.770 --> 00:04:14.479
F of one is one squared
minus one, which is zero.

00:04:14.479 --> 00:04:17.902
So this right over here is F of H of two.

00:04:18.394 --> 00:04:19.862
H of two is the input
into F, so the output

00:04:19.862 --> 00:04:27.062
is going to be F of our
input, F of H of two.

00:04:27.062 --> 00:04:29.526
Now we can go even further,
let's do a composite.

00:04:29.526 --> 00:04:35.111
Let's compose three of
these functions together.

00:04:35.111 --> 00:04:44.086
So let's take, and I'm doing
this on the fly a little bit,

00:04:44.086 --> 00:04:56.070
so I hope it's a good
result, G of F of two,

00:04:56.070 --> 00:04:59.175
and let me just think
about this for one second.

00:04:59.175 --> 00:05:03.590
So that's going to be G of F of two,

00:05:03.590 --> 00:05:09.704
and let's take H of G of
F of two, just for fun.

00:05:09.704 --> 00:05:12.167
Now we're really doing
a triple composition.

00:05:12.167 --> 00:05:13.718
So there's a bunch of
ways we could do this.

00:05:13.718 --> 00:05:19.398
One way is to just try to
evaluate what is F of two.

00:05:19.398 --> 00:05:26.183
Well F of two is going to be
equal to two squared minus one.

00:05:26.183 --> 00:05:28.711
It's going to be four minus one or three.

00:05:28.711 --> 00:05:32.870
So this is going to be equal to three.

00:05:32.870 --> 00:05:35.238
Now what is G of three?

00:05:35.238 --> 00:05:41.787
G of three is when T is equal
to three, G of three is four.

00:05:41.787 --> 00:05:47.531
So G of three, this whole thing, is four.

00:05:47.531 --> 00:05:50.267
F of two is three, three of G is four.

00:05:50.267 --> 00:05:52.427
What is H of four?

00:05:52.427 --> 00:05:55.804
Well we can just look back
to our original graph here.

00:05:55.804 --> 00:05:59.787
When X is four, H of four is negative one.

00:05:59.787 --> 00:06:07.915
So H of G of F of two, is
just equal to negative one.

00:06:07.915 --> 00:06:10.492
So hopefully this you
somewhat familiar with how

00:06:10.492 --> 00:06:13.548
to evaluate the composition of functions.


Creating new function from composition

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:03.523
Voiceover:When we first got introduced
to function composition,

00:00:03.523 --> 00:00:06.557
we looked at actually
evaluating functions at a point,

00:00:06.557 --> 00:00:08.822
or compositions of functions at a point.

00:00:08.822 --> 00:00:11.356
What I wanna do in this video is come up

00:00:11.356 --> 00:00:14.814
with expressions that define
a function composition.

00:00:14.814 --> 00:00:17.523
So, for example, I wanna figure out,

00:00:17.523 --> 00:00:21.694
what is, f of, g of x?

00:00:21.941 --> 00:00:26.598
f of, g of x.

00:00:26.891 --> 00:00:28.556
And I encourage you to pause the video,

00:00:28.556 --> 00:00:31.232
and try to think about it on your own.

00:00:31.603 --> 00:00:35.494
Well, g of x in this case,
is the input to f of x.

00:00:35.494 --> 00:00:36.908
So, wherever we see the x

00:00:36.908 --> 00:00:38.808
in this definition, that's the input.

00:00:38.808 --> 00:00:41.074
So we're going to replace
the input with g of x.

00:00:41.074 --> 00:00:43.159
We're going to replace the x with g of x.

00:00:43.159 --> 00:00:48.857
So, f of g of x is going to be
equal to the square root of-

00:00:48.857 --> 00:00:51.840
Well instead of an x,
we would write a g of x.

00:00:51.840 --> 00:00:56.317
g of x, g of x squared.

00:00:56.317 --> 00:00:59.588
g of x squared, minus one.

00:00:59.588 --> 00:01:01.273
Now what is g of x equal to?

00:01:01.273 --> 00:01:03.138
Well, g of x is this
thing right over here.

00:01:03.138 --> 00:01:06.673
So this is going to be
equal to the square root of,

00:01:06.673 --> 00:01:12.824
g of x, is x over 1 plus x.

00:01:12.824 --> 00:01:15.339
We're going to square that.

00:01:15.339 --> 00:01:18.506
We're going to square that, minus 1.

00:01:18.506 --> 00:01:22.272
So f of g of x, is also a function of x.

00:01:22.272 --> 00:01:24.522
So f of g of x is a square root of,

00:01:24.522 --> 00:01:27.104
and we could write this as x
squared over 1 plus x squared,

00:01:27.104 --> 00:01:28.856
but we could just leave it like this.

00:01:28.856 --> 00:01:30.772
It's equal to the square
root of this whole thing,

00:01:30.772 --> 00:01:34.722
x over 1 plus x, squared, minus one.

00:01:34.722 --> 00:01:36.621
Now let's go the other way round.

00:01:36.621 --> 00:01:41.687
What is g of f of x?

00:01:41.687 --> 00:01:43.556
What is g of f of x?

00:01:43.556 --> 00:01:45.839
And once again, I encourage
you to pause the video,

00:01:45.839 --> 00:01:48.166
and try to think about it on your own.

00:01:48.505 --> 00:01:52.737
Well, f of x is now the input into g of x.

00:01:52.737 --> 00:01:54.295
So everywhere we see the x here,

00:01:54.310 --> 00:01:56.183
we'll replace it with f of x.

00:01:56.183 --> 00:01:58.515
So this is going to be equal to,

00:01:58.515 --> 00:02:03.650
this is going to be
equal to, f of x, over-

00:02:03.650 --> 00:02:05.259
Let me do it in the same color,

00:02:05.259 --> 00:02:07.166
so you can appreciate it better.

00:02:07.166 --> 00:02:11.981
f of x over, one plus f of x.

00:02:11.981 --> 00:02:14.282
One plus f of x.

00:02:14.282 --> 00:02:16.049
And what's that equal to?

00:02:16.049 --> 00:02:19.932
Well, f of x is equal to the square root,

00:02:19.932 --> 00:02:21.964
of x squared minus one.

00:02:21.964 --> 00:02:24.450
x squared minus one.

00:02:24.450 --> 00:02:30.214
So it's gonna be that over
1, plus the square root.

00:02:30.214 --> 00:02:35.981
One plus the square root
of x squared minus one.

00:02:35.981 --> 00:02:40.049
So this is a composition f of
g of x, you get this thing.

00:02:40.049 --> 00:02:43.054
This is g of f of x,
where you get this thing.

00:02:43.054 --> 00:02:46.568
And to be clear, these are
very different expressions.

00:02:46.568 --> 00:02:49.269
So typically, you want
the composition one way.

00:02:49.269 --> 00:02:50.939
This isn't gonna be the
same as the composition

00:02:50.939 --> 00:02:52.240
the other way, unless the functions

00:02:52.240 --> 00:02:54.920
are designed in a fairly special way.


Modeling with function composition

WEBVTT
Kind: captions
Language: en

00:00:00.397 --> 00:00:02.844
Voiceover:Fuel is being
pumped into a storage tank.

00:00:02.844 --> 00:00:05.851
The volume, V, of the fuel in the tank

00:00:05.851 --> 00:00:08.824
depends on the depth, d,
according to the formula.

00:00:08.824 --> 00:00:12.099
V of d is equal to four
times three d squared

00:00:12.099 --> 00:00:13.792
plus five to the third power,

00:00:13.792 --> 00:00:16.073
where d is in measured in meters.

00:00:16.073 --> 00:00:18.996
Suppose that the depth, d, of the fuel

00:00:18.996 --> 00:00:22.029
depends on time measured in hours

00:00:22.029 --> 00:00:23.560
according to the formula.

00:00:23.560 --> 00:00:25.882
d of t is equal to one over
the square root of three

00:00:25.882 --> 00:00:28.068
times the square root of t minus five.

00:00:28.068 --> 00:00:29.680
That makes sense, that if I'm filling,

00:00:29.680 --> 00:00:32.216
if I'm pumping something
into a storage tank

00:00:32.216 --> 00:00:35.340
the depth of what I'm filling,

00:00:35.340 --> 00:00:37.410
of the fuel I guess is what I'm filling

00:00:37.410 --> 00:00:38.945
is going to depend on time

00:00:38.945 --> 00:00:41.989
and then the volume could
be dependent on the depth.

00:00:41.989 --> 00:00:45.687
Use function composition
to write the volume

00:00:45.687 --> 00:00:48.679
of the fuel of the tank
as a function of time.

00:00:48.679 --> 00:00:51.270
Simplify your answer as much as possible.

00:00:51.270 --> 00:00:53.167
Let me get my scratch pad out,

00:00:53.167 --> 00:00:55.187
so I've already written
what they told us here.

00:00:55.187 --> 00:00:57.083
This is volume as a function of depth

00:00:57.083 --> 00:00:59.477
and now this is depth
as a function of time.

00:00:59.477 --> 00:01:02.230
If we want volume as a function of time,

00:01:02.230 --> 00:01:05.192
what we really want to
do is figure out what is,

00:01:05.192 --> 00:01:09.799
so we could take V of d

00:01:09.799 --> 00:01:15.029
where we could take V of d of t.

00:01:17.182 --> 00:01:19.691
Now this would give me
volume as a function of time.

00:01:19.691 --> 00:01:21.221
Everywhere where I see a d,

00:01:21.221 --> 00:01:24.963
I should replace it with this description

00:01:24.963 --> 00:01:27.911
of how does d vary with respect to time.

00:01:27.911 --> 00:01:28.890
Let's do that.

00:01:28.890 --> 00:01:31.708
V of d of t is going to be equal to

00:01:31.708 --> 00:01:36.665
four times three, times d squared.

00:01:36.665 --> 00:01:38.460
Instead of writing d squared,

00:01:38.460 --> 00:01:42.211
I'm going to write all
of this business squared,

00:01:42.211 --> 00:01:45.732
so one over square root of three,

00:01:45.732 --> 00:01:49.160
times the square root of t minus five,

00:01:49.160 --> 00:01:55.658
all of that squared plus five

00:01:55.658 --> 00:01:57.658
to the third power.

00:01:57.658 --> 00:02:00.628
I really hope this simplifies
in a reasonable way.

00:02:00.628 --> 00:02:02.736
Let's see, let me actually
copy and paste this

00:02:02.736 --> 00:02:04.968
so I can have a little
bit more real estate.

00:02:07.352 --> 00:02:09.830
Let me just paste it down
here so I could work it out.

00:02:09.830 --> 00:02:14.295
All right, so then we get
this is going to be equal to,

00:02:14.295 --> 00:02:15.860
this is going to be equal to,

00:02:15.860 --> 00:02:17.669
let's see if we square
this right over here

00:02:17.669 --> 00:02:19.261
and I could do it all in.

00:02:19.261 --> 00:02:21.665
This is going to be four times three,

00:02:21.665 --> 00:02:27.174
times this whole thing squared.

00:02:27.174 --> 00:02:28.621
One over square root of three squared

00:02:28.621 --> 00:02:31.177
is going to be 1/3

00:02:31.177 --> 00:02:33.433
and then square root
of t minus five squared

00:02:33.433 --> 00:02:36.772
is going to be t minus five

00:02:36.772 --> 00:02:42.733
and then of course we have
plus five to the third power.

00:02:42.733 --> 00:02:45.137
Let's see three times 1/3,

00:02:45.137 --> 00:02:47.358
well that's just going to be equal to one.

00:02:47.358 --> 00:02:49.350
We are left with,

00:02:49.350 --> 00:02:55.697
this is equal to four times t minus five

00:02:55.697 --> 00:02:58.075
plus five, this does simplify nicely.

00:02:58.075 --> 00:02:59.486
Plus five,

00:02:59.486 --> 00:03:04.852
we're going to be left with four times t

00:03:04.852 --> 00:03:08.860
to the third power.

00:03:08.875 --> 00:03:13.172
Let's input that into our V of t

00:03:13.172 --> 00:03:14.649
and let me just make it clear.

00:03:14.649 --> 00:03:17.679
This is now V, this is giving our volume

00:03:17.679 --> 00:03:19.200
as a function of time.

00:03:19.200 --> 00:03:21.497
We can now say that this is V of t.

00:03:21.497 --> 00:03:26.913
V of t is four t squared.

00:03:26.913 --> 00:03:29.361
Once again this is the
volume as a function of time.

00:03:29.361 --> 00:03:31.535
You give me a time, I'll
tell you what the volume is.

00:03:31.535 --> 00:03:33.602
Four t to the third.

00:03:33.602 --> 00:03:36.482
Four t to the third power.

00:03:36.482 --> 00:03:38.967
V of t, did I say four t squared?

00:03:38.967 --> 00:03:40.371
I don't know, sometimes
I do strange things.

00:03:40.371 --> 00:03:45.022
I said four t to the third power,

00:03:45.022 --> 00:03:46.519
I did format it correctly

00:03:46.519 --> 00:03:48.591
and then there's a second
part to this question.

00:03:48.591 --> 00:03:51.092
How many cubic meters
of fuel are in the tank

00:03:51.092 --> 00:03:52.318
after two hours?

00:03:52.318 --> 00:03:54.015
Round your answer to the nearest 10th.

00:03:54.015 --> 00:03:57.277
Well two hours, that means t is equal two.

00:03:57.277 --> 00:03:59.225
They say t is measured in hours.

00:03:59.225 --> 00:04:00.953
When t is equal to two,

00:04:00.953 --> 00:04:06.099
the volume is going to be two
to the third power is eight.

00:04:06.099 --> 00:04:11.399
Eight times four is 32.

00:04:11.399 --> 00:04:13.544
Let's check our answer
and we got it right.


Solving equation involving function and inverse

WEBVTT
Kind: captions
Language: en

00:00:00.480 --> 00:00:03.107
Voiceover:We have two
pieces of information here.

00:00:03.107 --> 00:00:05.334
We're told that 11 is equal to eight plus

00:00:05.334 --> 00:00:07.961
f inverse of x minus three

00:00:07.961 --> 00:00:10.303
and that f of three is equal to 10.

00:00:10.303 --> 00:00:13.794
What I would like you to
try is pause the video

00:00:13.794 --> 00:00:17.257
and figure out the value of x.

00:00:19.779 --> 00:00:22.035
It looks a little daunting at first,

00:00:22.035 --> 00:00:24.886
you're like wow f
inverse of x minus three.

00:00:24.886 --> 00:00:26.648
Whenever you see something like this

00:00:26.648 --> 00:00:28.951
at least my impulse is at
least just try to simplify

00:00:28.951 --> 00:00:29.987
the information you have

00:00:29.987 --> 00:00:31.865
and then see if we can make sense of it.

00:00:31.865 --> 00:00:35.069
Here we could definitely
simplify this equation

00:00:35.069 --> 00:00:37.220
by subtracting eight from both sides.

00:00:37.220 --> 00:00:40.583
Just so that we have f
inverse by itself on one side.

00:00:40.583 --> 00:00:42.978
If we subtract eight from both sides,

00:00:42.978 --> 00:00:45.297
the left hand side 11 minus eight is three

00:00:45.297 --> 00:00:51.316
and the right hand side is
f inverse of x minus three.

00:00:51.316 --> 00:00:56.733
Or I could write this as
f inverse of x minus three

00:00:56.733 --> 00:00:59.443
is equal to three.

00:00:59.443 --> 00:01:02.636
These are essentially our
two pieces of information

00:01:02.636 --> 00:01:05.389
once we simplify this over here.

00:01:05.389 --> 00:01:07.361
This seems pretty interesting

00:01:07.361 --> 00:01:09.322
because we say f of three is 10

00:01:09.322 --> 00:01:13.640
and then we're saying f inverse

00:01:13.640 --> 00:01:17.719
of this right over here is equal to three.

00:01:17.719 --> 00:01:20.429
There seems to be something
interesting going on,

00:01:20.429 --> 00:01:22.391
we could even just think
about what a function

00:01:22.391 --> 00:01:23.768
and an inverse does.

00:01:23.768 --> 00:01:25.802
This is the domain of the function.

00:01:26.709 --> 00:01:27.772
Domain,

00:01:28.864 --> 00:01:32.778
and if this is the range,

00:01:32.778 --> 00:01:35.502
we know that the function maps

00:01:35.502 --> 00:01:40.661
from three in the domain
to 10 in the range.

00:01:40.661 --> 00:01:42.158
That's what the function does.

00:01:42.158 --> 00:01:45.018
If the function does that
then the inverse function

00:01:45.018 --> 00:01:47.215
would map from 10 in the range

00:01:47.215 --> 00:01:49.165
to three in the domain.

00:01:50.211 --> 00:01:53.047
This would be f inverse,

00:01:53.047 --> 00:01:55.808
so f inverse of 10 is equal to three.

00:01:55.808 --> 00:01:57.032
Let me write that down,

00:01:57.032 --> 00:02:00.189
f inverse of 10 is equal to three.

00:02:01.096 --> 00:02:03.477
What this thing right over here

00:02:03.477 --> 00:02:05.996
is the same thing as that
thing right over here.

00:02:05.996 --> 00:02:11.163
10 must be equal to x minus three.

00:02:11.163 --> 00:02:13.610
We could write x minus three

00:02:13.610 --> 00:02:15.725
is equal to 10,

00:02:15.725 --> 00:02:17.463
add three to both sides,

00:02:17.463 --> 00:02:21.952
x is equal to 13.


Restricting function domain to make invertible

WEBVTT
Kind: captions
Language: en

00:00:00.794 --> 00:00:02.875
Voiceover:It's useful to
make an invertible function

00:00:02.875 --> 00:00:04.731
from one that is not invertible

00:00:04.731 --> 00:00:06.524
by restricting it's domain.

00:00:06.524 --> 00:00:09.011
The graph of f(x) is shown below.

00:00:09.011 --> 00:00:10.318
To which intervals could we

00:00:10.318 --> 00:00:14.199
restrict f(x) to get
an invertible function?

00:00:14.308 --> 00:00:17.716
So in order for a
function to be invertible,

00:00:17.716 --> 00:00:21.940
it's inverse relationship
has to be a function as well.

00:00:21.940 --> 00:00:23.459
What do I mean by that?

00:00:23.459 --> 00:00:24.731
Well let's show that this one

00:00:24.731 --> 00:00:26.990
right over here is not invertible.

00:00:26.990 --> 00:00:28.931
And there's a couple of ways
you could think about it.

00:00:28.931 --> 00:00:32.019
F is clearly a function, it
passes the vertical line test.

00:00:32.019 --> 00:00:33.996
Vertical line test, you take
a vertical line anywhere

00:00:33.996 --> 00:00:37.235
in this graph and it doesn't
intersect more than one time.

00:00:37.235 --> 00:00:39.059
So here it only intersects once.

00:00:39.059 --> 00:00:40.699
Why does that help us think

00:00:40.699 --> 00:00:42.315
about whether this is a function?

00:00:42.315 --> 00:00:44.019
Well imagine if it did intersect.

00:00:44.019 --> 00:00:46.214
Imagine if the function was
a circle right over here.

00:00:46.214 --> 00:00:47.395
So imagine if the vertical line

00:00:47.395 --> 00:00:49.740
did intersect the
function twice like this.

00:00:49.740 --> 00:00:51.401
Well then what would f of 6 be?

00:00:51.401 --> 00:00:53.001
Is it 2 or is it negative 2?

00:00:53.001 --> 00:00:55.061
F of 6 needs to map to a unique value,

00:00:55.061 --> 00:00:56.825
you need to know what f of 6 is

00:00:56.825 --> 00:00:58.805
and here you wouldn't now.

00:00:58.805 --> 00:01:00.335
So that's the vertical line test

00:01:00.335 --> 00:01:02.709
for testing whether
something is a function

00:01:02.709 --> 00:01:05.925
but you could also apply
a horizontal line test

00:01:05.925 --> 00:01:08.493
to test whether it's inverse,

00:01:08.493 --> 00:01:10.413
the inverse relationship is a function.

00:01:10.413 --> 00:01:12.645
Or whether this is invertible.

00:01:12.645 --> 00:01:13.917
One way to think about it is

00:01:13.917 --> 00:01:16.157
if you swap the x and y values here,

00:01:16.157 --> 00:01:17.753
the graph of this function

00:01:17.753 --> 00:01:20.237
is going to look something like this.

00:01:20.237 --> 00:01:21.605
The graph of this function is

00:01:21.605 --> 00:01:23.638
going to look something like this.

00:01:23.638 --> 00:01:26.837
So let me see, actually it's
not going to look like that.

00:01:26.837 --> 00:01:31.059
It's going to look, it's
going to look like this.

00:01:31.333 --> 00:01:34.761
It's going to look like this,

00:01:34.938 --> 00:01:37.173
let me draw this as neatly as I can.

00:01:37.173 --> 00:01:38.893
I think you get the idea.

00:01:38.893 --> 00:01:44.464
So like that and and like that.

00:01:44.464 --> 00:01:45.629
So it's going to look like that

00:01:45.629 --> 00:01:48.685
if you were to take the
inverse relationship,

00:01:48.685 --> 00:01:51.038
if you were to swap the x and y's

00:01:51.038 --> 00:01:54.989
and now this clearly would not
pass the vertical line test.

00:01:54.989 --> 00:01:58.077
That clearly would not pass
the vertical line test.

00:01:58.077 --> 00:01:59.573
Or another way to think about it

00:01:59.573 --> 00:02:02.765
is we could just apply
a horizontal line test,

00:02:02.765 --> 00:02:04.949
we could apply a horizontal line test,

00:02:04.949 --> 00:02:06.781
let me clean it up a little bit,

00:02:06.781 --> 00:02:10.398
to our original function to
test whether it is invertible.

00:02:10.398 --> 00:02:13.485
The horizontal line
test, it clearly fails.

00:02:13.485 --> 00:02:14.753
Now once again, why does that

00:02:14.753 --> 00:02:16.405
show that that's not invertible?

00:02:16.405 --> 00:02:18.829
Actually let me do it on
a value that's easier to,

00:02:18.829 --> 00:02:21.509
even here we see a horizontal line

00:02:21.509 --> 00:02:26.774
is intersecting the function
there, there, there and there.

00:02:26.774 --> 00:02:28.677
Now why is that a problem?

00:02:28.677 --> 00:02:30.013
Well, let's just remind ourselves

00:02:30.013 --> 00:02:31.965
what a function and an
inverse function do.

00:02:31.965 --> 00:02:33.309
They're just mappings from the...

00:02:33.309 --> 00:02:34.509
The function is a mapping

00:02:34.509 --> 00:02:36.421
from the domain to the range.

00:02:36.421 --> 00:02:38.782
So let's say this is the domain.

00:02:38.782 --> 00:02:40.277
Domain.

00:02:40.277 --> 00:02:42.903
And this is the range.

00:02:44.164 --> 00:02:45.707
This is the range.

00:02:45.707 --> 00:02:46.883
And so our function is clearly,

00:02:46.883 --> 00:02:48.147
let's just take 2 of these values.

00:02:48.147 --> 00:02:50.835
Our function takes 2 in the domain

00:02:50.835 --> 00:02:54.419
and maps it to 2 in the range.

00:02:54.419 --> 00:02:56.411
So the function does that, f of 2 is 2.

00:02:56.411 --> 00:02:58.172
We see that right over here.

00:02:58.172 --> 00:03:00.379
And it takes 6 and it also maps it to 2.

00:03:00.379 --> 00:03:06.401
F of 6 is also, is also, equal to 2.

00:03:06.401 --> 00:03:08.532
Now this is fine for f being a function

00:03:08.532 --> 00:03:10.100
but it introduces a problem when

00:03:10.100 --> 00:03:12.372
we're trying to take the
inverse relationship.

00:03:12.372 --> 00:03:14.308
The inverse relationship all of a sudden,

00:03:14.308 --> 00:03:16.356
if i'm trying to set up f inverse,

00:03:16.356 --> 00:03:21.301
should f inverse, what should
f inverse of 2 be equal to?

00:03:21.301 --> 00:03:23.932
Should it be equal to 6?

00:03:23.932 --> 00:03:25.860
So should it be equal to 6?

00:03:25.860 --> 00:03:29.220
Or should it be equal or
should it be equal to 2?

00:03:29.220 --> 00:03:30.996
And so this is the problem.

00:03:30.996 --> 00:03:33.060
When you have multiple
elements in the domain

00:03:33.060 --> 00:03:35.884
mapping to the same element in the range,

00:03:35.884 --> 00:03:38.460
that's okay for being a function

00:03:38.460 --> 00:03:40.772
but then when you're trying to find to map

00:03:40.772 --> 00:03:41.984
from that element to the range

00:03:41.984 --> 00:03:43.444
to those multiple elements in the domain,

00:03:43.444 --> 00:03:45.148
that is no longer a function.

00:03:45.148 --> 00:03:47.380
This is not invertible.

00:03:47.380 --> 00:03:49.524
So essentially when we
look at the intervals

00:03:49.524 --> 00:03:50.780
that we could restrict through

00:03:50.780 --> 00:03:53.660
to get an invertible function
it essentially would be

00:03:53.660 --> 00:03:55.444
intervals where the thing would

00:03:55.444 --> 00:03:57.548
pass that horizontal line test.

00:03:57.548 --> 00:03:59.636
Where over that interval, every member

00:03:59.636 --> 00:04:02.020
of that domain, that restricted domain,

00:04:02.020 --> 00:04:06.820
maps to a only, there's
a one to one mapping

00:04:06.820 --> 00:04:08.892
between each member of that domain

00:04:08.892 --> 00:04:12.948
and a member of the new restricted range,

00:04:12.948 --> 00:04:14.693
or potentially restricted range.

00:04:14.693 --> 00:04:16.548
So let's think about these, these,

00:04:16.548 --> 00:04:20.676
let's think about these
different restricted domains.

00:04:20.676 --> 00:04:23.732
So if we think about from negative 2 to 2,

00:04:23.732 --> 00:04:26.516
so from negative 2 to 2
including negative 2 and 2.

00:04:26.516 --> 00:04:27.852
So this is a closed interval.

00:04:27.852 --> 00:04:28.900
We have brackets here.

00:04:28.900 --> 00:04:30.404
Actually all of these
are closed intervals.

00:04:30.404 --> 00:04:32.156
Notice it does not pass
the horizontal line test.

00:04:32.156 --> 00:04:35.908
F of negative 2 and f of 2 both map to 2

00:04:35.908 --> 00:04:38.244
and so we don't know what
f inverse of 2 should be.

00:04:38.244 --> 00:04:40.204
And you could put a horizontal
line right over there,

00:04:40.204 --> 00:04:41.700
makes it very clear that this

00:04:41.700 --> 00:04:44.204
is not passing the horizontal line test.

00:04:44.204 --> 00:04:46.316
So let me clear this up a little.

00:04:46.316 --> 00:04:49.437
So let's rule out the first interval.

00:04:49.437 --> 00:04:51.869
Now the second one, between 0 and 4.

00:04:51.869 --> 00:04:54.048
0 and 4.

00:04:54.048 --> 00:04:57.330
Once again, not passing
the horizontal line test.

00:04:57.330 --> 00:04:58.962
Rule that out.

00:04:58.962 --> 00:05:00.171
Negative 8 to 8.

00:05:00.171 --> 00:05:01.498
Well that doesn't even restrict the domain

00:05:01.498 --> 00:05:02.658
relative to what we had before

00:05:02.658 --> 00:05:04.406
so that's not going to work.

00:05:04.406 --> 00:05:06.977
6 to 8, now this is interesting.

00:05:06.977 --> 00:05:10.624
Between 6 and 8, you do
have a 1 to 1 mapping

00:05:10.624 --> 00:05:13.568
between elements of the domain
and elements of the range.

00:05:13.568 --> 00:05:17.272
Notice between 6 and 8
it does, it does pass,

00:05:17.272 --> 00:05:20.096
it does pass the horizontal line test.

00:05:20.096 --> 00:05:22.800
So this is a legitimate
restriction of the domain.

00:05:22.800 --> 00:05:24.504
So if we restricted the
domain of the function

00:05:24.504 --> 00:05:27.456
to just that, it would be invertible.

00:05:27.456 --> 00:05:28.343
It would be invertible.

00:05:28.343 --> 00:05:30.395
In fact, we could draw it's inverse.

00:05:30.395 --> 00:05:31.952
It's inverse would look like this.

00:05:31.952 --> 00:05:37.925
So, f of 6 is 2, so f
inverse of 2 should be 6

00:05:38.269 --> 00:05:43.517
and f of 8 is 0 so f
inverse of 0 should be 8.

00:05:43.937 --> 00:05:46.776
So it's inverse will look like this,

00:05:46.776 --> 00:05:48.432
it's inverse would look like this

00:05:48.432 --> 00:05:53.378
which is clearly a reflection
over the line, y equals x.

00:05:53.378 --> 00:05:55.312
Clearly a reflection over that line

00:05:55.312 --> 00:05:57.584
you see it over here, it's a reflection.

00:05:57.584 --> 00:06:02.048
And this, this inverse
function is clearly a function.

00:06:02.048 --> 00:06:04.192
It passes the vertical line
test for the same reason

00:06:04.192 --> 00:06:05.851
this passes the horizontal line test.

00:06:05.851 --> 00:06:07.512
So this is really good.

00:06:07.512 --> 00:06:10.632
So let's see negative 8 to negative 6.

00:06:10.632 --> 00:06:13.168
Well, yeah that's the same
thing on the other side.

00:06:13.168 --> 00:06:16.128
So negative 8 to negative 6, once again

00:06:16.128 --> 00:06:19.200
over this interval we do pass
the horizontal line test.

00:06:19.200 --> 00:06:20.800
So that looks good as well.

00:06:20.800 --> 00:06:23.048
And then 0 to 2.

00:06:23.048 --> 00:06:26.440
From 0 to 2, yeah it does
look like over that interval

00:06:26.440 --> 00:06:29.464
we pass the horizontal line test.

00:06:29.464 --> 23:59:59.999
So that looks good as well.


Solving exponential equation with logarithm

WEBVTT
Kind: captions
Language: en

00:00:00.722 --> 00:00:03.028
Voiceover:Solve the
equation for T and express

00:00:03.028 --> 00:00:06.189
your answer in terms
of base 10 logarithms.

00:00:06.189 --> 00:00:10.580
And this equation is 10 to
the 2T - 3 is equal to 7.

00:00:10.580 --> 00:00:14.182
We want to solve for T in
terms of base 10 logarithms.

00:00:14.182 --> 00:00:16.214
So let me get my little
scratchpad out and I've

00:00:16.214 --> 00:00:18.607
copied and pasted the same problem.

00:00:18.607 --> 00:00:21.388
So I'm just going to
rewrite it, so they have 10

00:00:21.388 --> 00:00:25.028
to the 2T-3 is equal to 7.

00:00:25.028 --> 00:00:27.069
Actually, let me color
code this a little bit.

00:00:27.069 --> 00:00:36.989
So 10 to the 2T - 3 is equal to 7.

00:00:36.989 --> 00:00:40.053
So this is clearly an
exponential form right over here.

00:00:40.053 --> 00:00:41.960
if we want to write it
in logarithmic form,

00:00:41.960 --> 00:00:43.977
where we could, that'll
essentially allow us to solve

00:00:43.977 --> 00:00:47.146
for the exponent, so we could say,

00:00:47.146 --> 00:00:50.191
this is the exact same
truth about the universe

00:00:50.191 --> 00:01:06.205
as saying that the log base
10 of 7 is equal to 2T - 3.

00:01:06.205 --> 00:01:07.850
Let's just make sure that makes sense,

00:01:07.850 --> 00:01:10.724
this is saying 10 to the 2T - 3 = 7

00:01:10.724 --> 00:01:13.362
This is saying that the power
that I need to raise 10 to

00:01:13.362 --> 00:01:19.021
to get to 7 is 2T - 3, or 10
to 2T - 3 power is equal to 7.

00:01:19.021 --> 00:01:20.880
So these are equivalent statements.

00:01:20.880 --> 00:01:23.130
What this form does is it
starts to put it into a form

00:01:23.130 --> 00:01:25.835
that's easier to solve for
T, now if we want to solve

00:01:25.835 --> 00:01:29.307
for T, we can add 3 to both sides.

00:01:29.307 --> 00:01:34.698
So if we add 3 to both
sides, we are going to get,

00:01:34.698 --> 00:01:46.717
log base 10 of 7 + 3, plus 3.

00:01:46.717 --> 00:01:48.813
And this +3 is of course
outside of the logarithm

00:01:48.813 --> 00:01:51.321
to make it clear, it's just like that.

00:01:51.321 --> 00:01:55.427
And on the right hand side
this is going to be equal to,

00:01:55.427 --> 00:01:58.788
this is going to be equal to just 2T.

00:01:58.788 --> 00:02:02.302
And now to solve for T we
just divide both sides by 2.

00:02:02.302 --> 00:02:06.932
So if we divide both sides by 2 we get,

00:02:06.932 --> 00:02:14.290
T is equal to all of this
business, log base 10 of 7 + 3

00:02:14.290 --> 00:02:19.311
all of that over 2, so let me
see if I can remember this,

00:02:19.311 --> 00:02:25.959
and write it in the actual
box that they gave us.

00:02:25.959 --> 00:02:37.488
So we want to do log base
10 of 7 and then that + 3

00:02:37.488 --> 00:02:40.646
yup it's formatting it
right, divided by 2.

00:02:40.646 --> 00:02:43.518
So that's what I'm
claiming the T is equal to

00:02:43.518 --> 00:02:46.517
in terms of Base 10 logarithms.

00:02:46.517 --> 23:59:59.999
So let me check my answer
and I got it right.


Deducing with logarithmic and exponential form

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:02.233
Voiceover:We've got 2 tables over here.

00:00:02.233 --> 00:00:05.833
This first table tells us
that for any given value of X,

00:00:05.833 --> 00:00:07.566
what is the value of b to the x?

00:00:07.566 --> 00:00:09.033
For example, if we look right over here,

00:00:09.033 --> 00:00:14.166
if x is 1.585, b to the 1.585 is 3.

00:00:14.166 --> 00:00:20.899
This is telling us that b
to the 1.585 is equal to 3.

00:00:20.899 --> 00:00:22.653
Similarly ...

00:00:22.653 --> 00:00:24.101
I can never say that ...

00:00:24.101 --> 00:00:27.633
it is telling us that b to the 2.322 is 5.

00:00:27.633 --> 00:00:29.233
b to the 2.807 is 7.

00:00:29.233 --> 00:00:31.833
b to the 2.169 is 9.

00:00:31.833 --> 00:00:33.500
Now this table over here is telling us

00:00:33.500 --> 00:00:35.766
for any given value of y,

00:00:35.766 --> 00:00:39.234
what is going to be log base b of y?

00:00:39.234 --> 00:00:43.633
This tells us that log base b of a is 0.

00:00:43.633 --> 00:00:46.500
Log base b of 2 is 1.

00:00:46.500 --> 00:00:51.566
Log base b of 2c is 1.585.

00:00:51.566 --> 00:00:54.900
Log base b of 10d, so this is
literally, this is telling us

00:00:54.900 --> 00:01:02.033
that log base b of 10d is equal to 2.322.

00:01:02.033 --> 00:01:04.033
That's what this last column tells us.

00:01:04.033 --> 00:01:08.500
Now what I challenge you
to do is pause this video,

00:01:08.500 --> 00:01:10.166
and using just the information here,

00:01:10.166 --> 00:01:11.433
and you don't need a calculator;

00:01:11.433 --> 00:01:12.766
in fact, you can't use a calculator.

00:01:12.766 --> 00:01:14.306
I forbid you;

00:01:14.306 --> 00:01:18.902
try to figure out what a, b, c, and d are

00:01:18.902 --> 00:01:21.366
just using your powers of reasoning,

00:01:21.366 --> 00:01:22.833
no calculator involved.

00:01:22.833 --> 00:01:24.300
Just use your powers of reasoning.

00:01:24.300 --> 00:01:28.533
Can you figure out what
a, b, c, and d are?

00:01:28.966 --> 00:01:30.966
I'm assuming you've given a go at it,

00:01:30.966 --> 00:01:34.633
so let's see what we can deduce from this.

00:01:34.633 --> 00:01:36.500
Here, we have just a bunch of numbers.

00:01:36.500 --> 00:01:39.035
We need to figure out what b is.

00:01:39.434 --> 00:01:45.366
These are all b to the 1.585 power is 3.

00:01:45.366 --> 00:01:47.700
I don't really know what to
make sense of this stuff here.

00:01:47.700 --> 00:01:50.433
Maybe this table will help us.

00:01:50.433 --> 00:01:51.633
This first ...

00:01:51.633 --> 00:01:53.300
Let me do these in different colors.

00:01:53.300 --> 00:01:56.233
This first column right over here tells us

00:01:56.233 --> 00:02:00.700
that log base b of a,

00:02:00.700 --> 00:02:02.500
so now y is equal to a,

00:02:02.500 --> 00:02:04.966
that that is equal to 0.

00:02:04.966 --> 00:02:07.500
Now this is an equivalent
statement to saying

00:02:07.500 --> 00:02:12.346
that b to the a power is equal to ...

00:02:12.346 --> 00:02:13.700
oh sorry, not b to the a power.

00:02:13.700 --> 00:02:15.792
This is an equivalent statement to saying

00:02:15.792 --> 00:02:22.633
b to the 0 power is equal to a.

00:02:22.633 --> 00:02:23.967
This is saying what exponent

00:02:23.967 --> 00:02:26.168
do I need to raise b to to get a?

00:02:26.168 --> 00:02:28.100
You raise it to the 0 power.

00:02:28.100 --> 00:02:30.934
This is saying b to the
0 power is equal to a.

00:02:31.300 --> 00:02:33.566
Now what is anything to the 0 power,

00:02:33.566 --> 00:02:35.167
assuming that it's not 0?

00:02:35.167 --> 00:02:37.966
If we're assuming that b is not 0,

00:02:38.119 --> 00:02:40.181
if we're assuming that b is not 0,

00:02:40.642 --> 00:02:42.500
so we're going to assume that,

00:02:42.500 --> 00:02:43.633
and we can assume,

00:02:43.633 --> 00:02:45.100
and I think that's a safe assumption

00:02:45.100 --> 00:02:47.500
because where we're raising b
to all of these other powers,

00:02:47.500 --> 00:02:49.500
we're getting a non-0 value.

00:02:49.500 --> 00:02:52.168
Since we know that b is not 0,

00:02:52.168 --> 00:02:54.568
anything with a 0 power is going to be 1.

00:02:54.568 --> 00:02:59.318
This tells us that a is equal to 1.

00:02:59.694 --> 00:03:01.700
We got one figured out.

00:03:01.700 --> 00:03:03.801
a is equal to 1.

00:03:04.166 --> 00:03:06.768
Now let's look at this
next piece of information

00:03:06.768 --> 00:03:07.694
right over here.

00:03:07.694 --> 00:03:09.100
What does that tell us?

00:03:09.100 --> 00:03:17.940
That tells us that log
base b of 2 is equal to 1.

00:03:17.940 --> 00:03:20.769
This is equivalent to saying
the power that I needed

00:03:20.769 --> 00:03:24.766
to raise b to get to to 2 is 1.

00:03:24.766 --> 00:03:26.166
Or if I want to write in exponential form,

00:03:26.166 --> 00:03:27.300
I could write this as saying

00:03:27.300 --> 00:03:33.133
that b to the first power is equal to 2.

00:03:33.233 --> 00:03:35.500
I'm raising something to the
first power and I'm getting 2?

00:03:35.500 --> 00:03:36.966
What is this thing?

00:03:36.966 --> 00:03:38.766
That means that b must be 2.

00:03:38.766 --> 00:03:40.700
2 to the first power is 2.

00:03:40.700 --> 00:03:46.366
So b is equal to 2.

00:03:46.366 --> 00:03:47.434
b to the first power is equal to 2.

00:03:47.434 --> 00:03:49.633
You could say b to the first
is equal to 2 to the first.

00:03:49.633 --> 00:03:51.038
That's also equal to 2.

00:03:51.038 --> 00:03:54.201
So b must be equal to 2.

00:03:54.366 --> 00:03:55.900
We've been able to figure that out.

00:03:55.900 --> 00:03:57.433
This is a 2 right over here.

00:03:57.433 --> 00:03:58.233
It actually makes sense.

00:03:58.233 --> 00:04:00.374
2 to the 1.585 power,
yeah, that feels right,

00:04:00.374 --> 00:04:02.233
that that's about 3.

00:04:02.233 --> 00:04:04.901
Now let's see what else we can do.

00:04:04.901 --> 00:04:07.567
Let's see if we can figure out c.

00:04:08.274 --> 00:04:09.933
Let's look at this column.

00:04:10.033 --> 00:04:12.266
Let's see what this column is telling us.

00:04:12.434 --> 00:04:15.766
That column we could read as log base b.

00:04:15.766 --> 00:04:17.768
Now our y is 2c.

00:04:17.768 --> 00:04:23.600
Log base b of 2c is equal to 1.585.

00:04:23.701 --> 00:04:25.837
Or we could read this as b,

00:04:25.837 --> 00:04:27.033
if we write in exponential form,

00:04:27.033 --> 00:04:34.200
b to the 1.585 is equal to 2c.

00:04:34.633 --> 00:04:37.376
Now what's b to the 1.585?

00:04:37.376 --> 00:04:42.366
They tell us right over here
that b to the 1.585 is 3,

00:04:42.366 --> 00:04:43.834
is 3,

00:04:43.834 --> 00:04:47.700
so this right over here is equal to 3.

00:04:47.700 --> 00:04:50.567
We get 2c is equal to 3,

00:04:50.567 --> 00:04:52.968
or divide both sides by 2,

00:04:52.968 --> 00:04:56.433
we would get c is equal to 1.5.

00:04:56.433 --> 00:04:58.066
This is working out pretty well.

00:04:58.167 --> 00:05:01.015
Now we have this last column,

00:05:01.015 --> 00:05:03.267
which I will circle in purple,

00:05:03.767 --> 00:05:05.556
and we can write this

00:05:05.556 --> 00:05:15.400
as log base b of 10d is equal to 2.322.

00:05:15.984 --> 00:05:19.967
This is saying the power I need
to raise b to to get to 10d

00:05:19.967 --> 00:05:23.300
is 2.322, or in exponential form,

00:05:23.300 --> 00:05:31.133
b to the 2.322 is equal to 10d.

00:05:31.167 --> 00:05:33.308
Now what is b to the 2.322?

00:05:33.308 --> 00:05:34.234
They tell us over here.

00:05:34.234 --> 00:05:39.702
b to the 2.322 is 5.

00:05:39.702 --> 00:05:41.369
This is equal to 5.

00:05:41.369 --> 00:05:44.567
We could write 10d is equal to 5,

00:05:44.567 --> 00:05:46.500
or divide both sides by 10.

00:05:46.500 --> 00:05:49.833
d is equal to 0.5,

00:05:49.833 --> 00:05:50.767
and we're done.

00:05:50.767 --> 00:05:55.233
We were able to figure out
what a, b, c, and d are

00:05:55.233 --> 00:05:58.201
without the use of a calculator.


Plotting points of logarithmic function

WEBVTT
Kind: captions
Language: en

00:00:00.397 --> 00:00:02.359
Voiceover:The three points plotted below

00:00:02.359 --> 00:00:05.022
are on the graph of y is
equal to b to the x power.

00:00:05.022 --> 00:00:07.245
Based only on these three points,

00:00:07.245 --> 00:00:09.692
plot the three corresponding points

00:00:09.692 --> 00:00:11.095
that must be on the graph

00:00:11.095 --> 00:00:13.390
of y is equal to log base b of x

00:00:13.390 --> 00:00:15.811
by clicking on the graph.

00:00:15.811 --> 00:00:17.835
I've actually copy and pasted this problem

00:00:17.835 --> 00:00:19.812
on my little scratch
pad so I can mark it up

00:00:19.812 --> 00:00:20.576
a little bit.

00:00:20.576 --> 00:00:22.615
What is this first function?

00:00:22.615 --> 00:00:24.845
This first function is telling us

00:00:24.845 --> 00:00:31.349
so x, and this is y is
equal to b to the x power.

00:00:31.349 --> 00:00:37.209
When x is equal to
zero, y is equal to one.

00:00:37.209 --> 00:00:39.003
That's this point right over here.

00:00:39.003 --> 00:00:44.188
When x is equal to one, b to the one power

00:00:44.188 --> 00:00:46.273
or b to the first power is equal to four.

00:00:46.273 --> 00:00:48.414
y is equal to four.

00:00:48.414 --> 00:00:49.956
Another way of thinking of this

00:00:49.956 --> 00:00:54.204
y or four is equal to b to the first power

00:00:54.204 --> 00:00:56.576
and actually we can do
[said] must be four.

00:00:56.576 --> 00:00:58.719
That's this point right over there.

00:00:58.719 --> 00:01:02.741
This point is telling us
that b to the second power

00:01:02.741 --> 00:01:04.170
is equal to 16.

00:01:04.170 --> 00:01:05.737
When x is equal to two,

00:01:05.737 --> 00:01:09.103
b to the second power, y is equal to 16.

00:01:09.103 --> 00:01:12.719
Now we want to plot the
three corresponding points

00:01:12.719 --> 00:01:13.872
on this function.

00:01:13.872 --> 00:01:16.112
Let me draw another table here.

00:01:16.112 --> 00:01:19.234
Now it's essentially the inverse function

00:01:19.234 --> 00:01:20.808
where this is going to be x

00:01:20.808 --> 00:01:26.648
and we want to calculate y
is equal to log base b of x.

00:01:28.094 --> 00:01:30.106
What are the possibilities here?

00:01:30.106 --> 00:01:31.877
What I want to do is think ...

00:01:31.877 --> 00:01:33.604
Let's take these values

00:01:33.604 --> 00:01:36.494
because these are
essentially inverse functions

00:01:36.494 --> 00:01:38.663
log is the inverse of exponents.

00:01:38.663 --> 00:01:49.038
If we take the points one, four, and 16.

00:01:51.714 --> 00:01:53.592
What is y going to be here?

00:01:53.592 --> 00:01:59.424
y is going to be log base b of one.

00:01:59.424 --> 00:02:02.494
This is saying what power
I need to raise b to

00:02:02.494 --> 00:02:03.636
to get to one.

00:02:03.636 --> 00:02:05.651
If we assume that b is non zero

00:02:05.651 --> 00:02:06.974
and that's a reasonable assumption

00:02:06.974 --> 00:02:09.328
because b to different
powers are non zero,

00:02:09.328 --> 00:02:13.965
this is going to be
zero for any non zero b.

00:02:13.965 --> 00:02:16.760
This is going to be zero
right there, over here.

00:02:16.760 --> 00:02:18.542
We have the point one comma zero,

00:02:18.542 --> 00:02:19.928
so it's that point over there.

00:02:19.928 --> 00:02:22.159
Notice this point
corresponds to this point,

00:02:22.159 --> 00:02:25.022
we have essentially
swapped the x's and y's.

00:02:25.022 --> 00:02:26.577
In general when you're taking an inverse

00:02:26.577 --> 00:02:32.398
you're going to reflect over
the line, y is equal to x

00:02:32.398 --> 00:02:34.884
and this is clearly
reflection over that line.

00:02:34.884 --> 00:02:36.566
Now let's look over here,

00:02:36.566 --> 00:02:40.616
when x is equal to four
what is log base b of four.

00:02:40.616 --> 00:02:42.811
What is the power I need to raise b to

00:02:42.811 --> 00:02:43.973
to get to four.

00:02:43.973 --> 00:02:45.232
We see right over here,

00:02:45.232 --> 00:02:47.642
b to the first power is equal to four.

00:02:47.642 --> 00:02:49.093
We already figured that out,

00:02:49.093 --> 00:02:51.775
when I take b to the first
power is equal to four.

00:02:51.775 --> 00:02:55.195
This right over here is
going to be equal to one.

00:02:55.195 --> 00:02:59.623
When x is equal to
four, y is equal to one.

00:02:59.623 --> 00:03:02.564
Notice once again, it is
a reflection over the line

00:03:02.564 --> 00:03:04.041
y is equal to x.

00:03:05.164 --> 00:03:09.214
When x is equal to 16

00:03:10.121 --> 00:03:14.956
then y is equal to log base b of 16.

00:03:14.956 --> 00:03:18.186
The power I need to
raise b to, to get to 16.

00:03:18.186 --> 00:03:19.400
Well we already know,

00:03:19.400 --> 00:03:22.341
if we take b squared, we get to 16,

00:03:22.341 --> 00:03:24.475
so this is equal to two.

00:03:24.475 --> 00:03:28.830
When x is equal to 16, y is equal to two.

00:03:28.830 --> 00:03:31.503
Notice we essentially just
swapped the x and y values

00:03:31.503 --> 00:03:33.151
for each of these points.

00:03:33.151 --> 00:03:34.669
This is y, this is a reflection

00:03:34.669 --> 00:03:37.423
over the line y is equal to x.

00:03:37.423 --> 00:03:40.816
Now, let's actually do that
on the actual interface.

00:03:40.816 --> 00:03:43.167
The whole reason is to
give you this appreciation

00:03:43.167 --> 00:03:45.090
that these are inverse
functions of each other.

00:03:45.090 --> 00:03:47.710
Let's plot the points.

00:03:47.710 --> 00:03:50.030
That point corresponded to that point,

00:03:50.030 --> 00:03:52.721
so x zero, y one

00:03:52.721 --> 00:03:54.710
corresponds to x one, y zero.

00:03:54.710 --> 00:03:56.697
Here x is one, y is four

00:03:56.697 --> 00:03:58.902
that corresponds to x four, y one.

00:03:58.902 --> 00:04:01.378
Here x is two, y is 16

00:04:01.378 --> 00:04:06.262
that corresponds to x 16, y is two.

00:04:07.262 --> 00:04:08.352
We got it right.


Writing in logarithmic and exponential form

WEBVTT
Kind: captions
Language: en

00:00:00.867 --> 00:00:01.956
Voiceover:Rewrite the following

00:00:01.956 --> 00:00:03.892
equation in logarithmic form.

00:00:03.892 --> 00:00:07.011
So they wrote 100 is equal to 10

00:00:07.011 --> 00:00:08.888
to the second power.

00:00:08.888 --> 00:00:10.460
So if we wanna write the same

00:00:10.460 --> 00:00:12.700
information, really, in logarithmic form,

00:00:12.700 --> 00:00:17.510
we could say that the power that I need

00:00:17.510 --> 00:00:21.580
to raise 10 to to get
to 100 is equal to 2,

00:00:21.580 --> 00:00:33.627
or log base 10 of 100 is equal to 2.

00:00:33.627 --> 00:00:35.155
Notice these are equivalent statements.

00:00:35.155 --> 00:00:36.699
This is just in exponential form.

00:00:36.699 --> 00:00:38.204
This is is logarithmic form.

00:00:38.204 --> 00:00:39.812
This is saying that the power I need

00:00:39.812 --> 00:00:43.099
to raise 10 to to get
to 100 is equal to 2.

00:00:43.099 --> 00:00:44.304
Which is the same thing as saying

00:00:44.304 --> 00:00:46.757
that 10 to the 2nd power is 100.

00:00:46.757 --> 00:00:49.140
10 to the second power is 100.

00:00:49.140 --> 00:00:50.599
And the way that I specify the base

00:00:50.599 --> 00:00:52.367
is by doing this
underscore right over here.

00:00:52.367 --> 00:00:54.951
So underscore 10, log base 10 of 100

00:00:54.951 --> 00:00:56.528
is equal to 2.

00:00:56.528 --> 00:00:59.376
Here they ask us to rewrite the following

00:00:59.376 --> 00:01:01.598
equation in exponential form.

00:01:01.598 --> 00:01:04.447
So this is log base 5 of 1 over 125

00:01:04.447 --> 00:01:06.168
is equal to negative 3.

00:01:06.168 --> 00:01:08.207
This is one way to
think about it is saying

00:01:08.207 --> 00:01:09.840
the power that I need to raise 5

00:01:09.840 --> 00:01:14.472
to to get to 1 over 125
is equal to negative 3

00:01:14.472 --> 00:01:22.240
or that 5 to the negative 3 power is

00:01:22.240 --> 00:01:25.496
equal to 1 over 125.

00:01:25.496 --> 00:01:27.583
And we can verify that this has formatted

00:01:27.583 --> 00:01:28.263
it the right way.

00:01:28.263 --> 00:01:30.729
5 to the negative 3 power is 1 over 125.

00:01:30.729 --> 00:01:33.855
The exact same truth about the universe,

00:01:33.855 --> 00:01:35.152
just in different forms.

00:01:35.152 --> 00:01:37.743
Logarithmic form and exponential form.

00:01:37.743 --> 00:01:39.096
So let me check my answer

00:01:39.096 --> 00:01:40.032
and make sure I got it right.

00:01:40.032 --> 00:01:41.610
And I did.


Function invertibility

WEBVTT
Kind: captions
Language: en

00:00:00.908 --> 00:00:04.308
Voiceover:We've seen that
a function maps elements

00:00:04.308 --> 00:00:06.101
from its domain.

00:00:06.101 --> 00:00:08.644
If you give it an element in its domain

00:00:08.644 --> 00:00:11.708
it will then map it to
an element in its range.

00:00:11.708 --> 00:00:13.109
So for example,

00:00:13.109 --> 00:00:14.775
if this is the domain.

00:00:14.775 --> 00:00:16.840
Let me do this in this orange color.

00:00:16.840 --> 00:00:19.567
If this is the domain of a function

00:00:19.567 --> 00:00:20.909
and let me just pick out

00:00:20.909 --> 00:00:22.377
some random elements right over here.

00:00:22.377 --> 00:00:24.967
So let's say that this is one.

00:00:24.967 --> 00:00:26.975
This is negative one.

00:00:26.975 --> 00:00:28.773
This is seven.

00:00:28.773 --> 00:00:30.440
And this is pi.

00:00:30.440 --> 00:00:31.842
Let's say these are all members

00:00:31.842 --> 00:00:33.640
of the domain of f.

00:00:33.640 --> 00:00:35.710
And let's say that f and let's say

00:00:35.710 --> 00:00:38.245
that this right over here is the range.

00:00:38.245 --> 00:00:41.639
So this is right over here is the range.

00:00:41.639 --> 00:00:45.372
And so let's say that f of one, f of one.

00:00:45.372 --> 00:00:47.508
So if you input one into the function,

00:00:47.508 --> 00:00:54.836
the function maps one to
the number negative three.

00:00:54.836 --> 00:00:58.305
And let's say it maps
the number negative one

00:00:58.305 --> 00:01:02.179
So f of negative one is equal to,

00:01:02.179 --> 00:01:03.978
is equal to seven.

00:01:03.978 --> 00:01:06.779
And let's say that f of seven,

00:01:06.779 --> 00:01:11.380
f of seven is equal to 10.

00:01:11.380 --> 00:01:13.510
But let's says that f of pi

00:01:13.510 --> 00:01:16.103
is also equal to 10.

00:01:16.103 --> 00:01:17.378
So this is the function map.

00:01:17.378 --> 00:01:18.644
Again this is a legitimate function.

00:01:18.644 --> 00:01:20.641
Remember in order to be a function

00:01:20.641 --> 00:01:23.907
for any input it has to map it to a unique

00:01:23.907 --> 00:01:26.302
it needs to map it to only one output.

00:01:26.302 --> 00:01:27.707
This wouldn't be a function

00:01:27.707 --> 00:01:29.235
if it did something like this.

00:01:29.235 --> 00:01:32.105
If you input pi instead of
saying you're gonna get 10

00:01:32.105 --> 00:01:35.979
instead of saying that
f of pi is equal to 10.

00:01:35.979 --> 00:01:36.839
This is clearly a function.

00:01:36.839 --> 00:01:38.443
You're mapping from pi to 10.

00:01:38.443 --> 00:01:41.446
For the input pi you know
you're gonna get the output 10.

00:01:41.446 --> 00:01:42.245
But if you were doing some type

00:01:42.245 --> 00:01:44.367
of a relationship where you said oh well

00:01:44.367 --> 00:01:47.304
if you put in a pi sometimes it's a 10

00:01:47.304 --> 00:01:49.437
and sometimes it's an 11.

00:01:49.437 --> 00:01:51.045
Now this would not be a function

00:01:51.045 --> 00:01:53.243
cause you're mapping pi

00:01:53.243 --> 00:01:55.380
to two different values right over here.

00:01:55.380 --> 00:01:57.377
So that's not allowed if you're thinking

00:01:57.377 --> 00:01:58.238
about a function.

00:01:58.238 --> 00:02:02.313
For any input you have
to map it to only one,

00:02:02.313 --> 00:02:03.166
only one output.

00:02:03.166 --> 00:02:06.175
Now you could have two inputs
that map to the same output

00:02:06.175 --> 00:02:07.033
because this is predictable.

00:02:07.033 --> 00:02:08.500
You know that f of seven is 10

00:02:08.500 --> 00:02:11.043
and you know that f of pi is 10.

00:02:11.043 --> 00:02:13.369
Well what I wanna do is now think about

00:02:13.369 --> 00:02:17.174
when does a function not
when it is not invertible?

00:02:17.174 --> 00:02:18.645
We've already seen that the inverse

00:02:18.645 --> 00:02:24.578
of a function should map
between an element of its range

00:02:24.578 --> 00:02:28.038
and the corresponding
element in the domain.

00:02:28.038 --> 00:02:31.435
So the inverse of f should map three,

00:02:31.435 --> 00:02:33.166
negative three to one.

00:02:33.166 --> 00:02:34.366
So f inverse.

00:02:34.366 --> 00:02:39.372
So because because f of one, f of one.

00:02:39.372 --> 00:02:40.839
Actually let me write it this way.

00:02:40.839 --> 00:02:41.509
Actually I'll just write it.

00:02:41.509 --> 00:02:44.710
Because f of one was
equal to negative three

00:02:44.710 --> 00:02:49.643
f inverse, f inverse of negative three.

00:02:49.643 --> 00:02:51.912
And the inverse function
is also a function.

00:02:51.912 --> 00:02:53.444
So it has to have the same
properties of a function.

00:02:53.444 --> 00:02:55.445
It should map to only one thing.

00:02:55.445 --> 00:02:59.179
F inverse of negative three
is going to be equal to one.

00:02:59.179 --> 00:03:05.246
F inverse of seven is going
to be equal to negative one.

00:03:05.246 --> 00:03:08.377
F inverse, this is f
inverse right over here.

00:03:08.377 --> 00:03:10.307
But now we have something
strange happening

00:03:10.307 --> 00:03:11.778
if we think about let me write this down.

00:03:11.778 --> 00:03:15.444
F inverse of seven is
equal to negative one.

00:03:15.444 --> 00:03:16.975
That makes sense because we already knew

00:03:16.975 --> 00:03:21.712
that f of f of negative
one is equal to seven.

00:03:21.712 --> 00:03:24.307
But now something interesting
happens over here.

00:03:24.307 --> 00:03:28.706
What would f inverse of 10 be?

00:03:28.706 --> 00:03:29.979
Well f inverse of 10

00:03:29.979 --> 00:03:31.635
if I'm trying to map it 10

00:03:31.635 --> 00:03:33.507
there's two elements of the domain

00:03:33.507 --> 00:03:34.772
that when I applied the function

00:03:34.772 --> 00:03:37.573
got us to 10 in the range.

00:03:37.573 --> 00:03:39.037
So if you're taking, if you're trying

00:03:39.037 --> 00:03:40.439
to see whether f is invertible

00:03:40.439 --> 00:03:44.167
then the inverse function
should map this element

00:03:44.167 --> 00:03:45.976
to only one element of the domain.

00:03:45.976 --> 00:03:48.840
But its mapping it to two
elements of the domain.

00:03:48.840 --> 00:03:55.906
So is this equal to seven
or is this equal to pi?

00:03:55.906 --> 00:03:58.445
And because the inverse
mapping can't be done

00:03:58.445 --> 00:04:01.039
where you only map it with a unique thing

00:04:01.039 --> 00:04:03.501
this function is not invertible.

00:04:03.501 --> 00:04:06.712
This is not not invertible.

00:04:06.712 --> 00:04:09.774
The inverse mapping, the inverse mapping

00:04:09.774 --> 00:04:11.175
from the range to the domain

00:04:11.175 --> 00:04:13.906
the inverse relationship
is not a function.

00:04:13.906 --> 00:04:17.242
It does not it can't take
an element in our range

00:04:17.242 --> 00:04:19.833
and map it to a unique
element to the domain.

00:04:19.833 --> 00:04:20.567
For 10 we don't know.

00:04:20.567 --> 00:04:23.566
Does that map to pi or
does that map to seven?

00:04:23.566 --> 00:04:25.500
So this function f is a function.

00:04:25.500 --> 00:04:27.572
For any member of its
domain it goes to a unique

00:04:27.572 --> 00:04:31.712
it goes to its a very well
defined element in the range.

00:04:31.712 --> 00:04:33.704
But the other way around isn't true.

00:04:33.704 --> 00:04:35.105
For the inverse function

00:04:35.105 --> 00:04:37.103
for any element in this set

00:04:37.103 --> 00:04:39.505
it doesn't go to only
one element in that set.

00:04:39.505 --> 00:04:42.375
This 10 could be mapped
to either pi or seven

00:04:42.375 --> 00:04:44.441
so f inverse so another way

00:04:44.441 --> 00:04:47.501
to think about this is
not an invertible f.

00:04:47.501 --> 00:04:49.113
F is not invertible.

00:04:49.113 --> 00:04:52.828
F is a function but the inverse
mapping is not a function

00:04:52.828 --> 00:04:56.351
so we do not so we would say
that f is not invertible.