Subtracting rational expressions: unlike denominators

WEBVTT
Kind: captions
Language: en

00:00:00.188 --> 00:00:00.885
- [Voiceover] So right over here, we have

00:00:00.885 --> 00:00:03.160
one rational expression being subtracted

00:00:03.160 --> 00:00:04.810
from another rational expression.

00:00:04.810 --> 00:00:06.364
I encourage you to pause the video

00:00:06.364 --> 00:00:08.384
and see what this would result in.

00:00:08.384 --> 00:00:10.649
So actually do the subtraction.

00:00:10.649 --> 00:00:12.041
Alright, let's do this together.

00:00:12.041 --> 00:00:15.547
And if we're subtracting
two rational expressions,

00:00:15.547 --> 00:00:18.588
we'd like to have them
have the same denominator.

00:00:18.588 --> 00:00:20.864
And they clearly don't
have the same denominator

00:00:20.864 --> 00:00:23.011
and so we need to find
a common denominator.

00:00:23.011 --> 00:00:24.544
And a common denominator is one that

00:00:24.544 --> 00:00:27.504
is going to be divisible
by either of these

00:00:27.504 --> 00:00:30.423
and then we can multiply them by

00:00:30.423 --> 00:00:32.478
an appropriate expression or number

00:00:32.478 --> 00:00:34.759
so that it becomes the common denominator.

00:00:34.759 --> 00:00:35.978
So the easiest common denominator

00:00:35.978 --> 00:00:38.787
I can think of especially
because these factors,

00:00:38.787 --> 00:00:40.784
these two expressions have no factors

00:00:40.784 --> 00:00:43.059
in common, would just be their product.

00:00:43.059 --> 00:00:45.683
So this is going to be equal to,

00:00:45.683 --> 00:00:48.121
so we could just multiply these two.

00:00:48.121 --> 00:00:50.780
So this is going to be, so let me do,

00:00:50.780 --> 00:00:54.877
let me do this one right
over here, in magenta.

00:00:54.877 --> 00:00:56.608
So this is going to be equal to

00:00:56.608 --> 00:00:57.698
the common denominator.

00:00:57.698 --> 00:00:59.637
If I say, if I want to just multiply

00:00:59.637 --> 00:01:01.146
those two denominators.

00:01:01.146 --> 00:01:04.896
For this one, I'll have my 8x plus seven

00:01:04.896 --> 00:01:08.379
and now I'm going to
multiply by 3x plus one.

00:01:08.379 --> 00:01:10.630
Multiplying it by the other denominator

00:01:10.630 --> 00:01:13.288
and I had negative 5x in the numerator

00:01:13.288 --> 00:01:14.867
but if I'm going to
multiply the denominator

00:01:14.867 --> 00:01:16.411
by 3x plus one and I don't want to change

00:01:16.411 --> 00:01:18.269
the value of the expression, we'll have

00:01:18.269 --> 00:01:21.496
to multiply the numerator
by 3x plus one, as well.

00:01:21.496 --> 00:01:24.166
Notice, 3x plus one divided
by 3x plus one is just one

00:01:24.166 --> 00:01:26.430
and you'll be left with
what we started with.

00:01:26.430 --> 00:01:31.190
And from that, we are going
to subtract all of this.

00:01:31.190 --> 00:01:32.025
Now there's a couple of ways you

00:01:32.025 --> 00:01:33.267
can think of this subtraction.

00:01:33.267 --> 00:01:35.995
I could just write a
minus sign right over here

00:01:35.995 --> 00:01:40.396
and do the same thing that I
just did for the first term.

00:01:40.396 --> 00:01:41.591
Or another way to think about it

00:01:41.591 --> 00:01:43.065
and actually for this particular case,

00:01:43.065 --> 00:01:45.387
I like thinking about it better this way,

00:01:45.387 --> 00:01:48.035
is to just add the negative of this.

00:01:48.035 --> 00:01:50.332
So if I just multiply negative
one times this expression,

00:01:50.332 --> 00:01:53.653
I'd get negative 6x
third over 3x plus one.

00:01:53.653 --> 00:01:55.532
If I had more terms up
here, in the numerator,

00:01:55.532 --> 00:01:57.368
I would have to be careful to distribute

00:01:57.368 --> 00:02:00.154
that negative sign, but
here, I only have one term.

00:02:00.154 --> 00:02:02.313
So I just made a negative and so I can say

00:02:02.313 --> 00:02:07.026
this is going to be plus
and, let me do this,

00:02:07.026 --> 00:02:11.090
in a new color, do this in green.

00:02:11.090 --> 00:02:13.411
Our common denominator,
we already established,

00:02:13.411 --> 00:02:15.942
is the product of our two denominators

00:02:15.942 --> 00:02:20.942
so it is going to be 8x plus
seven times 3x plus one.

00:02:24.544 --> 00:02:26.541
Now if we multiply the denominator here

00:02:26.541 --> 00:02:29.826
was 3x plus one, we're
multiplying it by 8x plus seven.

00:02:29.826 --> 00:02:30.744
So that mean's we have to multiply

00:02:30.744 --> 00:02:33.089
the numerator by 8x plus seven as well.

00:02:33.089 --> 00:02:38.089
8x plus seven times negative
6x to the third power.

00:02:39.265 --> 00:02:42.329
Notice, 8x plus seven divided
by 8x plus seven is one.

00:02:42.329 --> 00:02:44.372
If you were to do that, you would get back

00:02:44.372 --> 00:02:47.333
to your original
expression right over here,

00:02:47.333 --> 00:02:50.212
the negative 6x to the
third over 3x plus one.

00:02:50.212 --> 00:02:51.744
And now, we're ready to add.

00:02:51.744 --> 00:02:53.648
This is all going to be equal to,

00:02:53.648 --> 00:02:56.749
I'll write the denominator in white.

00:02:56.749 --> 00:02:58.721
So we have our common denominator,

00:02:58.721 --> 00:03:03.701
8x plus seven times 3x plus one.

00:03:03.701 --> 00:03:05.849
Now, in the magenta, I would want

00:03:05.849 --> 00:03:08.902
to distribute the negative 5x,

00:03:08.902 --> 00:03:13.902
so negative 5x times positive
3x is negative 15x squared.

00:03:16.603 --> 00:03:21.583
And then, negative 5x
times one is minus 5x.

00:03:21.583 --> 00:03:24.486
And then, in the green, I would have,

00:03:24.486 --> 00:03:26.645
let's see, I'll distribute the negative 6x

00:03:26.645 --> 00:03:29.304
to the third power, so
negative 6x to the third

00:03:29.304 --> 00:03:32.206
times positive 8x is going to be,

00:03:32.206 --> 00:03:37.206
negative 48x to the fourth power.

00:03:37.441 --> 00:03:38.881
And then negative 6x to the third

00:03:38.881 --> 00:03:43.282
times positive seven is
going to be negative 42,

00:03:43.282 --> 00:03:47.402
negative 42x to the third.

00:03:47.402 --> 00:03:50.908
And I think I'm done
because there's no more,

00:03:50.908 --> 00:03:53.404
there's, you know, I only
have one fourth-degree term,

00:03:53.404 --> 00:03:55.807
one third-degree term,
one second-degree term,

00:03:55.807 --> 00:03:57.538
one first-degree term, and that's it.

00:03:57.538 --> 00:03:59.441
There's no more simplification here.

00:03:59.441 --> 00:04:02.940
Some of you might want to just write it

00:04:02.940 --> 00:04:05.536
in descending degree order,
so you could write it

00:04:05.536 --> 00:04:10.536
as negative 48x to the fourth minus 42x

00:04:11.422 --> 00:04:16.422
to the third minus 15x squared minus 5x.

00:04:18.120 --> 00:04:23.120
All of that over 8x plus
seven times 3x plus one.

00:04:27.930 --> 00:04:29.624
But either way, we are all done.

00:04:29.624 --> 00:04:31.493
I mean, it looks like up here, yeah,

00:04:31.493 --> 00:04:33.513
there's no, nothing to factor out.

00:04:33.513 --> 00:04:34.965
These two are divisible by five.

00:04:34.965 --> 00:04:37.972
These are divisible by six but even

00:04:37.972 --> 00:04:40.107
if I were to factor that
out, nothing over here,

00:04:40.107 --> 00:04:42.824
down here, no five or six to factor out.

00:04:42.824 --> 00:04:45.645
Yeah, so it looks like we are all done.


Adding rational expression: unlike denominators

WEBVTT
Kind: captions
Language: en

00:00:00.259 --> 00:00:02.233
- [Voiceover] Pause the
video and try to add

00:00:02.233 --> 00:00:04.601
these two rational expressions.

00:00:04.601 --> 00:00:06.343
Okay, I'm assuming you've had a go at it.

00:00:06.343 --> 00:00:09.082
Now we can work through this together.

00:00:09.082 --> 00:00:10.777
So the first thing that you might have hit

00:00:10.777 --> 00:00:12.101
when you tried to do it,

00:00:12.101 --> 00:00:14.493
is you realized that they
have different denominators

00:00:14.493 --> 00:00:15.932
and it's hard to add fractions

00:00:15.932 --> 00:00:17.766
when they have different denominators.

00:00:17.766 --> 00:00:18.695
You need to rewrite them

00:00:18.695 --> 00:00:21.017
so that you have a common denominator.

00:00:21.017 --> 00:00:23.269
And the easiest way to
get a common denominator

00:00:23.269 --> 00:00:25.336
is you can just multiply
the two denominators,

00:00:25.336 --> 00:00:26.474
especially in case like this

00:00:26.474 --> 00:00:28.958
where they don't seem
to share any factors.

00:00:28.958 --> 00:00:31.257
Both of these are about
as factor as you can get

00:00:31.257 --> 00:00:33.486
and they don't share anything in common.

00:00:33.486 --> 00:00:36.667
And so let's set up a common denominator.

00:00:36.667 --> 00:00:39.222
So this is going to be equal to

00:00:39.222 --> 00:00:42.682
it's going to be equal to something

00:00:42.682 --> 00:00:44.261
let's see, it's going
to be equal to something

00:00:44.261 --> 00:00:46.118
over our common denominator.

00:00:46.118 --> 00:00:47.574
Let's make it

00:00:47.574 --> 00:00:49.454
let's make it 2x,

00:00:49.454 --> 00:00:50.987
I'm going to do this in another color.

00:00:50.987 --> 00:00:54.903
So we're going to make it 2x-3

00:00:54.903 --> 00:00:57.666
times 3x+1

00:00:57.666 --> 00:01:00.961
times 3x+1

00:01:00.961 --> 00:01:03.771
and then plus

00:01:03.771 --> 00:01:07.315
plus something else

00:01:07.315 --> 00:01:10.240
over 2x-3

00:01:10.240 --> 00:01:13.096
2x-3

00:01:13.096 --> 00:01:14.872
times 3x+1.

00:01:14.872 --> 00:01:19.202
Times 3x+1.

00:01:19.202 --> 00:01:21.287
And so to go from 2x,

00:01:21.287 --> 00:01:24.328
to go from just a 2x-3
here the denominator

00:01:24.328 --> 00:01:27.416
to a (2x-3)(3x+1)

00:01:27.416 --> 00:01:29.844
we multiply the denominator by 3x+1.

00:01:29.844 --> 00:01:31.082
So if we do that to the denominator,

00:01:31.082 --> 00:01:32.455
we don't want to change the value

00:01:32.455 --> 00:01:33.593
of the rational expression.

00:01:33.593 --> 00:01:36.216
We'd also have to do
that to the numerator.

00:01:36.216 --> 00:01:39.849
So the original numerator was 5x.

00:01:39.849 --> 00:01:41.683
I'll do that in blue color.

00:01:41.683 --> 00:01:45.119
So the original numerator was 5x

00:01:45.119 --> 00:01:47.952
and now we're going to
multiply it by the 3x+1,

00:01:47.952 --> 00:01:50.831
so times 3x+1.

00:01:50.831 --> 00:01:53.525
Notice I didn't change the
value of this expression.

00:01:53.525 --> 00:01:57.512
I multiplied it by 3x+1 over 3x+1,

00:01:57.512 --> 00:02:02.232
which is 1 as long as
3x+1 does not equal zero.

00:02:02.232 --> 00:02:04.438
So let's do the same thing over here.

00:02:04.438 --> 00:02:08.640
Over here I have a denominator of 3x+1,

00:02:08.640 --> 00:02:10.962
I multiplied it by 2x-3,

00:02:10.962 --> 00:02:12.286
so I would take my numerator,

00:02:12.286 --> 00:02:15.652
which is -4x²

00:02:15.652 --> 00:02:19.321
and I would also multiply it by 2x-3.

00:02:19.321 --> 00:02:22.270
2x-3.

00:02:22.270 --> 00:02:24.081
Let me put parentheses around this

00:02:24.081 --> 00:02:27.726
so it doesn't look like
I'm subtracting 4x².

00:02:27.726 --> 00:02:30.094
And so then I can rewrite
all of this business

00:02:30.094 --> 00:02:33.042
as being equal to,

00:02:33.042 --> 00:02:35.118
well, in the numerator,

00:02:35.118 --> 00:02:38.647
in the numerator I'm
going to have 5x times 3x

00:02:38.647 --> 00:02:42.269
which is 15x²

00:02:42.269 --> 00:02:47.215
5x times 1, which is + 5x,

00:02:47.215 --> 00:02:49.412
and then over here,

00:02:49.412 --> 00:02:51.107
let me do this in green,

00:02:51.107 --> 00:02:55.727
let's see, I could do -4x times 2x

00:02:55.727 --> 00:02:59.860
which would be -8x²

00:02:59.860 --> 00:03:02.542
and then -4x times -3

00:03:02.542 --> 00:03:05.592
which is +12x².

00:03:05.592 --> 00:03:06.924
Did I do that right?

00:03:06.924 --> 00:03:08.224
Negative...

00:03:08.224 --> 00:03:09.547
Oh, let me be very careful.

00:03:09.547 --> 00:03:12.984
- 4, my spider sense could tell
that I did something shady.

00:03:12.984 --> 00:03:14.934
In fact, if you want to pause
the video you could see,

00:03:14.934 --> 00:03:17.395
try to figure out what
I just did that's wrong.

00:03:17.395 --> 00:03:20.792
So -4x² times 2x

00:03:20.792 --> 00:03:23.722
is -8x to the third power.

00:03:23.722 --> 00:03:28.534
- 8x³

00:03:28.534 --> 00:03:32.720
and then -4x² times -3 is 12x²

00:03:32.720 --> 00:03:36.064
and then our entire denominator

00:03:36.064 --> 00:03:37.201
our entire denominator

00:03:37.201 --> 00:03:38.548
we have a common denominator now

00:03:38.548 --> 00:03:40.777
so we were able to just add everything

00:03:40.777 --> 00:03:43.426
is 2x-3

00:03:43.426 --> 00:03:46.064
2x-3

00:03:46.064 --> 00:03:49.004
times 3x+1

00:03:49.004 --> 00:03:51.871
times 3x+1

00:03:51.871 --> 00:03:54.123
and let's see, how can we simplify this?

00:03:54.123 --> 00:03:57.350
So this is all going to be equal to,

00:03:57.350 --> 00:03:59.022
let me draw,

00:03:59.022 --> 00:04:01.762
make sure we recognize
it's a rational expression,

00:04:01.762 --> 00:04:05.291
and so let's see, we can look at, we can,

00:04:05.291 --> 00:04:09.796
our highest degree term here is the -8x³

00:04:09.796 --> 00:04:11.894
so it's -8

00:04:11.894 --> 00:04:15.404
- 8x³

00:04:15.404 --> 00:04:18.688
and then we have a 15x²

00:04:18.688 --> 00:04:21.057
and we also have a 12x².

00:04:21.057 --> 00:04:23.143
We could add those two together

00:04:23.143 --> 00:04:26.853
to get a 27x²

00:04:26.853 --> 00:04:30.631
so we've already taken care of this,

00:04:30.631 --> 00:04:32.999
we've taken care, let me
do it in that green color,

00:04:32.999 --> 00:04:35.901
so we've taken of this,

00:04:35.901 --> 00:04:38.850
we've taken care of those two
and we're just left with a 5x,

00:04:38.850 --> 00:04:42.890
so + 5x

00:04:42.890 --> 00:04:45.677
and then all of that is over

00:04:45.677 --> 00:04:50.677
2x-3 times 3x+1

00:04:51.984 --> 00:04:53.981
3x+1

00:04:53.981 --> 00:04:55.630
and we are

00:04:55.630 --> 00:04:58.161
and we are all done.

00:04:58.161 --> 00:05:00.065
It doesn't seem like there's any easy way

00:05:00.065 --> 00:05:01.806
to simplify this further.

00:05:01.806 --> 00:05:03.617
You could factor out an
x out of the numerator

00:05:03.617 --> 00:05:05.057
but that's not going to cancel out

00:05:05.057 --> 00:05:06.357
with anything in the denominator

00:05:06.357 --> 00:05:09.115
and it looks like we are all done.


Solving square-root equations: no solution

WEBVTT
Kind: captions
Language: en

00:00:00.027 --> 00:00:02.195
- [Voiceover] Let's say that
we have the radical equation,

00:00:02.195 --> 00:00:07.146
the square root of three x minus seven

00:00:07.146 --> 00:00:10.213
plus the square root

00:00:10.213 --> 00:00:13.517
of two x minus one

00:00:13.517 --> 00:00:15.701
is equal to zero.

00:00:15.701 --> 00:00:16.888
I encourage you to pause the video

00:00:16.888 --> 00:00:18.639
and see if you can solve for x

00:00:18.639 --> 00:00:21.899
before we work through it together.

00:00:21.899 --> 00:00:24.344
Alright, so one thing we could do is

00:00:24.344 --> 00:00:25.913
we could try to isolate
each of the radicals

00:00:25.913 --> 00:00:27.604
on either side of the equation.

00:00:27.604 --> 00:00:31.518
So let's subtract this one from both sides

00:00:31.518 --> 00:00:33.903
so I can get it onto the right-hand side,

00:00:33.903 --> 00:00:35.925
or a version of it on the right-hand side.

00:00:35.925 --> 00:00:38.038
So I'm subtracting it
from the left-hand side

00:00:38.038 --> 00:00:42.826
and from the right-hand side.

00:00:42.826 --> 00:00:46.048
And so, this is going to get us,

00:00:46.048 --> 00:00:49.036
that is going to get us,

00:00:49.036 --> 00:00:52.326
on the left-hand side I
just have square root,

00:00:52.326 --> 00:00:54.017
these cancel out so I'm just left with

00:00:54.017 --> 00:00:57.910
the square root of three x minus seven

00:00:57.910 --> 00:00:59.812
is going to be equal to this,

00:00:59.812 --> 00:01:04.803
the negative of the square
root of two x minus one.

00:01:04.803 --> 00:01:08.516
So now we can square both sides.

00:01:08.516 --> 00:01:10.720
And we always have to be
careful when we're doing that

00:01:10.720 --> 00:01:12.621
because whether we're
squaring the positive

00:01:12.621 --> 00:01:14.232
or the negative square root
here, we're going to get

00:01:14.232 --> 00:01:15.469
the same value.

00:01:15.469 --> 00:01:17.461
So the solution we might
get might be the version

00:01:17.461 --> 00:01:19.745
when we're solving for
the positive square root,

00:01:19.745 --> 00:01:21.043
not when we take the negative of it.

00:01:21.043 --> 00:01:22.483
So we have to test our
solutions at the end

00:01:22.483 --> 00:01:24.173
to make sure that they are actually valid

00:01:24.173 --> 00:01:25.964
for our original equation.

00:01:25.964 --> 00:01:28.817
But if you square both
sides, on the left-hand side

00:01:28.817 --> 00:01:32.354
we're going to get three x minus seven,

00:01:32.354 --> 00:01:34.528
and on the right-hand
side, a negative square's

00:01:34.528 --> 00:01:35.785
just a positive, and the square root of

00:01:35.785 --> 00:01:40.303
two x minus one squared is
going to be two x minus one.

00:01:40.303 --> 00:01:42.920
Now let's see, we can
subtract two x from both sides

00:01:42.920 --> 00:01:45.455
to get all of our xs on one side,

00:01:45.455 --> 00:01:47.840
so I'm trying to get rid of this.

00:01:47.840 --> 00:01:50.436
And we can add seven to both sides

00:01:50.436 --> 00:01:52.790
because I'm trying to get
rid of the negative seven.

00:01:52.790 --> 00:01:55.688
So add seven to both sides.

00:01:55.688 --> 00:01:58.304
And we are going to get,

00:01:58.304 --> 00:02:01.775
we are going to get
three x minus two x is x,

00:02:01.775 --> 00:02:04.311
is equal to negative one plus seven.

00:02:04.311 --> 00:02:06.544
x is equal to six.

00:02:06.544 --> 00:02:08.929
Now let's verify that this actually works.

00:02:08.929 --> 00:02:10.710
So, if we look at our original equation,

00:02:10.710 --> 00:02:14.816
the square root of three times six

00:02:14.816 --> 00:02:19.737
minus seven, minus seven, needs to plus,

00:02:19.737 --> 00:02:24.146
plus, the square root of
two times six minus one,

00:02:24.146 --> 00:02:26.832
needs to be equals to zero.

00:02:26.832 --> 00:02:28.462
So does this actually work out?

00:02:28.462 --> 00:02:30.243
Three times six minus seven.

00:02:30.243 --> 00:02:33.681
So this is going to be
the square root of eleven

00:02:33.681 --> 00:02:36.813
plus the square root of eleven

00:02:36.813 --> 00:02:38.081
needs to be equal to zero,

00:02:38.081 --> 00:02:39.550
which clearly is not
going to be equal to zero.

00:02:39.550 --> 00:02:41.905
This is two square roots of eleven,

00:02:41.905 --> 00:02:43.776
which does not equal zero.

00:02:43.776 --> 00:02:45.137
So this does not work, and you might say,

00:02:45.137 --> 00:02:46.253
wait, how did this happen?

00:02:46.253 --> 00:02:48.004
I did all of this nice, neat algebra,

00:02:48.004 --> 00:02:49.242
I didn't make any mistakes,

00:02:49.242 --> 00:02:51.023
but I got something that doesn't work.

00:02:51.023 --> 00:02:53.136
Well, this right here is
an extraneous solution.

00:02:53.136 --> 00:02:54.886
Why is it an extraneous solution?

00:02:54.886 --> 00:02:59.293
Because it's actually the
solution to the equation,

00:02:59.293 --> 00:03:00.803
it's a solution to the equation,

00:03:00.803 --> 00:03:04.183
the square root of three x minus seven

00:03:04.183 --> 00:03:08.068
minus the square root of two x minus one

00:03:08.068 --> 00:03:09.426
is equal to zero.

00:03:09.426 --> 00:03:11.388
And you might say, well,
if it's a solution to that,

00:03:11.388 --> 00:03:13.773
if it's the solution to
this thing right over here,

00:03:13.773 --> 00:03:14.829
how did I get the answer

00:03:14.829 --> 00:03:16.641
while I'm trying to do
algebraic steps there?

00:03:16.641 --> 00:03:18.421
Well, the key is when we added,

00:03:18.421 --> 00:03:20.927
when we took this on the
right-hand side and squared it,

00:03:20.927 --> 00:03:22.828
well it all boiled down to this,

00:03:22.828 --> 00:03:24.851
regardless of what starting
point you started with,

00:03:24.851 --> 00:03:26.441
due to the exact same thing
you would've gotten to that

00:03:26.441 --> 00:03:28.303
say, this point right over here.

00:03:28.303 --> 00:03:30.576
So the solution to this ended up

00:03:30.576 --> 00:03:32.413
being the solution to this starting point,

00:03:32.413 --> 00:03:35.176
versus the one that we
originally started with.

00:03:35.176 --> 00:03:38.619
So this one, interestingly,
has no solutions.

00:03:38.619 --> 00:03:39.902
And it'd actually be fun to think about

00:03:39.902 --> 00:03:41.268
why it has no solutions.

00:03:41.268 --> 00:03:42.386
We've shown, to a certain degree,

00:03:42.386 --> 00:03:43.719
the only solution you got

00:03:43.719 --> 00:03:45.377
by taking reasonable, algebraic steps

00:03:45.377 --> 00:03:46.579
is an extraneous one.

00:03:46.579 --> 00:03:48.302
It's a solution to a different equation,

00:03:48.302 --> 00:03:51.158
that gets, that has a
common intermediate step,

00:03:51.158 --> 00:03:53.385
but it's also fun to think about why,

00:03:53.385 --> 00:03:57.730
why this right over here is impossible.


Solving square-root equations: two solutions

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:01.982
- [Voiceover] Let's say
that we have the equation

00:00:01.982 --> 00:00:06.841
six plus three w is equal to

00:00:06.841 --> 00:00:11.822
the square root of two w plus 12

00:00:11.822 --> 00:00:14.207
plus two w.

00:00:14.207 --> 00:00:17.648
See if you can pause the
video and solve for w,

00:00:17.648 --> 00:00:19.248
and it might have more than one solution,

00:00:19.248 --> 00:00:20.877
so keep that in mind.

00:00:20.877 --> 00:00:22.779
Alright, now let's work
through this together.

00:00:22.779 --> 00:00:23.836
So the first thing I'd like to do

00:00:23.836 --> 00:00:25.274
whenever I see one of
these radical equations

00:00:25.274 --> 00:00:28.112
is just isolate the radical
on one side of the equation.

00:00:28.112 --> 00:00:31.492
So let's subtract two w from both sides.

00:00:31.492 --> 00:00:33.696
I want to get rid of that two
w from the right-hand side.

00:00:33.696 --> 00:00:35.266
I just want the radical sign.

00:00:35.266 --> 00:00:38.133
And if I subtract two w from both sides,

00:00:38.133 --> 00:00:39.552
what am I left with?

00:00:39.552 --> 00:00:42.276
Well, on the left-hand
side, I am left with

00:00:42.276 --> 00:00:45.343
six plus three w minus two w.

00:00:45.343 --> 00:00:47.300
Well, three of something
take away two of 'em,

00:00:47.300 --> 00:00:48.910
you're going to be left with w.

00:00:48.910 --> 00:00:51.144
Six plus w is equal to,

00:00:51.144 --> 00:00:52.955
these cancel out, we're left with

00:00:52.955 --> 00:00:57.242
the square root of two w plus 12.

00:00:57.242 --> 00:00:58.660
And to get rid of the radical,

00:00:58.660 --> 00:01:00.743
we're going to square both sides,

00:01:00.743 --> 00:01:03.158
and we've seen before that
this process right over here,

00:01:03.158 --> 00:01:04.063
it's a little bit tricky,

00:01:04.063 --> 00:01:04.949
because when you're squaring a radical

00:01:04.949 --> 00:01:06.911
in a radical equation like this,

00:01:06.911 --> 00:01:11.016
and then you solve, you might
find an extraneous solution.

00:01:11.016 --> 00:01:12.133
What do I mean by that?

00:01:12.133 --> 00:01:13.370
Well, we're going to get the same result

00:01:13.370 --> 00:01:16.068
whether we square this or
whether we square that,

00:01:16.068 --> 00:01:18.454
because when you square a
negative, it becomes a positive.

00:01:18.454 --> 00:01:21.080
But those are fundamentally
two different equations.

00:01:21.080 --> 00:01:22.800
We only want the solutions that satisfy

00:01:22.800 --> 00:01:24.672
the one that doesn't
have the negative there.

00:01:24.672 --> 00:01:26.181
So that's why we're going
to test our solutions

00:01:26.181 --> 00:01:29.260
to make sure they're valid
for our original equation.

00:01:29.260 --> 00:01:31.041
So, if we square both sides,

00:01:31.041 --> 00:01:34.995
on the left-hand side,
we're going to have,

00:01:34.995 --> 00:01:37.319
well, it's gonna be w squared

00:01:37.319 --> 00:01:40.459
plus two times their product.

00:01:40.459 --> 00:01:43.603
So, two times six times w, so that's 12 w,

00:01:43.603 --> 00:01:46.476
plus six squared, 36,

00:01:46.476 --> 00:01:50.159
Is equal to, now, if
you take the square root

00:01:50.159 --> 00:01:53.962
and square it, you're going to
be left with two plus twelve.

00:01:53.962 --> 00:01:58.429
Now, we can subtract two
w and 12 from both sides,

00:01:58.429 --> 00:01:59.648
so let's do that.

00:01:59.648 --> 00:02:02.385
So then we can get it into
a standard quadratic form.

00:02:02.385 --> 00:02:05.827
So let's subtract two w from both sides

00:02:05.827 --> 00:02:09.479
and let's subtract 12 from both sides.

00:02:09.479 --> 00:02:11.562
So, subtract 12 from the right.

00:02:11.562 --> 00:02:12.558
Subtract 12 here.

00:02:12.558 --> 00:02:13.795
And once again, I just
want to get rid of this

00:02:13.795 --> 00:02:15.104
on the right-hand side,

00:02:15.104 --> 00:02:17.670
and I'm going to be left with,

00:02:17.670 --> 00:02:20.266
I am going to be left with,

00:02:20.266 --> 00:02:22.259
on the left-hand side, it's gonna be

00:02:22.259 --> 00:02:23.677
w squared.

00:02:23.677 --> 00:02:27.300
See, 12 w minus two w is plus ten w,

00:02:27.300 --> 00:02:30.700
and then 36 minus twelve is plus 24,

00:02:30.700 --> 00:02:34.112
is equal to zero.

00:02:34.112 --> 00:02:35.440
Now, let's see, to solve this,

00:02:35.440 --> 00:02:36.427
let's see, is this factorable?

00:02:36.427 --> 00:02:38.892
Are there two numbers that up to ten

00:02:38.892 --> 00:02:40.734
and whose product is 24?

00:02:40.734 --> 00:02:43.299
What jumps out at me is six and four.

00:02:43.299 --> 00:02:46.318
So we can rewrite this as w plus four

00:02:46.318 --> 00:02:50.393
times w plus six is equal to zero.

00:02:50.393 --> 00:02:52.476
And so, if I have the
product of two things

00:02:52.476 --> 00:02:53.713
equaling zero,

00:02:53.713 --> 00:02:55.856
well to solve this,
either one or both of them

00:02:55.856 --> 00:02:56.732
could be equal zero.

00:02:56.732 --> 00:02:58.301
Zero times anything is gonna be zero.

00:02:58.301 --> 00:03:01.848
So, w plus four is equal to zero,

00:03:01.848 --> 00:03:05.572
or w plus six is equal to zero.

00:03:05.572 --> 00:03:07.835
And over here, if you
subtract four from both sides,

00:03:07.835 --> 00:03:10.401
you get w is equal to negative four.

00:03:10.401 --> 00:03:12.811
Or, subtract six from both sides here.

00:03:12.811 --> 00:03:14.678
w is equal to negative six.

00:03:14.678 --> 00:03:17.847
Now let's verify that these
actually are solutions

00:03:17.847 --> 00:03:18.873
to our original equation.

00:03:18.873 --> 00:03:20.141
Remember, our original equation was six.

00:03:20.141 --> 00:03:22.073
I'll rewrite it here.

00:03:22.073 --> 00:03:25.001
Our original equation was six plus three w

00:03:25.001 --> 00:03:29.981
is equal to the square
root of two w plus 12

00:03:29.981 --> 00:03:32.125
plus two w.

00:03:32.125 --> 00:03:35.022
So let's see, if w is
equal to negative four,

00:03:35.022 --> 00:03:38.571
if w is equal to negative four right over,

00:03:38.571 --> 00:03:40.050
let me do something different,

00:03:40.050 --> 00:03:41.680
is equal to negative four.

00:03:41.680 --> 00:03:45.574
So, that's going to be six
plus three times negative four

00:03:45.574 --> 00:03:50.574
is equal to the square root
of two times negative four

00:03:50.705 --> 00:03:54.599
plus 12 plus two times negative four.

00:03:54.599 --> 00:03:58.644
So this would be, this
is, negative twelve here.

00:03:58.644 --> 00:04:00.867
This is negative eight here.

00:04:00.867 --> 00:04:02.937
This is negative eight here,

00:04:02.937 --> 00:04:06.789
so you have six plus negative
twelve, which is negative six,

00:04:06.789 --> 00:04:11.769
is equal to the square root
of negative eight plus 12,

00:04:11.769 --> 00:04:15.814
is four, plus negative eight.

00:04:15.814 --> 00:04:17.565
So that would be negative six is equal to

00:04:17.565 --> 00:04:19.647
two plus negative eight,

00:04:19.647 --> 00:04:21.187
which is absolutely true.

00:04:21.187 --> 00:04:23.602
So this is definitely a solution.

00:04:23.602 --> 00:04:26.952
And let's try w is equal to negative six.

00:04:26.952 --> 00:04:30.303
So, we is equal to negative six.

00:04:30.303 --> 00:04:33.100
So, we're gonna get, if we look
up here, we're going to have

00:04:33.100 --> 00:04:36.469
six plus three times negative six

00:04:36.469 --> 00:04:40.819
is equal to the square root
of two times negative six

00:04:40.819 --> 00:04:45.083
plus 12 plus two w.

00:04:45.083 --> 00:04:47.888
So, this is gonna be negative 18.

00:04:47.888 --> 00:04:49.929
This is going to be negative 12.

00:04:49.929 --> 00:04:50.663
This is negative 12.

00:04:50.663 --> 00:04:51.925
Negative 12 plus 12 is zero.

00:04:51.925 --> 00:04:54.483
Square root of zero, this is always zero.

00:04:54.483 --> 00:04:55.896
And then two times,

00:04:55.896 --> 00:04:58.432
and actually let me, I shouldn't
have written a w there,

00:04:58.432 --> 00:05:00.364
I should have written
two times negative six.

00:05:00.364 --> 00:05:01.662
So back to what I was doing,

00:05:01.662 --> 00:05:04.559
this right over here is negative eighteen.

00:05:04.559 --> 00:05:08.000
This is two times negative six plus 12.

00:05:08.000 --> 00:05:10.351
This is all zero.

00:05:10.351 --> 00:05:11.474
Square root of zero is zero.

00:05:11.474 --> 00:05:14.552
And then this is negative twelve.

00:05:14.552 --> 00:05:16.193
So you get six plus negative of eighteen

00:05:16.193 --> 00:05:17.844
which is negative 12.

00:05:17.844 --> 00:05:19.806
Is equal to zero plus negative 12.

00:05:19.806 --> 00:05:20.923
Negative 12.

00:05:20.923 --> 00:05:22.643
Which is absolutely correct.

00:05:22.643 --> 00:05:24.485
So these are actually both solutions

00:05:24.485 --> 00:05:27.453
to our original radical equation.


Modeling with composite functions

WEBVTT
Kind: captions
Language: en

00:00:00.550 --> 00:00:01.788
- [Voiceover] "Carter has noticed a few

00:00:01.788 --> 00:00:03.425
"quantitative relationships related

00:00:03.425 --> 00:00:05.558
"to the success of his football team

00:00:05.558 --> 00:00:07.955
"and has modeled them with
the following functions."

00:00:07.955 --> 00:00:09.296
All right, this is interesting.

00:00:09.296 --> 00:00:10.394
So he has this function,

00:00:10.394 --> 00:00:12.425
which he denotes with the capital N

00:00:12.425 --> 00:00:13.522
and it's the winning,

00:00:13.522 --> 00:00:16.103
and the input of it is
the winning percentage, W

00:00:16.103 --> 00:00:19.902
and the output is the average
number of fans per game.

00:00:19.902 --> 00:00:21.812
So, he's making some
type of model that says,

00:00:21.812 --> 00:00:23.356
look the number of fans per game

00:00:23.356 --> 00:00:25.347
are gonna be in some way dependent

00:00:25.347 --> 00:00:26.850
on what your winning percentage is.

00:00:26.850 --> 00:00:27.866
And, I'm assuming he's modeled

00:00:27.866 --> 00:00:28.923
the higher the winning percentage,

00:00:28.923 --> 00:00:31.276
the more fans are gonna show up at a game.

00:00:31.276 --> 00:00:33.494
Now this is, W,

00:00:33.494 --> 00:00:37.356
the input is the average
daily practice time, x,

00:00:37.356 --> 00:00:39.427
and the output is the winning percentage.

00:00:39.427 --> 00:00:40.456
All right, that makes sense.

00:00:40.456 --> 00:00:41.296
Probably once again,

00:00:41.296 --> 00:00:44.425
probably some type of a positive
effect of practicing more

00:00:44.425 --> 00:00:47.514
is going to create a
higher winning percentage.

00:00:47.514 --> 00:00:48.651
And this other function,

00:00:48.651 --> 00:00:50.927
number of rainy days, r

00:00:50.927 --> 00:00:52.633
and then average practice time.

00:00:52.633 --> 00:00:55.518
Yup, well the more rainy days you have

00:00:55.518 --> 00:00:58.119
well that's going to lower
your average practice time.

00:00:58.119 --> 00:01:00.638
So, I definitely see how practice time, P,

00:01:00.638 --> 00:01:03.402
would be a function of
number of rainy days.

00:01:03.402 --> 00:01:08.196
"The expression N(W(x)) represents
which of the following?"

00:01:08.196 --> 00:01:09.416
Well before we even look at the choices,

00:01:09.416 --> 00:01:10.737
let's think about what's happening.

00:01:10.737 --> 00:01:12.402
This is another way of denoting

00:01:12.402 --> 00:01:14.028
we're gonna take x,

00:01:14.028 --> 00:01:16.425
we're gonna take x right over here,

00:01:16.425 --> 00:01:20.143
and we are going to input it into W

00:01:20.143 --> 00:01:23.597
and we're going to get out W(x)

00:01:23.597 --> 00:01:28.230
and then we're going to input
that into the function N.

00:01:28.230 --> 00:01:33.230
And, we are going to get out, N(W(x)).

00:01:33.443 --> 00:01:36.542
So, what does the function W do?

00:01:36.542 --> 00:01:39.590
What does the function
W do, right over here?

00:01:39.590 --> 00:01:41.012
Well, that's winning percentage

00:01:41.012 --> 00:01:43.105
as a function of practice time.

00:01:43.105 --> 00:01:45.567
So, you input practice,

00:01:45.567 --> 00:01:49.027
practice time,

00:01:49.027 --> 00:01:50.196
and it gives you,

00:01:50.196 --> 00:01:53.528
it somehow predicts a winning percentage,

00:01:53.528 --> 00:01:57.286
winning percentage.

00:01:57.286 --> 00:01:58.846
And, then you take that winning percentage

00:01:58.846 --> 00:02:01.187
and you input it into function N.

00:02:01.187 --> 00:02:05.149
Function N is going to output
the number of fans per game,

00:02:05.149 --> 00:02:06.836
based on winning percentage.

00:02:06.836 --> 00:02:10.931
So this is number of fans.

00:02:10.931 --> 00:02:12.890
So when you take the composite function,

00:02:12.890 --> 00:02:14.353
you're actually creating a function

00:02:14.353 --> 00:02:17.076
that starts with practice
time as the input

00:02:17.076 --> 00:02:18.538
and shows the number of fans

00:02:18.538 --> 00:02:22.094
that are gonna be dependent
on your practice time.

00:02:22.094 --> 00:02:23.394
So this is interesting.

00:02:23.394 --> 00:02:25.101
So, we should look for a choice that says,

00:02:25.101 --> 00:02:27.539
how does the number of
fans that show up at a game

00:02:27.539 --> 00:02:32.539
how is that dependent on practice time, x?

00:02:32.639 --> 00:02:33.573
All right.

00:02:33.573 --> 00:02:35.930
"The team's winning
percentage as a function

00:02:35.930 --> 00:02:38.593
"of the average daily practice time."

00:02:38.593 --> 00:02:40.644
Now that would be just W(x).

00:02:40.644 --> 00:02:43.407
If they said just W(x)
that'd be winning percentage

00:02:43.407 --> 00:02:45.337
as a function of average
daily practice time.

00:02:45.337 --> 00:02:49.644
So, I can cross that one out.

00:02:49.644 --> 00:02:51.879
The average number of fans per game,

00:02:51.879 --> 00:02:52.794
all right this is interesting

00:02:52.794 --> 00:02:55.008
because that's what the
final output's going to be

00:02:55.008 --> 00:02:57.203
in terms of the average
number of fans per game,

00:02:57.203 --> 00:02:59.864
that is the output of the function N,

00:02:59.864 --> 00:03:01.876
the function N right over here.

00:03:01.876 --> 00:03:03.176
"The average number of fans per game

00:03:03.176 --> 00:03:07.586
"as a function of the number
of rainy days in a season,"

00:03:07.586 --> 00:03:08.215
Nope.

00:03:08.215 --> 00:03:09.048
We're not doing that.

00:03:09.048 --> 00:03:10.754
We're doing it as a
function of practice time.

00:03:10.754 --> 00:03:11.811
You could construct that.

00:03:11.811 --> 00:03:13.700
In fact, if you wanted to do this

00:03:13.700 --> 00:03:17.641
that would be N as a function of W

00:03:17.641 --> 00:03:22.121
as a function of P of r.

00:03:22.121 --> 00:03:23.818
So, that would have been this choice

00:03:23.818 --> 00:03:27.628
where you input the number of rainy days

00:03:27.628 --> 00:03:29.833
from that you're able to
figure out practice time

00:03:29.833 --> 00:03:31.560
and then you input practice time

00:03:31.560 --> 00:03:33.286
to figure out win perecentage

00:03:33.286 --> 00:03:35.135
and then you input win percentage

00:03:35.135 --> 00:03:37.635
to figure out the number
of fans in the crowd.

00:03:37.635 --> 00:03:38.874
But that's not what we're doing here.

00:03:38.874 --> 00:03:42.226
We're just starting
with daily practice time

00:03:42.226 --> 00:03:44.237
and getting to fans per game.

00:03:44.237 --> 00:03:47.123
So let me rule this one out.

00:03:47.123 --> 00:03:48.628
And if you found this one a little bit,

00:03:48.628 --> 00:03:49.622
what I just did a little bit confusing

00:03:49.622 --> 00:03:51.369
I encourage you to try to set up a diagram

00:03:51.369 --> 00:03:52.682
like I just did in the beginning.

00:03:52.682 --> 00:03:56.448
Instead of saying, oh well,
we could start with r to get,

00:03:56.448 --> 00:03:58.663
use that as input to get
average daily practice time

00:03:58.663 --> 00:04:00.695
and then use that as an input into W

00:04:00.695 --> 00:04:01.914
to get winning percentage.

00:04:01.914 --> 00:04:04.860
Then use that as an input into N

00:04:04.860 --> 00:04:07.400
to get average number of fans per game,

00:04:07.400 --> 00:04:11.463
but that's not what they're
describing for N(W(x)).

00:04:11.463 --> 00:04:13.536
"The average number of fans per game

00:04:13.536 --> 00:04:16.969
"as a function of the team's
average daily practice time."

00:04:16.969 --> 00:04:18.249
Yeah, that's what's going on.

00:04:18.249 --> 00:04:19.773
You have your average practice time, x

00:04:19.773 --> 00:04:22.068
being inputted into the function W.

00:04:22.068 --> 00:04:25.807
So your average practice
time is going inputted into W

00:04:25.807 --> 00:04:28.001
and it outputs winning percentage,

00:04:28.001 --> 00:04:30.561
which you then input into N

00:04:30.561 --> 00:04:33.060
to get the average
number of fans per game.

00:04:33.060 --> 00:04:34.259
The average number of fans per game

00:04:34.259 --> 00:04:37.063
as function of the team's
average daily practice time.

00:04:37.063 --> 00:04:40.964
So, yup, I definitely like that choice.

00:04:40.964 --> 00:04:42.183
Let's do another one of these.

00:04:42.183 --> 00:04:44.602
This is interesting.

00:04:44.602 --> 00:04:48.441
"Deniz studied the park near her home

00:04:48.441 --> 00:04:51.205
"where she identified several
quantitative relationships

00:04:51.205 --> 00:04:54.069
"and modeled them with
the following functions."

00:04:54.069 --> 00:04:57.705
So, B, it inputs the height
of a tree in terms of x

00:04:57.705 --> 00:05:00.957
and it outputs the number of
birds nesting in that tree.

00:05:00.957 --> 00:05:04.877
H, input the average temperature
at a specific location

00:05:04.877 --> 00:05:07.580
and it outputs the height of
the tree at that location.

00:05:07.580 --> 00:05:11.948
And T, the altitude of a specific location

00:05:11.948 --> 00:05:13.655
and then, if that's the input,

00:05:13.655 --> 00:05:15.463
and then the output is
the average temperature

00:05:15.463 --> 00:05:16.154
at that location.

00:05:16.154 --> 00:05:17.109
All right, this is interesting.

00:05:17.109 --> 00:05:19.323
"According to Deniz's findings,

00:05:19.323 --> 00:05:20.911
"which of the following
expressions represents

00:05:20.911 --> 00:05:24.971
"the height of a tree as a
function of its altitude?"

00:05:24.971 --> 00:05:26.314
So we wanna figure out,

00:05:26.314 --> 00:05:29.766
we want an output, the height of a tree

00:05:29.766 --> 00:05:33.444
and we wanna input, the
altitude of a specific location.

00:05:33.444 --> 00:05:34.419
So, let's think about it.

00:05:34.419 --> 00:05:37.854
If we take our altitude
at a specific location, r

00:05:37.854 --> 00:05:40.941
and we input it into the function T,

00:05:40.941 --> 00:05:45.025
out of that we're going to get T(r).

00:05:45.025 --> 00:05:46.813
T, I'll be writing a little bit neater.

00:05:46.813 --> 00:05:49.088
We're gonna get T(r),

00:05:49.088 --> 00:05:52.584
which would represent average
temperature at that location,

00:05:52.584 --> 00:05:55.366
average temp,

00:05:55.366 --> 00:05:57.703
and then if we take the average
temperature at that location

00:05:57.703 --> 00:06:00.692
and input it into function H,

00:06:00.692 --> 00:06:02.762
and then we input it into function H,

00:06:02.762 --> 00:06:06.956
we are going to get the height
of a tree at that location.

00:06:06.956 --> 00:06:11.845
So, we're going to get H(T(r))

00:06:11.845 --> 00:06:16.701
and so this is going to be
height of tree at that location,

00:06:16.701 --> 00:06:20.114
height of tree.

00:06:20.114 --> 00:06:23.771
And so, there you have it, H(T(r)).

00:06:23.771 --> 00:06:26.184
You start with r, altitude
at a specific location.

00:06:26.184 --> 00:06:26.981
Input it into function T.

00:06:26.981 --> 00:06:29.683
T's gonna spit out the average
temperature of that location.

00:06:29.683 --> 00:06:31.105
You input that into H.

00:06:31.105 --> 00:06:33.054
It's gonna get you the height
of the tree at that location.

00:06:33.054 --> 00:06:35.087
So, H(T(r)).

00:06:35.087 --> 00:06:39.946
H(T(r)) is this choice right over there.


Finding inverse functions: rational

WEBVTT
Kind: captions
Language: en

00:00:00.166 --> 00:00:01.397
- [Voiceover] So we're told
that g of x is equal to

00:00:01.397 --> 00:00:05.129
two x minus one over x plus three.

00:00:05.129 --> 00:00:07.276
Based on this, pause the
video and see if you can

00:00:07.276 --> 00:00:09.117
figure out what the inverse of g is.

00:00:09.117 --> 00:00:10.777
g inverse of x.

00:00:10.777 --> 00:00:13.434
What is that going to be equal to?

00:00:13.434 --> 00:00:15.034
Alright, I'm assuming
you've had a go at it,

00:00:15.034 --> 00:00:16.331
and just as a little bit of a reminder

00:00:16.331 --> 00:00:18.123
of what we're talking about
when we're talking about

00:00:18.123 --> 00:00:21.263
inverse of functions, or
inverses of functions.

00:00:21.263 --> 00:00:23.406
If that's the domain,
the set of all inputs

00:00:23.406 --> 00:00:25.820
that we could input into g,

00:00:25.820 --> 00:00:28.265
so that's the domain,

00:00:28.265 --> 00:00:32.129
and if we consider the range
to be the set of outputs,

00:00:32.129 --> 00:00:36.446
so that is the range,

00:00:36.446 --> 00:00:40.732
for a given input, so for
a member of the domain, x,

00:00:40.732 --> 00:00:44.029
the function g, let me do it that color,

00:00:44.029 --> 00:00:49.029
the function g will map it to a point,

00:00:49.030 --> 00:00:51.183
let me do g in yellow,

00:00:51.183 --> 00:00:55.107
it will map it to a member of the range.

00:00:55.107 --> 00:00:56.848
So if this is x,

00:00:56.848 --> 00:01:00.319
this value right over here we
would consider to be g of x.

00:01:00.319 --> 00:01:02.221
That's what the function g does.

00:01:02.221 --> 00:01:05.300
Now, the inverse function
goes the other way.

00:01:05.300 --> 00:01:07.865
You start with a member of the range

00:01:07.865 --> 00:01:11.880
and you go back to a member of the domain.

00:01:11.880 --> 00:01:13.933
That's what g inverse does.

00:01:13.933 --> 00:01:15.985
And so, there's a couple
of ways to think about it.

00:01:15.985 --> 00:01:17.313
If we were to set,

00:01:17.313 --> 00:01:18.370
if we were to say, look,

00:01:18.370 --> 00:01:21.077
let's just set y is equal to g of x.

00:01:21.077 --> 00:01:24.488
So we could just call
that point y, as well.

00:01:24.488 --> 00:01:27.808
So this expression here,

00:01:27.808 --> 00:01:29.726
let's just, okay, if we have
an x, this is what we do.

00:01:29.726 --> 00:01:31.250
We multiply by two times
and we subtract one,

00:01:31.250 --> 00:01:32.759
divided by x plus three,

00:01:32.759 --> 00:01:34.933
and then that tells us
what the associated g of x,

00:01:34.933 --> 00:01:36.774
what the associated y is.

00:01:36.774 --> 00:01:39.340
But what if we're given
the y, or the g of x.

00:01:39.340 --> 00:01:40.758
How do we get x?

00:01:40.758 --> 00:01:43.445
Well, we could just solve for x.

00:01:43.445 --> 00:01:44.350
So let's do that.

00:01:44.350 --> 00:01:47.118
If we said that y is equal to

00:01:47.118 --> 00:01:51.454
two x minus one

00:01:51.454 --> 00:01:54.709
over x plus three, let's solve for x,

00:01:54.709 --> 00:01:56.266
so that we know for any y we get,

00:01:56.266 --> 00:01:58.680
what the corresponding x we'll get.

00:01:58.680 --> 00:01:59.526
So how do we do that?

00:01:59.526 --> 00:02:02.484
Well, we can multiply both
sides times x plus three.

00:02:02.484 --> 00:02:04.929
So if we do that, we're going to get y,

00:02:04.929 --> 00:02:08.543
we're going to get y times

00:02:08.543 --> 00:02:10.687
x plus three,

00:02:10.687 --> 00:02:14.429
x plus three, is equal to,

00:02:14.429 --> 00:02:18.022
is equal to two x minus one.

00:02:18.022 --> 00:02:21.748
All I did was multiply both
sides times x plus three.

00:02:21.748 --> 00:02:24.223
x plus three, and I
multiplied x plus three

00:02:24.223 --> 00:02:25.521
on that side as well.

00:02:25.521 --> 00:02:29.174
The x plus three cancels
out with the x plus three.

00:02:29.174 --> 00:02:30.653
And then, let's see.

00:02:30.653 --> 00:02:32.222
Let's see what we can do now.

00:02:32.222 --> 00:02:35.392
Well, we could distribute the y.

00:02:35.392 --> 00:02:38.652
We could distribute the y,

00:02:38.652 --> 00:02:40.493
and I'll switch now to a different color

00:02:40.493 --> 00:02:41.731
because that's going to get complicated

00:02:41.731 --> 00:02:44.387
if I keep trying to separate
the ys in a different color.

00:02:44.387 --> 00:02:46.560
So we're going to get y times x,

00:02:46.560 --> 00:02:49.137
y x, plus three y,

00:02:49.137 --> 00:02:53.453
plus three y, is equal to two x minus one.

00:02:53.453 --> 00:02:56.864
And remember, we are now
trying to solve for x.

00:02:56.864 --> 00:02:59.702
So we can collect all of the x terms

00:02:59.702 --> 00:03:02.137
on one side of this equation,

00:03:02.137 --> 00:03:06.574
and then all of the non-x-involved
terms on the other side.

00:03:06.574 --> 00:03:09.170
So let's get all of our x
terms on the left-hand side

00:03:09.170 --> 00:03:11.524
and all of our non-x terms
on the right-hand side.

00:03:11.524 --> 00:03:13.034
So I want to get rid of this.

00:03:13.034 --> 00:03:16.837
This doesn't involve an x, so
let me subtract three y here.

00:03:16.837 --> 00:03:19.584
And so I'll subtract
three y from this side.

00:03:19.584 --> 00:03:21.153
Minus three y.

00:03:21.153 --> 00:03:23.115
And we want to get rid of
this from the right-hand side,

00:03:23.115 --> 00:03:24.745
so minus two x.

00:03:24.745 --> 00:03:25.953
So we can put it right over here.

00:03:25.953 --> 00:03:27.432
Minus two x.

00:03:27.432 --> 00:03:30.209
And so, we are going to get,

00:03:30.209 --> 00:03:32.080
let me scroll down a little bit.

00:03:32.080 --> 00:03:35.340
So, whoops. Sorry for that.

00:03:35.340 --> 00:03:37.544
So this is going to be y x minus two x

00:03:37.544 --> 00:03:42.334
is going to be y minus two times x.

00:03:42.334 --> 00:03:44.567
These cancel out, which is their goal.

00:03:44.567 --> 00:03:46.671
Is equal to two x minus two x,

00:03:46.671 --> 00:03:48.039
let's cancel that, which was the goal.

00:03:48.039 --> 00:03:49.911
And then you have one minus three y.

00:03:49.911 --> 00:03:52.446
One minus three y.

00:03:52.446 --> 00:03:53.563
And now, to solve for x,

00:03:53.563 --> 00:03:56.582
you just divide both sides by y minus two.

00:03:56.582 --> 00:03:58.815
y minus two.

00:03:58.815 --> 00:04:01.351
And we're going to get x is equal to

00:04:01.351 --> 00:04:04.190
one minus three y

00:04:04.190 --> 00:04:07.208
over y minus two.

00:04:07.208 --> 00:04:09.049
So one way you could think about it is,

00:04:09.049 --> 00:04:10.992
you could say, well x is equal to,

00:04:10.992 --> 00:04:13.829
this expression is equal to

00:04:13.829 --> 00:04:17.934
the inverse function as a function of y.

00:04:17.934 --> 00:04:19.866
You give me a y,

00:04:19.866 --> 00:04:22.734
you give me a y that's
sitting in this range,

00:04:22.734 --> 00:04:26.054
I can input it into this
function definition here

00:04:26.054 --> 00:04:28.318
and I can give you the corresponding x.

00:04:28.318 --> 00:04:31.185
I can give you the corresponding x.

00:04:31.185 --> 00:04:33.721
Now, we didn't ask for g inverse of y.

00:04:33.721 --> 00:04:36.044
We asked for g inverse of x.

00:04:36.044 --> 00:04:38.119
But the important thing to remember is

00:04:38.119 --> 00:04:40.506
this variable that we use inside of,

00:04:40.506 --> 00:04:42.547
the variable that we use inside of

00:04:42.547 --> 00:04:44.086
functions right over here,

00:04:44.086 --> 00:04:46.622
they're chosen somewhat arbitrarily.

00:04:46.622 --> 00:04:49.157
They just say, okay, I'm
going to call the input y,

00:04:49.157 --> 00:04:50.727
and if you're going to call the input y,

00:04:50.727 --> 00:04:52.840
well, this is how I would
give you the output.

00:04:52.840 --> 00:04:54.349
But we could call the input a,

00:04:54.349 --> 00:04:57.850
and that would be one minus
three a over a minus twp.

00:04:57.850 --> 00:04:59.713
Or we could even call the input,

00:04:59.713 --> 00:05:01.192
so let me make this clear.

00:05:01.192 --> 00:05:04.421
So we have g inverse of y.

00:05:04.421 --> 00:05:05.991
Actually, since I did it in blue up here,

00:05:05.991 --> 00:05:07.893
let me write that in blue as well.

00:05:07.893 --> 00:05:10.230
So if we did,

00:05:10.230 --> 00:05:13.369
if we did g inverse,

00:05:13.369 --> 00:05:16.026
g inverse of y,

00:05:16.026 --> 00:05:17.806
I'm just rewriting what we already have,

00:05:17.806 --> 00:05:20.040
is equal to one minus three y

00:05:20.040 --> 00:05:21.942
over y minus two.

00:05:21.942 --> 00:05:24.085
This choice of variable is arbitrary.

00:05:24.085 --> 00:05:26.873
We could say g inverse of

00:05:26.873 --> 00:05:29.932
of smiley face,

00:05:29.932 --> 00:05:31.863
of smiley face, is equal to

00:05:31.863 --> 00:05:36.029
one minus three times smiley face,

00:05:36.029 --> 00:05:37.357
three times smiley face,

00:05:37.357 --> 00:05:40.768
over smiley face minus two.

00:05:40.768 --> 00:05:42.428
Or, if we want it in terms of,

00:05:42.428 --> 00:05:43.998
if we want to call the input x,

00:05:43.998 --> 00:05:46.956
we would just write g inverse of x

00:05:46.956 --> 00:05:49.401
is equal to one minus three x

00:05:49.401 --> 00:05:51.846
over x minus two.

00:05:51.846 --> 00:05:53.385
And we'd be done.

00:05:53.385 --> 00:05:54.351
And I really want to point out,

00:05:54.351 --> 00:05:55.830
you might say, well, this is confusing,

00:05:55.830 --> 00:05:58.849
because now we're calling
the input to this x,

00:05:58.849 --> 00:05:59.886
while, you know, we normally associate

00:05:59.886 --> 00:06:01.415
y with the variables.

00:06:01.415 --> 00:06:03.539
That's true, but you have
to remember these variables,

00:06:03.539 --> 00:06:06.657
they're just labels
that we apply to things

00:06:06.657 --> 00:06:08.992
so that we can keep track of them.

00:06:08.992 --> 00:06:11.467
But they're just what we call the input.

00:06:11.467 --> 00:06:14.093
So we can call the input
into the inverse function,

00:06:14.093 --> 00:06:16.569
we can call it x, smiley face, y,

00:06:16.569 --> 00:06:19.617
you know, a, b, c, d, whatever we want.

00:06:19.617 --> 00:06:22.243
So they want it in
terms of g inverse of x,

00:06:22.243 --> 00:06:24.055
so we're going to define the input as x,

00:06:24.055 --> 00:06:25.866
whatever we input into it as x.

00:06:25.866 --> 00:06:28.401
And so this is going to be the expression

00:06:28.401 --> 00:06:31.480
that maps back.

00:06:31.480 --> 00:06:32.838
Now another way that you
could have done this,

00:06:32.838 --> 00:06:34.468
because if you think about
what just happened here,

00:06:34.468 --> 00:06:37.608
if we said now that y
is equal to g inverse,

00:06:37.608 --> 00:06:40.294
we've essentially swapped
the x and y variables.

00:06:40.294 --> 00:06:42.948
So, one way to solve
these is you can start

00:06:42.948 --> 00:06:45.445
where we started, right over there,

00:06:45.445 --> 00:06:47.619
and then you swap the xs and the ys,

00:06:47.619 --> 00:06:49.520
which is essentially what
we did in the last step

00:06:49.520 --> 00:06:50.728
the way I did it.

00:06:50.728 --> 00:06:54.230
And then you solve for y, after the swap,

00:06:54.230 --> 00:06:55.920
and you would get to this same result.

00:06:55.920 --> 00:06:58.486
And you could say, y
is equal to g inverse.

00:06:58.486 --> 00:07:00.506
So I encourage you to try either way.


Finding inverse functions: radical

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:02.744
- [Voiceover] So we're told
that h of x is equal to the

00:00:02.744 --> 00:00:07.399
negative cube root of
three x minus six plus 12.

00:00:07.399 --> 00:00:10.656
And what we wanna figure out
is, what is the inverse of h?

00:00:10.656 --> 00:00:11.631
So what is...

00:00:12.618 --> 00:00:16.947
What is h inverse of x
going to be equal to?

00:00:16.947 --> 00:00:17.928
And like always, pause the video

00:00:17.928 --> 00:00:19.153
and see if you could figure it out.

00:00:20.145 --> 00:00:22.412
Well in previous videos,
we've empathized that

00:00:22.412 --> 00:00:23.733
what an inverse does is...

00:00:24.876 --> 00:00:27.058
A function will map from a
domain to a range and you can

00:00:27.058 --> 00:00:29.635
think of the inverse as
mapping back from that point

00:00:29.635 --> 00:00:32.073
in the range to where you started from.

00:00:32.073 --> 00:00:34.244
So one way to think about it
is, we want to come up with

00:00:34.244 --> 00:00:37.437
an expression that unwinds
whatever this does.

00:00:37.437 --> 00:00:41.697
So if we say that y, if we
say that y is equal to h of x,

00:00:41.697 --> 00:00:45.424
or we could say that y is
equal to the negative of

00:00:45.424 --> 00:00:50.424
the cube root of three
x minus six plus 12.

00:00:50.930 --> 00:00:52.752
This gives us our y.

00:00:52.752 --> 00:00:54.812
And you can think of y
as a member of the range.

00:00:54.812 --> 00:00:58.109
A member of the range in
terms of what our input is.

00:00:58.109 --> 00:00:59.961
In terms of a member of the domain.

00:00:59.961 --> 00:01:01.958
We wanna go the other way around,

00:01:01.958 --> 00:01:04.883
so what we could do is we
could try to solve for x.

00:01:04.883 --> 00:01:07.414
If we solve for x, we're
gonna have some expression

00:01:07.414 --> 00:01:09.097
that's a function of y.

00:01:09.097 --> 00:01:11.175
We're gonna have that being equal to x.

00:01:11.175 --> 00:01:13.554
And so that would be the inverse mapping.

00:01:13.554 --> 00:01:14.937
Another way you could do that,

00:01:14.937 --> 00:01:18.975
is you could just swap x
and y and then solve for y.

00:01:18.975 --> 00:01:20.381
But that's a little bit less intuitive

00:01:20.381 --> 00:01:22.017
that this is actually the inverse.

00:01:22.017 --> 00:01:24.897
So actually, let's just solve for x here.

00:01:24.897 --> 00:01:26.836
So the first thing we might want to do is,

00:01:26.836 --> 00:01:29.552
let's isolate this cube root on,

00:01:29.552 --> 00:01:30.632
let's say to the right hand side.

00:01:30.632 --> 00:01:33.045
So let's subtract 12 from both sides.

00:01:33.045 --> 00:01:38.028
And we would get y minus 12
is equal to the cube root of,

00:01:38.028 --> 00:01:39.963
it's actually the negative cube root.

00:01:39.963 --> 00:01:40.833
Don't wanna lose track of that.

00:01:40.833 --> 00:01:44.014
Negative cube root of three x minus six

00:01:44.014 --> 00:01:46.825
and then we subtracted 12
from both sides so that 12

00:01:46.825 --> 00:01:49.624
is now, that 12 is now gone.

00:01:49.624 --> 00:01:52.668
And now what we would do, what
we could multiply both sides

00:01:52.668 --> 00:01:55.804
by negative one, that might
get rid of this negative here.

00:01:55.804 --> 00:01:58.174
So we multiply both sides by negative one.

00:02:00.345 --> 00:02:03.682
And then we multiply this
times a negative one.

00:02:03.682 --> 00:02:04.877
On the left hand side,

00:02:04.877 --> 00:02:07.582
we'll that's the same thing as 12 minus y.

00:02:07.582 --> 00:02:12.582
And on the right hand side,
we're gonna get the cube root of

00:02:12.904 --> 00:02:15.504
three x minus six.

00:02:15.504 --> 00:02:16.691
And now and this is gonna be

00:02:16.691 --> 00:02:18.803
a little bit algebraically hairy.

00:02:18.803 --> 00:02:21.337
We wanna cube both sides.

00:02:21.337 --> 00:02:22.742
So let's do that.

00:02:22.742 --> 00:02:24.761
So let's cube both sides.

00:02:24.761 --> 00:02:26.933
And actually it doesn't get
that algebraically hairy

00:02:26.933 --> 00:02:30.104
because I don't actually
have to figure what this,

00:02:30.104 --> 00:02:31.490
I don't have to expand it,

00:02:31.490 --> 00:02:33.424
I could just leave it as 12 minus y cubed.

00:02:33.424 --> 00:02:35.555
And so if we cube both
sides on the left hand side,

00:02:35.555 --> 00:02:39.108
we're just left with 12 minus y cubed.

00:02:39.108 --> 00:02:41.411
And on the right hand side,
well you take the cube

00:02:41.411 --> 00:02:43.397
of the cube root, you're
just gonna be left with

00:02:43.397 --> 00:02:46.811
what you originally had
under the cube root sign,

00:02:46.811 --> 00:02:48.029
I guess you could say.

00:02:48.029 --> 00:02:50.799
And now we wanna solve for x,
let's add six to both sides.

00:02:51.892 --> 00:02:56.759
So we're gonna get 12
minus y cubed plus six

00:02:56.759 --> 00:02:58.757
is equal to three x.

00:02:58.757 --> 00:03:02.211
Now we could divide both
sides by 3 and we're all done.

00:03:03.457 --> 00:03:05.977
Divide both sides by three and we get...

00:03:05.977 --> 00:03:07.830
We get x...

00:03:09.414 --> 00:03:13.783
Is equal to 12 minus y

00:03:13.783 --> 00:03:18.732
to the third power plus six over three.

00:03:19.792 --> 00:03:22.041
And so this, if you have a member of the,

00:03:22.041 --> 00:03:22.958
one way to think about it,

00:03:22.958 --> 00:03:24.849
if you have a member of the range y,

00:03:24.849 --> 00:03:27.659
this is going to map it back
to the x that would have gotten

00:03:27.659 --> 00:03:29.052
you to that member of the range.

00:03:29.052 --> 00:03:31.296
So this is the inverse
function so we could write,

00:03:31.296 --> 00:03:35.799
h inverse of y is equal to this business.

00:03:35.799 --> 00:03:38.664
12 minus y cubed

00:03:38.664 --> 00:03:42.017
plus six over three.

00:03:42.017 --> 00:03:43.912
And like we said in previous videos,

00:03:43.912 --> 00:03:46.856
this choice of calling y the input,

00:03:46.856 --> 00:03:48.938
well it could be anything,
we could call that star.

00:03:48.938 --> 00:03:52.986
We could say h inverse of
star and we're just naming

00:03:52.986 --> 00:03:55.962
our input star is equal to 12

00:03:55.962 --> 00:04:00.559
minus star cubed plus six over three.

00:04:00.559 --> 00:04:03.854
Or if we just want to call
the input x, we could just say

00:04:03.854 --> 00:04:06.165
h inverse of x and once again,

00:04:06.165 --> 00:04:09.166
this is just what we're calling the input,

00:04:09.166 --> 00:04:12.339
is equal to 12 minus y to the third

00:04:12.339 --> 00:04:15.350
plus six over three.

00:04:15.350 --> 00:04:17.588
Might be a little bit
confusing because now,

00:04:17.588 --> 00:04:20.425
in theory x could be considered
a member of the range

00:04:20.425 --> 00:04:23.731
and we're mapping back to
a member of the domain.

00:04:23.731 --> 00:04:27.987
But either way, we can
call the input function

00:04:27.987 --> 00:04:30.190
to a function partially anything.

00:04:30.190 --> 00:04:32.061
But there you have it, that
is our inverse function,

00:04:32.061 --> 00:04:35.298
that essentially unwinds what
our original function does.


Reflecting & compressing functions

WEBVTT
Kind: captions
Language: en

00:00:00.256 --> 00:00:01.139
- [Voiceover] So we're told g of x

00:00:01.139 --> 00:00:04.489
is a transformation of f of x.

00:00:04.489 --> 00:00:07.269
The graph below shows f
of x as a solid blue line.

00:00:07.269 --> 00:00:10.125
So this is the graph of
y is equal to f of x,

00:00:10.125 --> 00:00:11.820
and the g of x is a dotted red line.

00:00:11.820 --> 00:00:14.259
So that's the graph of
y is equal to g of x.

00:00:14.259 --> 00:00:16.255
What is g of x in terms of f of x?

00:00:16.255 --> 00:00:17.792
And they gave us some choices here,

00:00:17.792 --> 00:00:19.394
and I encourage you to pause the video

00:00:19.394 --> 00:00:21.991
and see if you can figure this out.

00:00:21.991 --> 00:00:24.522
Well there's a couple of ways
that you can think about it.

00:00:24.522 --> 00:00:26.588
One is if you just eyeball it,

00:00:26.588 --> 00:00:28.701
it looks like if you flipped f of x

00:00:28.701 --> 00:00:32.091
over the x-axis, it looks
a little bit like g(x),

00:00:32.091 --> 00:00:33.688
but g(x) looks like a version of that

00:00:33.688 --> 00:00:35.435
that's diminished a little bit.

00:00:35.435 --> 00:00:36.573
So for example, if you were to

00:00:36.573 --> 00:00:38.988
flip it perfectly over the x-axis,

00:00:38.988 --> 00:00:41.031
you would get something--

00:00:41.031 --> 00:00:42.935
you would get something that looks like--

00:00:42.935 --> 00:00:45.025
and I'm just going to sketch it.

00:00:45.025 --> 00:00:47.092
So if you just flipped
perfectly over the x-axis,

00:00:47.092 --> 00:00:50.644
you would get something
that looks like this,

00:00:50.644 --> 00:00:54.870
and trying my best to eyeball it.

00:00:54.870 --> 00:00:58.610
It would look something like this.

00:00:58.610 --> 00:01:02.764
If you perfectly just flipped it over.

00:01:02.764 --> 00:01:04.544
It would look something like that.

00:01:04.544 --> 00:01:06.294
I might not get it perfectly right.

00:01:06.294 --> 00:01:08.128
But that would be-- this right over here.

00:01:08.128 --> 00:01:09.336
So let me be clear.

00:01:09.336 --> 00:01:13.933
If this is y is equal to
f of x, then this line

00:01:13.933 --> 00:01:15.048
right over here that I just drew,

00:01:15.048 --> 00:01:18.508
that would be y is equal
to negative f of x.

00:01:18.508 --> 00:01:20.528
Because whatever f of x would give you,

00:01:20.528 --> 00:01:24.057
just take the negative of it
you're flipping over the x-axis

00:01:24.057 --> 00:01:26.658
and now g of x looks like a
diminished version of that,

00:01:26.658 --> 00:01:28.283
and if I were to just eyeball it,

00:01:28.283 --> 00:01:31.348
it looks like it's roughly 1/3 of this.

00:01:31.348 --> 00:01:32.788
So my initial guess - and we can

00:01:32.788 --> 00:01:35.399
verify this little bit - is that this

00:01:36.979 --> 00:01:40.953
right over here is 1/3 the value of this.

00:01:40.953 --> 00:01:42.694
So I would say it is...

00:01:42.694 --> 00:01:47.694
So g of x is equal to
1/3 of negative f of x

00:01:47.826 --> 00:01:52.826
or negative 1/3 of f of x.

00:01:53.166 --> 00:01:55.442
And that is not one of the choices,

00:01:55.442 --> 00:01:56.928
which makes me extra cautious,

00:01:56.928 --> 00:02:00.249
but let me just emphasize
why I like this choice.

00:02:00.249 --> 00:02:02.687
Because we said this is
negative f of x right there,

00:02:02.687 --> 00:02:04.846
and it looks like for any x value,

00:02:04.846 --> 00:02:06.982
what g of x is is 1/3 of that.

00:02:06.982 --> 00:02:08.376
So instead of getting to four,

00:02:08.376 --> 00:02:09.769
we're getting to a little bit over one.

00:02:09.769 --> 00:02:12.091
Instead of getting to
three right over here,

00:02:12.091 --> 00:02:13.739
we're getting to one.

00:02:13.739 --> 00:02:17.060
Instead of getting to one right over here,

00:02:17.060 --> 00:02:19.521
we are only getting to 1/3.

00:02:19.521 --> 00:02:22.749
So it looks like it is 1/3 of this line

00:02:22.749 --> 00:02:25.280
that I just hand drew,
which is negative f of x.

00:02:25.280 --> 00:02:26.905
So it would be 1/3 of negative f of x,

00:02:26.905 --> 00:02:29.947
which would be negative 1/3 f of x.

00:02:29.947 --> 00:02:31.410
Once again, not a choice here,

00:02:31.410 --> 00:02:33.151
but let's actually look at some values.

00:02:33.151 --> 00:02:35.752
So if we said, "Let's
see some values where

00:02:35.752 --> 00:02:39.165
"it looks like we're
hitting integer values."

00:02:42.055 --> 00:02:46.201
So for example, right at this point,

00:02:46.201 --> 00:02:51.201
right over here, it looks
like f of negative seven

00:02:52.377 --> 00:02:54.653
is equal to negative one.

00:02:54.653 --> 00:02:58.245
It looks like g of negative seven...

00:02:59.415 --> 00:03:02.916
So it looks like g of negative seven

00:03:02.916 --> 00:03:07.398
is equal to positive 1/3, positive 1/3,

00:03:07.398 --> 00:03:11.136
if I am just eyeballing it.

00:03:11.136 --> 00:03:12.646
So that seems consistent with this,

00:03:12.646 --> 00:03:15.780
because if f of negative
seven is negative one,

00:03:15.780 --> 00:03:17.800
you take the negative 1/3 times that,

00:03:17.800 --> 00:03:19.403
and you get positive 1/3.

00:03:19.403 --> 00:03:21.144
We can do that in a
couple of other places.

00:03:21.144 --> 00:03:23.721
If we look right over...

00:03:23.721 --> 00:03:27.896
If we look right over here,

00:03:27.896 --> 00:03:30.291
it looks like f of negative one.

00:03:30.291 --> 00:03:34.378
f of negative one is
equal to negative three,

00:03:34.378 --> 00:03:38.906
and it looks like g of one
is negative 1/3 times that.

00:03:38.906 --> 00:03:42.621
Negative 1/3 times that is
that point right over there.

00:03:42.621 --> 00:03:46.290
Negative 1/3 times negative
three is positive one.

00:03:46.290 --> 00:03:51.097
So g of... g is colored in that red color.

00:03:51.097 --> 00:03:54.022
So we get for that point right over there,

00:03:54.022 --> 00:03:58.388
we get g of negative one
is equal to positive one.

00:03:58.388 --> 00:04:00.779
So once again, this is negative 1/3

00:04:00.779 --> 00:04:02.289
times this right over here.

00:04:02.289 --> 00:04:04.239
So I feel pretty confident in my answer.

00:04:04.239 --> 00:04:07.513
g of x is equal to
negative 1/3 times f of x.

00:04:07.513 --> 00:04:09.626
Once again, negative f of
x would just flipped over,

00:04:09.626 --> 00:04:12.273
and then multiplying it by
the 1/3 diminishes that,

00:04:12.273 --> 00:04:14.061
which we see right over there.

00:04:14.061 --> 00:04:17.567
So I feel pretty confident with my

00:04:17.567 --> 00:04:18.769
none of the above.


Compressing functions

WEBVTT
Kind: captions
Language: en

00:00:00.306 --> 00:00:03.151
- [Voiceover] G of x is a
transformation of f of x.

00:00:03.151 --> 00:00:05.677
The graph here shows this
is y is equal to f of x,

00:00:05.677 --> 00:00:06.675
the solid blue line,

00:00:06.675 --> 00:00:09.918
This is y is equal to g
of x as a dash red line.

00:00:09.918 --> 00:00:12.804
And it asked us, "What is
g of x in terms of f of x?"

00:00:12.804 --> 00:00:14.511
And like always, pause the
video and see if you can

00:00:14.511 --> 00:00:17.041
give a go at it and then
we're going to do it together.

00:00:17.041 --> 00:00:18.872
All right, so when you
immediately look at it,

00:00:18.872 --> 00:00:22.772
it looks like g of x is kinda
a thin dot version of f of x.

00:00:22.772 --> 00:00:24.603
It seems like if you were to compress it

00:00:24.603 --> 00:00:27.494
to towards the center,
that's what g of x looks like

00:00:27.494 --> 00:00:29.850
but let's go a little
more mid on that bone

00:00:29.850 --> 00:00:32.479
and see if we can identify
corresponding points.

00:00:32.479 --> 00:00:36.314
So for example, if we will
look at f of negative six.

00:00:36.314 --> 00:00:39.054
That's, so f of negative six.

00:00:39.054 --> 00:00:42.792
That seems like it corresponds
or it gives us the same value

00:00:42.792 --> 00:00:45.288
as f of negative six.

00:00:45.288 --> 00:00:46.414
So we wanna find the corresponding points.

00:00:46.414 --> 00:00:47.308
We hit this minimum point.

00:00:47.308 --> 00:00:48.434
We're coming back up.

00:00:48.434 --> 00:00:49.108
Hit the minimum point.

00:00:49.108 --> 00:00:50.454
We're coming back up.

00:00:50.454 --> 00:00:52.695
It seems like the corresponding
point right over there

00:00:52.695 --> 00:00:55.330
is g of negative three.

00:00:55.330 --> 00:00:56.514
So let's write that down.

00:00:56.514 --> 00:01:00.113
Let's see, it looks like f of negative six

00:01:00.113 --> 00:01:02.829
and it is equal to g of negative three.

00:01:02.829 --> 00:01:03.955
These are corresponding points.

00:01:03.955 --> 00:01:07.210
If you apply the transformation
at the point f equals,

00:01:07.210 --> 00:01:10.612
at the point negative six
comma f of negative six.

00:01:10.612 --> 00:01:14.315
You get to the point negative
three, g of negative three

00:01:14.315 --> 00:01:15.666
right over there.

00:01:15.666 --> 00:01:16.998
Let's do a couple of more.

00:01:16.998 --> 00:01:19.750
If you look at f is,

00:01:19.750 --> 00:01:22.030
so f of two,

00:01:22.030 --> 00:01:25.872
looks like it corresponds to g of one,

00:01:25.872 --> 00:01:28.487
f of two corresponds to g of one.

00:01:28.487 --> 00:01:29.753
So let's write that down.

00:01:29.753 --> 00:01:33.909
F of two looks like it
corresponds to g of one.

00:01:33.909 --> 00:01:35.708
And once again, I'm looking
at the where the functions

00:01:35.708 --> 00:01:38.494
of the same value and also
optically I'm just looking at,

00:01:38.494 --> 00:01:40.154
well, looks like it's the
same part of the function

00:01:40.154 --> 00:01:45.088
if we assumed g of x is
squeezed version of f of x.

00:01:45.088 --> 00:01:49.019
And so in general, it looks like

00:01:49.019 --> 00:01:50.799
for given x.

00:01:50.799 --> 00:01:55.130
So we could say f of x
is going to be equal to

00:01:55.130 --> 00:01:57.476
g of, well whatever you have it here,

00:01:57.476 --> 00:01:59.473
it seems like we have
half the value over here.

00:01:59.473 --> 00:02:02.979
So g of x over two.

00:02:02.979 --> 00:02:04.790
If you wanna think of it the other way,

00:02:04.790 --> 00:02:06.310
if you wanna think of the other way,

00:02:06.310 --> 00:02:08.934
if you wanna say g of x

00:02:08.934 --> 00:02:11.639
is going to be f of.

00:02:11.639 --> 00:02:12.989
Well, whatever we have here,

00:02:12.989 --> 00:02:14.800
it's f of twice x.

00:02:14.800 --> 00:02:17.342
So f of two x.

00:02:17.342 --> 00:02:19.223
And we see that that
is one of the choices.

00:02:19.223 --> 00:02:22.543
The g of x is equal to f of two x.

00:02:22.543 --> 00:02:24.824
Whatever the x that you input into g of x,

00:02:24.824 --> 00:02:27.522
you get that same value
out of the function

00:02:27.522 --> 00:02:30.378
when you input two times that into f of x.

00:02:30.378 --> 00:02:32.665
These seem to validate that.

00:02:32.665 --> 00:02:33.942
It looks optically like that

00:02:33.942 --> 00:02:35.486
and we shrunk it down.

00:02:35.486 --> 00:02:36.646
One way to think about it

00:02:36.646 --> 00:02:37.900
when you take,

00:02:37.900 --> 00:02:41.557
when you multiply the
input into a function

00:02:41.557 --> 00:02:44.425
by a number larger than one,

00:02:44.425 --> 00:02:45.597
it's going to compress.

00:02:45.597 --> 00:02:47.756
It's going to make things happen faster.

00:02:47.756 --> 00:02:50.577
The input to the function
is going to increase

00:02:50.577 --> 00:02:52.284
or become negative faster

00:02:52.284 --> 00:02:54.474
so it thins it up.

00:02:54.474 --> 00:02:56.598
And if that doesn't make into any sense,

00:02:56.598 --> 00:02:58.962
you can also just try
some of these values.

00:02:58.962 --> 00:03:00.239
And I encourage you to try more.

00:03:00.239 --> 00:03:01.702
Find the corresponding points

00:03:01.702 --> 00:03:04.337
where the f's and the
g's seem to match up.

00:03:04.337 --> 00:03:06.090
And you'll see over and over again

00:03:06.090 --> 00:03:08.180
that to get the same value,

00:03:08.180 --> 00:03:10.258
you have to put two times as much into f

00:03:10.258 --> 00:03:12.452
as you have to put into g.

00:03:12.452 --> 00:03:14.356
Hopefully that helps.


Shifting functions

WEBVTT
Kind: captions
Language: en

00:00:00.248 --> 00:00:01.363
- [Voiceover] So we have these two graphs

00:00:01.363 --> 00:00:02.500
that look pretty similar,

00:00:02.500 --> 00:00:05.681
y equals f of x and y is equal to g of x.

00:00:05.681 --> 00:00:06.831
And what they asked us to do is

00:00:06.831 --> 00:00:10.499
write a formula for the
function g in terms of f.

00:00:10.499 --> 00:00:11.880
So let's think about how to do it

00:00:11.880 --> 00:00:13.030
and like always pause the video

00:00:13.030 --> 00:00:15.716
and see if you can work
through it on your own.

00:00:15.716 --> 00:00:17.070
All right, well, what I'd like to do is

00:00:17.070 --> 00:00:19.102
I'd like to focus on this minimum point

00:00:19.102 --> 00:00:22.700
because I think that's a
very easy thing to look at

00:00:22.700 --> 00:00:25.220
because of them have that minimum point

00:00:25.220 --> 00:00:26.241
right over there.

00:00:26.241 --> 00:00:28.203
And so we can think,
but how do we shift f?

00:00:28.203 --> 00:00:30.107
Especially this minimum point,

00:00:30.107 --> 00:00:33.984
how do we shift it to get
to overlapping with g?

00:00:33.984 --> 00:00:36.504
Well, the first thing that
might jump out of this is

00:00:36.504 --> 00:00:40.218
that we would want to shift to the left.

00:00:40.218 --> 00:00:43.144
And we'd wanna shift to the left four.

00:00:43.144 --> 00:00:46.824
So let me do this in a new color.

00:00:46.824 --> 00:00:50.051
So I would wanna shift to the left

00:00:50.051 --> 00:00:52.169
by four.

00:00:53.529 --> 00:00:55.542
We have shifted to the left by four

00:00:55.542 --> 00:00:58.342
or you could say we
shifted by negative four.

00:00:58.342 --> 00:00:59.967
Either way, you could think about it.

00:00:59.967 --> 00:01:02.465
And then we're gonna shift down.

00:01:02.465 --> 00:01:03.884
So we're gonna shift, we need to go

00:01:03.884 --> 00:01:08.138
from y equals two to y is
equal to negative five.

00:01:08.138 --> 00:01:10.263
So let me do that.

00:01:10.263 --> 00:01:13.026
So let's shift down.

00:01:13.026 --> 00:01:14.455
So we shift down by seven

00:01:14.455 --> 00:01:17.810
or you could say we have
a negative seven shift.

00:01:17.810 --> 00:01:19.215
So, how do you express g of x

00:01:19.215 --> 00:01:22.454
if it's a version of f of x
that shifted to the left by four

00:01:22.454 --> 00:01:24.009
and shifted down by seven.

00:01:24.009 --> 00:01:26.633
Or, you could say I have a
negative four horizontal shift.

00:01:26.633 --> 00:01:29.106
I have a negative seven vertical shift.

00:01:29.106 --> 00:01:33.146
Well, one thing to think
about it is g of x,

00:01:33.146 --> 00:01:37.548
g of x is going to be equal to

00:01:37.548 --> 00:01:39.644
f of,

00:01:39.644 --> 00:01:41.569
let me do it in a little darker color,

00:01:41.569 --> 00:01:46.104
it's going to be equal to f of x

00:01:46.104 --> 00:01:48.949
minus your horizontal shift,

00:01:48.949 --> 00:01:52.327
all right, horizontal shift.

00:01:52.327 --> 00:01:55.008
So x minus your horizontal shift

00:01:55.008 --> 00:01:58.637
plus your vertical shift.

00:01:58.637 --> 00:02:03.637
So plus your vertical

00:02:03.808 --> 00:02:05.689
shift.

00:02:05.689 --> 00:02:08.709
Well, what is our horizontal shift here?

00:02:08.709 --> 00:02:10.648
Well, we're shifting to the left

00:02:10.648 --> 00:02:12.192
so it's a negative shift.

00:02:12.192 --> 00:02:15.327
So our horizontal shift is negative four.

00:02:15.327 --> 00:02:16.894
Now, what's our vertical shift?

00:02:16.894 --> 00:02:18.670
Well, we went down so
our vertical shift is

00:02:18.670 --> 00:02:20.574
negative seven.

00:02:20.574 --> 00:02:22.257
So it's negative seven.

00:02:22.257 --> 00:02:23.476
So there you have it.

00:02:23.476 --> 00:02:26.227
We get g of x.

00:02:26.227 --> 00:02:28.544
Let me do that in the same color.

00:02:28.544 --> 00:02:33.544
We get g of x is equal to

00:02:33.834 --> 00:02:37.464
f of x minus negative four

00:02:37.464 --> 00:02:40.444
or x plus four

00:02:40.444 --> 00:02:43.314
and then we have plus negative seven

00:02:43.314 --> 00:02:46.146
or you can just say minus seven.

00:02:46.146 --> 00:02:47.382
And we're done.

00:02:47.382 --> 00:02:48.787
And when I look at things like this,

00:02:48.787 --> 00:02:51.966
the negative seven is somewhat,

00:02:51.966 --> 00:02:53.412
it's more intuitive to me.

00:02:53.412 --> 00:02:54.890
As I shifted it down, it made sense

00:02:54.890 --> 00:02:56.256
that I have a negative seven.

00:02:56.256 --> 00:02:57.424
But first, when you work on these,

00:02:57.424 --> 00:02:59.014
you say, "I shifted to the left.

00:02:59.014 --> 00:03:02.741
"Why is it a plus four?"

00:03:02.741 --> 00:03:04.455
And though I think about it is

00:03:04.455 --> 00:03:07.705
in order to get the same
value out of the function,

00:03:07.705 --> 00:03:09.830
instead of inputting,
so if you want to get

00:03:09.830 --> 00:03:13.150
the value of x of zero,
you now have to put

00:03:13.150 --> 00:03:15.890
x equals negative four in.

00:03:15.890 --> 00:03:17.910
And then you get that same value.

00:03:17.910 --> 00:03:20.313
You still get to zero.

00:03:20.313 --> 00:03:22.205
So that's, I know if that helps or hurts

00:03:22.205 --> 00:03:24.225
in terms of your understanding,

00:03:24.225 --> 00:03:27.151
but it often helps to try out
some different values for x

00:03:27.151 --> 00:03:30.122
and see how it actually
does shift the function.

00:03:30.122 --> 00:03:32.026
And if you just try to get
your head on this piece,

00:03:32.026 --> 00:03:34.023
the horizontal shift, I
recommended, you know,

00:03:34.023 --> 00:03:35.068
not even using this example.

00:03:35.068 --> 00:03:37.529
Use an example that only
has a horizontal shift.

00:03:37.529 --> 00:03:39.050
It will become a little more intuitive.

00:03:39.050 --> 00:03:40.605
And we do, we have many videos that go

00:03:40.605 --> 00:03:42.625
into much more depth that explain that.

00:03:42.625 --> 00:03:45.272
Let's do another example of this.

00:03:45.272 --> 00:03:48.999
So here, we have

00:03:48.999 --> 00:03:50.926
y is equal to g of x in purple

00:03:50.926 --> 00:03:52.981
and y is equal to f of x in blue.

00:03:52.981 --> 00:03:54.768
And they say given that f of x

00:03:54.768 --> 00:03:58.820
is equal to square root
of x plus four minus two,

00:03:58.820 --> 00:04:03.820
write an expression for
g of x in terms of x.

00:04:03.823 --> 00:04:05.162
So first, let's me just try to,

00:04:05.162 --> 00:04:07.774
expression for g of x in terms of f of x.

00:04:07.774 --> 00:04:08.633
We can see once again,

00:04:08.633 --> 00:04:10.978
it's just a shifted version of f of x.

00:04:10.978 --> 00:04:13.381
And remember, I'll just
write it in general.

00:04:13.381 --> 00:04:17.578
So g of x is going to be equal to

00:04:17.578 --> 00:04:22.465
f of x minus your horizontal shift

00:04:24.735 --> 00:04:29.613
plus your vertical shift.

00:04:29.613 --> 00:04:31.749
And so to go from f to g,

00:04:31.749 --> 00:04:34.414
what is your horizontal shift?

00:04:34.414 --> 00:04:36.597
Well, your horizontal shift is,

00:04:36.597 --> 00:04:38.176
if you take this point right over here,

00:04:38.176 --> 00:04:42.100
which should map to that point
once we shift everything.

00:04:42.100 --> 00:04:45.675
Your horizontal shift is two to the left

00:04:45.675 --> 00:04:48.907
or you could say it's a
negative two horizontal shift.

00:04:48.907 --> 00:04:50.961
So that should be negative two.

00:04:50.961 --> 00:04:53.116
And then, what is a vertical shift?

00:04:53.116 --> 00:04:54.962
Well, a vertical shift is removed,

00:04:54.962 --> 00:04:58.360
we go from y equals negative
two to the y equals three.

00:04:58.360 --> 00:05:01.672
So we're shifting five up.

00:05:01.672 --> 00:05:04.687
So this is a vertical
shift of positive five.

00:05:04.687 --> 00:05:06.823
So your vertical shift is five.

00:05:06.823 --> 00:05:08.095
So if we just want to write g of x

00:05:08.095 --> 00:05:11.628
in terms of f of x like we just
did in the previous example,

00:05:11.628 --> 00:05:16.047
we could say g of x is
going to be equal to f of x

00:05:16.047 --> 00:05:19.751
minus negative two which is x plus two.

00:05:19.751 --> 00:05:22.039
And then we have plus five.

00:05:22.039 --> 00:05:23.583
But that's not what they asked us to do.

00:05:23.583 --> 00:05:25.023
They asked us to write.

00:05:25.023 --> 00:05:26.207
They, whoops.

00:05:26.207 --> 00:05:28.482
They asked us to write an expression

00:05:28.482 --> 00:05:30.607
for g of x in terms of x.

00:05:30.607 --> 00:05:32.104
And so here we're actually gonna use

00:05:32.104 --> 00:05:34.786
the definition of f of x.

00:05:34.786 --> 00:05:36.342
So let me make it clear.

00:05:36.342 --> 00:05:40.147
We know that f of x is going

00:05:40.147 --> 00:05:42.377
to be equal to square root

00:05:42.377 --> 00:05:46.833
of x plus four minus two.

00:05:46.833 --> 00:05:49.635
So given that,

00:05:49.635 --> 00:05:52.513
what is f of x plus two?

00:05:52.513 --> 00:05:56.619
Well, f of x plus two
is going to be equal to,

00:05:56.619 --> 00:05:58.372
everywhere we see an x,
we're gonna replace it

00:05:58.372 --> 00:05:59.672
with an x plus two.

00:05:59.672 --> 00:06:04.672
Square root of x plus two plus four

00:06:05.190 --> 00:06:06.565
minus two

00:06:06.565 --> 00:06:09.350
which is equal to the square root of

00:06:09.350 --> 00:06:13.043
x plus six minus two.

00:06:13.043 --> 00:06:14.228
Well, that's fair enough.

00:06:14.228 --> 00:06:15.691
That's just f of x plus two.

00:06:15.691 --> 00:06:19.310
Now, what is f of x plus two plus five?

00:06:19.310 --> 00:06:24.310
So f of x plus two plus five

00:06:24.652 --> 00:06:28.368
is going to be in this thing
right over here plus five.

00:06:28.368 --> 00:06:31.131
So it's going to be
equal to square root of

00:06:31.131 --> 00:06:33.903
x plus six minus two

00:06:33.903 --> 00:06:37.029
and I'm going to add five,
with a different color.

00:06:37.029 --> 00:06:38.523
So plus five,

00:06:38.523 --> 00:06:40.543
so plus five.

00:06:40.543 --> 00:06:43.249
And so what we end up with

00:06:43.249 --> 00:06:46.255
is going to be square root of

00:06:46.255 --> 00:06:49.157
x plus six minus two plus five

00:06:49.157 --> 00:06:50.840
is going to be plus three.

00:06:50.840 --> 00:06:53.302
So that is equal to g of x.

00:06:53.302 --> 00:06:54.997
Just as a reminder, what do we do here?

00:06:54.997 --> 00:06:58.482
First, I expressed g of
x in terms of f of x.

00:06:58.482 --> 00:07:00.293
Let's say, hey, to get
from f of x to g of x,

00:07:00.293 --> 00:07:01.853
I shift two to the left.

00:07:01.853 --> 00:07:03.068
Two to the left.

00:07:03.068 --> 00:07:05.056
So we're kinda intuitive, plus two

00:07:05.056 --> 00:07:07.305
makes it a shift of two to the left.

00:07:07.305 --> 00:07:10.103
This was minus two would be
shifted two to the right.

00:07:10.103 --> 00:07:12.285
But like I just said in
the previous example,

00:07:12.285 --> 00:07:13.748
it's good to try it some x's

00:07:13.748 --> 00:07:15.559
and to see why that makes sense.

00:07:15.559 --> 00:07:17.219
And then we shifted five up.

00:07:17.219 --> 00:07:19.367
So this was g of x in terms of f of x.

00:07:19.367 --> 00:07:22.547
Then they told us what f of
x actually is in terms of x.

00:07:22.547 --> 00:07:25.543
So we said, "Okay, well,
what is f of x plus two?"

00:07:25.543 --> 00:07:28.248
F of x plus two, we
substituted x plus two for x

00:07:28.248 --> 00:07:29.571
and we got this.

00:07:29.571 --> 00:07:32.961
But g of x is f of x plus two plus five.

00:07:32.961 --> 00:07:35.445
So we took what we figured
out f of x plus two is

00:07:35.445 --> 00:07:38.081
and then we added five
and that's what g of x is.

00:07:38.081 --> 00:07:39.955
And then we are all done.


Evaluating composite functions

WEBVTT
Kind: captions
Language: en

00:00:00.590 --> 00:00:01.997
- [Voiceover] So, we're told that g of x

00:00:01.997 --> 00:00:05.140
is equal to x squared
plus five x minus three

00:00:05.140 --> 00:00:08.141
and h of y is equal to three times

00:00:08.141 --> 00:00:11.030
y minus one squared, minus five.

00:00:11.030 --> 00:00:12.538
And then, we're asked,

00:00:12.538 --> 00:00:16.718
what is h of g of negative six?

00:00:16.718 --> 00:00:20.092
And the way it's written might
look a little strange to you.

00:00:20.092 --> 00:00:24.859
This little circle that we have
in between the h and the g,

00:00:24.859 --> 00:00:27.581
that's our function composition symbol.

00:00:27.581 --> 00:00:29.901
So, function,

00:00:29.901 --> 00:00:32.971
function composition,

00:00:32.971 --> 00:00:36.161
composition,

00:00:36.161 --> 00:00:39.591
composition symbol.

00:00:39.591 --> 00:00:40.935
And one way to rewrite this,

00:00:40.935 --> 00:00:42.408
it might make a little bit more sense.

00:00:42.408 --> 00:00:46.294
So, this h of g of negative six.

00:00:46.294 --> 00:00:48.678
You could rewrite this as,

00:00:48.678 --> 00:00:51.276
this is going to be the same thing as

00:00:51.276 --> 00:00:54.342
g of negative six,

00:00:54.342 --> 00:00:57.072
and then h of that.

00:00:57.072 --> 00:01:00.511
So, h of g of negative six.

00:01:00.511 --> 00:01:04.160
Notice, I spoke this out the
same way that I said this.

00:01:04.160 --> 00:01:07.307
This is h of g of negative six.

00:01:07.307 --> 00:01:10.305
This is h of g of negative six.

00:01:10.305 --> 00:01:12.905
I find the second notation
far more intuitive,

00:01:12.905 --> 00:01:14.379
but it's good to become familiar with this

00:01:14.379 --> 00:01:16.718
function composition
notation, this little circle,

00:01:16.718 --> 00:01:18.379
because you might see that sometime and

00:01:18.379 --> 00:01:20.263
you shouldn't stress, it's
just the same thing as

00:01:20.263 --> 00:01:21.981
what we have right over here.

00:01:21.981 --> 00:01:24.620
Now, what is h of g of negative six?

00:01:24.620 --> 00:01:26.683
Well, we just have to remind
ourselves that this means

00:01:26.683 --> 00:01:28.764
that we're going to take
the number negative six,

00:01:28.764 --> 00:01:31.995
we're going to input
it into our function g,

00:01:31.995 --> 00:01:35.493
and then that will
output g of negative six,

00:01:35.493 --> 00:01:37.831
whatever that number is, we'll
figure it out in a second,

00:01:37.831 --> 00:01:40.857
and then we're going to input
that into our function h.

00:01:40.857 --> 00:01:44.084
We're going to input
that into our function h.

00:01:44.084 --> 00:01:46.165
And then, what we output is going to be

00:01:46.165 --> 00:01:50.208
h of g of negative six, which
is what we want to figure out.

00:01:50.208 --> 00:01:54.107
h of g

00:01:54.107 --> 00:01:56.467
of negative six.

00:01:56.467 --> 00:01:57.853
So, we just have to do
it one step at a time.

00:01:57.853 --> 00:01:59.459
A lot of times, when you
first start looking at

00:01:59.459 --> 00:02:02.027
these function composition,
it seems really convoluted

00:02:02.027 --> 00:02:03.842
and confusing, but you just have to,

00:02:03.842 --> 00:02:06.235
I want you to take a breath
and take it one step at a time.

00:02:06.235 --> 00:02:08.209
Well, let's figure out
what g of negative six is.

00:02:08.209 --> 00:02:10.575
It's going to evaluate
to a number in this case.

00:02:10.575 --> 00:02:13.348
And then, we input that
number into the function h,

00:02:13.348 --> 00:02:14.292
and then we'll figure out another ...

00:02:14.292 --> 00:02:16.521
that's going to map to another number.

00:02:16.521 --> 00:02:18.622
So, g of negative six.

00:02:18.622 --> 00:02:20.042
Let's figure that out.

00:02:20.042 --> 00:02:23.297
g of negative six is equal to

00:02:23.297 --> 00:02:26.447
negative six squared,

00:02:26.447 --> 00:02:31.204
plus five times negative six, minus three,

00:02:31.204 --> 00:02:36.204
which is equal to positive
36, minus 30, minus three.

00:02:36.652 --> 00:02:37.593
So, that's equal to what?

00:02:37.593 --> 00:02:41.777
36 minus 33, which is equal to three.

00:02:41.777 --> 00:02:44.802
So, g of negative six is equal to three.

00:02:44.802 --> 00:02:46.636
g of negative six is equal to three.

00:02:46.636 --> 00:02:48.332
g of negative six is equal to three.

00:02:48.332 --> 00:02:51.140
You input negative six
into g, it outputs three.

00:02:51.140 --> 00:02:54.104
And so, h of g of negative
six has now simplified

00:02:54.104 --> 00:02:58.733
to just h of three because
g of negative six is three.

00:02:58.733 --> 00:03:01.464
So, let's figure out what h of three is.

00:03:01.464 --> 00:03:06.464
h of three ... notice,
whatever we outputted from g,

00:03:06.488 --> 00:03:08.544
we're inputting that now into h.

00:03:08.544 --> 00:03:11.647
So, that's the number three,
so h of three is going to be

00:03:11.647 --> 00:03:15.937
three times three minus one,

00:03:15.937 --> 00:03:20.127
three minus one squared, minus five,

00:03:20.127 --> 00:03:23.815
which is equal to three times two squared,

00:03:23.815 --> 00:03:26.595
this is two right over here, minus five,

00:03:26.595 --> 00:03:30.727
which is equal to three
times four minus five,

00:03:30.727 --> 00:03:35.230
which is equal to 12 minus five,

00:03:35.230 --> 00:03:37.753
which is equal to seven.

00:03:37.753 --> 00:03:39.064
And we're done.

00:03:39.064 --> 00:03:43.011
So, you input negative
six into g, you get three.

00:03:43.011 --> 00:03:44.892
And then, you take that output from g

00:03:44.892 --> 00:03:48.119
and you put it into h and you get seven.

00:03:48.119 --> 00:03:49.919
So, this right over here is seven.

00:03:49.919 --> 00:03:52.288
All of this has come out
to be equal to seven.

00:03:52.288 --> 00:03:57.094
So, h of g of negative
six is equal to seven.

00:03:57.094 --> 00:04:01.343
h of g of negative six is equal to seven.

00:04:01.343 --> 00:04:04.510
Input negative six into
g, take that output

00:04:04.510 --> 00:04:09.080
and input it into h, and
you're gonna get seven.


Writing exponential functions

WEBVTT
Kind: captions
Language: en

00:00:00.399 --> 00:00:04.021
- [Voiceover] g is an exponential function

00:00:04.021 --> 00:00:07.248
with an initial value of -2.

00:00:07.248 --> 00:00:09.733
So, an initial value of -2,

00:00:09.733 --> 00:00:13.007
and a common ratio of 1/7,

00:00:13.007 --> 00:00:15.445
common ratio of 1/7.

00:00:15.445 --> 00:00:18.858
Write the formula for g(t).

00:00:18.858 --> 00:00:20.901
Well, the fact that it's
an exponential function,

00:00:20.901 --> 00:00:22.642
we know that its formula is going to be

00:00:22.642 --> 00:00:26.194
of the form g(t) is equal

00:00:26.194 --> 00:00:28.274
to our initial value

00:00:28.274 --> 00:00:29.910
which we could call A,

00:00:29.910 --> 00:00:33.602
times our common ratio
which we could call r,

00:00:33.602 --> 00:00:35.320
to the t power.

00:00:35.320 --> 00:00:36.574
It's going to have that form.

00:00:36.574 --> 00:00:38.060
And they tell us what
the initial value is.

00:00:38.060 --> 00:00:39.708
It's -2.

00:00:39.708 --> 00:00:43.354
So this right over here is -2.

00:00:43.354 --> 00:00:46.372
And we know that the common ratio is 1/7.

00:00:46.372 --> 00:00:48.485
So this is 1/7.

00:00:48.485 --> 00:00:50.914
So let me just write it
again a little bit neater.

00:00:50.914 --> 00:00:53.844
g(t) is going to be equal to

00:00:53.844 --> 00:00:56.854
our initial value, -2,

00:00:56.854 --> 00:00:58.983
times

00:00:58.983 --> 00:01:03.883
times our common ratio, 1/7,

00:01:03.883 --> 00:01:06.842
to the t power.

00:01:06.842 --> 00:01:08.862
And hopefully this makes sense.

00:01:08.862 --> 00:01:10.812
Initial value is this number.

00:01:10.812 --> 00:01:12.647
Well, if t is equal to 0,

00:01:12.647 --> 00:01:15.015
then 1/7 to the zero power is 1.

00:01:15.015 --> 00:01:17.221
And so g(0), you could do that

00:01:17.221 --> 00:01:18.726
at time as being equal to zero

00:01:18.726 --> 00:01:21.257
if your t is time, would be equal to -2.

00:01:21.257 --> 00:01:23.021
So that would be our initial value.

00:01:23.021 --> 00:01:24.159
And then if you think about it, every time

00:01:24.159 --> 00:01:25.854
you increase t by one,

00:01:25.854 --> 00:01:29.151
you're going to mulitply by 1/7 again.

00:01:29.151 --> 00:01:31.845
And so, the ratio between successive terms

00:01:31.845 --> 00:01:33.307
is going to be 1/7.

00:01:33.307 --> 00:01:34.488
And so that's why we call that

00:01:34.488 --> 00:01:36.613
the common ratio.

00:01:36.613 --> 00:01:39.080
Hopefully, you found that interesting.