Finding zeros of polynomials (1 of 2)

WEBVTT
Kind: captions
Language: en

00:00:00.504 --> 00:00:02.547
- [Voiceover] So, we have a
fifth-degree polynomial here,

00:00:02.547 --> 00:00:04.869
p of x, and we're asked
to do several things.

00:00:04.869 --> 00:00:07.330
First, find the real roots.

00:00:07.330 --> 00:00:09.466
And let's sort of remind
ourselves what roots are.

00:00:09.466 --> 00:00:12.365
So root is the same thing as a zero,

00:00:12.365 --> 00:00:15.749
and they're the x-values
that make the polynomial

00:00:15.749 --> 00:00:17.415
equal to zero.

00:00:17.415 --> 00:00:19.679
So the real roots are the x-values

00:00:19.679 --> 00:00:22.793
where p of x is equal to zero.

00:00:22.793 --> 00:00:24.976
So, the x-values that satisfy this

00:00:24.976 --> 00:00:27.785
are going to be the roots, or the zeros,

00:00:27.785 --> 00:00:29.015
and we want the real ones.

00:00:29.015 --> 00:00:31.325
As you'll learn in the future,
there's also going to be

00:00:31.325 --> 00:00:34.240
imaginary roots, or
zeros, or there might be.

00:00:34.890 --> 00:00:37.014
Then we want to think
about how many times,

00:00:37.014 --> 00:00:39.881
how many times we intercept the x-axis.

00:00:39.881 --> 00:00:43.319
Well as we'll see, however
many real roots we have

00:00:43.319 --> 00:00:46.767
that's how many times we
are going to intercept...

00:00:46.767 --> 00:00:48.802
However many unique real roots we have,

00:00:48.802 --> 00:00:50.195
that's however many times we're going to

00:00:50.195 --> 00:00:51.475
intercept the x-axis.

00:00:51.475 --> 00:00:53.063
How do I know that?

00:00:53.063 --> 00:00:54.578
Well, let's just think about

00:00:54.578 --> 00:00:57.103
an arbitrary polynomial here.

00:00:57.103 --> 00:00:59.436
So those are my axes.

00:00:59.436 --> 00:01:03.813
This is the x-axis, that's my y-axis.

00:01:03.813 --> 00:01:07.458
And let me just graph an
arbitrary polynomial here.

00:01:07.458 --> 00:01:09.257
So, let's say it looks like that.

00:01:09.257 --> 00:01:11.370
Well, what's going on right over here.

00:01:11.370 --> 00:01:15.155
At this x-value, we see, based
on the graph of the function,

00:01:15.155 --> 00:01:17.894
that p of x is going to be equal to zero.

00:01:17.894 --> 00:01:19.369
So that's going to be a root.

00:01:19.369 --> 00:01:21.157
This is also going to be a root,

00:01:21.157 --> 00:01:23.785
because at this x-value, the
function is equal to zero.

00:01:23.785 --> 00:01:25.777
At this x-value the
function's equal to zero.

00:01:25.777 --> 00:01:27.750
At this x-value the
function is equal zero.

00:01:27.750 --> 00:01:30.815
If we're on the x-axis
then the y-value is zero.

00:01:30.815 --> 00:01:32.615
So the function is going
to be equal to zero.

00:01:32.615 --> 00:01:35.587
This is a graph of y is equal,

00:01:35.587 --> 00:01:36.829
y is equal to p of x.

00:01:36.829 --> 00:01:38.081
Not necessarily this p of x,

00:01:38.081 --> 00:01:40.593
but I'm just drawing
some arbitrary p of x.

00:01:40.593 --> 00:01:42.671
So there's some x-value
that makes the function

00:01:42.671 --> 00:01:43.936
equal to zero.

00:01:43.936 --> 00:01:45.016
Well, that's going to be a point

00:01:45.016 --> 00:01:47.477
at which we are intercepting the x-axis.

00:01:47.477 --> 00:01:48.336
So we want to know how many times

00:01:48.336 --> 00:01:49.810
we are intercepting the x-axis.

00:01:49.810 --> 00:01:51.273
As we'll see, it's
gonna be the same number

00:01:51.273 --> 00:01:54.593
of real roots, or the same
number of real zeros we have.

00:01:54.593 --> 00:01:56.502
And then they want us to
figure out the smallest

00:01:56.502 --> 00:01:59.928
of those x-intercepts,
and we'll figure it out

00:01:59.928 --> 00:02:03.729
for this particular polynomial.

00:02:03.729 --> 00:02:06.272
So, let me give myself
a little bit more space.

00:02:06.272 --> 00:02:07.549
So, let's get to it.

00:02:07.549 --> 00:02:12.239
So we really want to solve
p of x is equal to zero.

00:02:12.239 --> 00:02:14.723
So we really want to set,
that right over there,

00:02:14.723 --> 00:02:18.027
equal to zero, and solve this.

00:02:18.027 --> 00:02:20.969
So we want to solve this equation.

00:02:20.969 --> 00:02:22.640
The x-values that make this equal to zero,

00:02:22.640 --> 00:02:23.627
if I input them into the function

00:02:23.627 --> 00:02:25.949
I'm gonna get the function equaling zero.

00:02:25.949 --> 00:02:26.889
All right.

00:02:26.889 --> 00:02:28.433
So the first thing that
might jump out at you

00:02:28.433 --> 00:02:31.243
is that all of these
terms are divisible by x.

00:02:31.243 --> 00:02:33.623
So I like to factor that
out from the get-go.

00:02:33.623 --> 00:02:37.010
So, we can rewrite this as x times

00:02:37.010 --> 00:02:40.051
x to the fourth power

00:02:40.391 --> 00:02:43.189
plus nine x-squared

00:02:43.189 --> 00:02:45.731
minus two x-squared

00:02:45.731 --> 00:02:48.691
minus 18 is equal to zero.

00:02:48.691 --> 00:02:49.527
Now there's something else

00:02:49.527 --> 00:02:51.106
that might have jumped out at you.

00:02:51.106 --> 00:02:52.406
It actually just jumped out of me

00:02:52.406 --> 00:02:53.289
as I was writing this down is

00:02:53.289 --> 00:02:55.924
that we have two third-degree terms.

00:02:55.924 --> 00:02:57.027
After we've factored out an x,

00:02:57.027 --> 00:02:58.861
we have two second-degree terms.

00:02:58.861 --> 00:03:01.426
Now, it might be tempting to
just add these two together,

00:03:01.426 --> 00:03:03.518
and actually that it would be
a completely legitimate way

00:03:03.518 --> 00:03:07.219
of trying to factor this so
that we can solve this equation.

00:03:07.219 --> 00:03:09.588
But instead of doing it that way,

00:03:09.588 --> 00:03:10.625
we might take this as a clue

00:03:10.625 --> 00:03:13.790
that maybe we can factor by grouping.

00:03:13.790 --> 00:03:15.926
Remember, factor by grouping, you split up

00:03:15.926 --> 00:03:18.352
that middle degree term
and see if you can reverse

00:03:18.352 --> 00:03:20.326
the distributive property twice.

00:03:20.326 --> 00:03:21.847
So, let's see if we can do that.

00:03:21.847 --> 00:03:25.100
Can we group together
these first two terms

00:03:25.910 --> 00:03:27.930
and factor something interesting out?

00:03:27.930 --> 00:03:31.529
And group together these second two terms

00:03:31.529 --> 00:03:32.806
and factor something interesting out?

00:03:32.806 --> 00:03:34.605
And then maybe we can factor
something out after that.

00:03:34.605 --> 00:03:35.986
What am I talking about?

00:03:35.986 --> 00:03:40.050
Well, this is going to be
the same thing as x times...

00:03:40.050 --> 00:03:42.708
Well, this one, actually let
me write a big parenthesis,

00:03:42.708 --> 00:03:46.829
here, this one right over
here is the same thing as...

00:03:46.829 --> 00:03:48.710
I can factor out an x-squared.

00:03:48.710 --> 00:03:52.426
So, it's gonna be x-squared plus...

00:03:53.376 --> 00:03:54.356
Sorry.

00:03:54.356 --> 00:03:56.569
It's gonna be x-squared, if
I factor out an x-squared,

00:03:56.569 --> 00:04:00.231
I'm gonna get an x-squared plus nine.

00:04:00.841 --> 00:04:03.268
And then over here, if I factor out

00:04:03.268 --> 00:04:05.740
a, let's see, negative two.

00:04:05.740 --> 00:04:06.901
I don't want to...

00:04:06.901 --> 00:04:09.308
If I factor out a, yep, negative two,

00:04:09.308 --> 00:04:12.753
I'm gonna get, so minus two times...

00:04:12.753 --> 00:04:16.966
I'm gonna get an x-squared
plus nine, again.

00:04:16.966 --> 00:04:18.464
Now this is interesting,
because this is telling us

00:04:18.464 --> 00:04:23.084
maybe we can factor out
an x-squared plus nine.

00:04:23.084 --> 00:04:26.950
So, let me factor out
an x-squared plus nine

00:04:26.950 --> 00:04:29.966
from each of these terms,
and I'm going to get,

00:04:29.966 --> 00:04:32.487
I am going to get x...

00:04:32.487 --> 00:04:34.798
I'll leave these big green
parentheses here for now,

00:04:34.798 --> 00:04:37.108
If we factor out an x-squared plus nine,

00:04:37.108 --> 00:04:41.559
it's going to be x-squared plus nine

00:04:41.949 --> 00:04:45.746
times x-squared,

00:04:47.126 --> 00:04:49.551
x-squared minus two.

00:04:49.971 --> 00:04:52.490
X-squared minus two,

00:04:52.490 --> 00:04:55.729
and I gave myself a
little bit too much space.

00:04:55.729 --> 00:04:57.934
So, let me delete that.

00:04:57.934 --> 00:05:01.150
So let me delete that right over there

00:05:01.150 --> 00:05:03.679
and then close the parentheses.

00:05:04.679 --> 00:05:08.846
Then close the parentheses.

00:05:08.846 --> 00:05:11.900
Actually, I can even get rid
of those green parentheses now,

00:05:11.900 --> 00:05:15.464
if I want to, optimally, make
this a little bit simpler.

00:05:15.464 --> 00:05:17.167
So far we've been able to factor it as

00:05:17.167 --> 00:05:19.605
x times x-squared plus nine
times x-squared minus two.

00:05:19.605 --> 00:05:20.975
And the whole point
that I'm factoring this

00:05:20.975 --> 00:05:23.111
is if I can find the product of a bunch

00:05:23.111 --> 00:05:24.435
of expressions equaling zero,

00:05:24.435 --> 00:05:26.524
then I can say, "Well, the
product of those expressions

00:05:26.524 --> 00:05:28.370
"are going to be zero if one
or more of those expressions

00:05:28.370 --> 00:05:31.005
"are equal to zero",
and I can solve for x.

00:05:31.005 --> 00:05:31.934
Well, let's see.

00:05:31.934 --> 00:05:33.547
This one's completely factored.

00:05:33.547 --> 00:05:35.510
This one is completely
factored if we're thinking

00:05:35.510 --> 00:05:37.866
about real roots.

00:05:37.866 --> 00:05:40.153
This one, you can view it
as a difference of squares

00:05:40.153 --> 00:05:43.949
if you view two as a
square root of two-squared.

00:05:43.949 --> 00:05:46.573
So, we can rewrite this as,

00:05:46.573 --> 00:05:48.256
and of course all of
this is equal to zero.

00:05:48.256 --> 00:05:50.926
Let me just write equals.

00:05:50.926 --> 00:05:54.552
So we could write this as equal to x times

00:05:55.078 --> 00:05:57.381
times x-squared

00:05:57.381 --> 00:06:00.361
plus nine times...

00:06:00.361 --> 00:06:02.849
Let's see, I can factor this business

00:06:02.849 --> 00:06:07.074
into x plus the square root of two

00:06:07.074 --> 00:06:09.303
times x minus the square root of two.

00:06:09.303 --> 00:06:11.454
I'm just recognizing this
as a difference of squares.

00:06:11.454 --> 00:06:13.582
And, once again, we just
want to solve this whole,

00:06:13.582 --> 00:06:16.373
all of this business, equaling zero.

00:06:16.373 --> 00:06:18.405
All of this equaling zero.

00:06:18.405 --> 00:06:19.531
So how can this equal to zero?

00:06:19.531 --> 00:06:21.028
Well any one of these expressions,

00:06:21.028 --> 00:06:23.571
if I take the product, and if
any one of them equals zero

00:06:23.571 --> 00:06:24.825
then I'm gonna get zero.

00:06:24.825 --> 00:06:27.188
So, x could be equal to zero.

00:06:28.168 --> 00:06:29.271
X could be equal to zero,

00:06:29.271 --> 00:06:31.210
and that actually gives us a root.

00:06:31.210 --> 00:06:33.531
When x is equal to zero, this
polynomial is equal to zero,

00:06:33.531 --> 00:06:35.667
and that's pretty easy to verify.

00:06:35.667 --> 00:06:38.550
Let's see, can x-squared
plus nine equal zero?

00:06:39.870 --> 00:06:42.122
X-squared plus nine equal zero.

00:06:42.122 --> 00:06:43.585
Well, if you subtract
nine from both sides,

00:06:43.585 --> 00:06:45.407
you get x-squared is
equal to negative nine.

00:06:45.407 --> 00:06:48.066
And that's why I said, there's
no real solution to this.

00:06:48.066 --> 00:06:50.078
So, no real, let me write that,

00:06:50.078 --> 00:06:53.458
no real solution.

00:06:54.168 --> 00:06:57.925
There are some imaginary
solutions, but no real solutions.

00:06:57.925 --> 00:07:02.106
Now, can x plus the square
root of two equal zero?

00:07:02.106 --> 00:07:05.166
X plus the square root of two equal zero.

00:07:05.166 --> 00:07:07.348
Sure, if we subtract square
root of two from both sides,

00:07:07.348 --> 00:07:10.924
you get x is equal to the
negative square root of two.

00:07:10.924 --> 00:07:14.917
And can x minus the square
root of two equal zero?

00:07:14.917 --> 00:07:17.170
Sure, you add square root
of two to both sides,

00:07:17.170 --> 00:07:21.128
you get x is equal to
the square root of two.

00:07:21.128 --> 00:07:22.092
So, there we have it.

00:07:22.092 --> 00:07:24.448
We have figured out our zeros.

00:07:24.448 --> 00:07:26.027
X could be equal to zero.

00:07:26.027 --> 00:07:28.047
P of zero is zero.

00:07:28.047 --> 00:07:31.008
P of negative square root of two is zero,

00:07:31.008 --> 00:07:33.503
and p of square root of
two is equal to zero.

00:07:33.503 --> 00:07:35.221
So, those are our zeros.

00:07:35.221 --> 00:07:38.368
Their zeros are at zero,
negative squares of two,

00:07:38.368 --> 00:07:40.504
and positive squares of two.

00:07:40.504 --> 00:07:42.431
And so those are going
to be the three times

00:07:42.431 --> 00:07:44.904
that we intercept the x-axis.

00:07:44.904 --> 00:07:47.005
And what is the smallest
of those intercepts?

00:07:47.005 --> 00:07:48.421
Well, the smallest number here

00:07:48.421 --> 00:07:50.429
is negative square root,

00:07:50.429 --> 00:07:53.149
negative square root of two.

00:07:53.149 --> 00:07:54.771
And you could tackle it the other way.

00:07:54.771 --> 00:07:58.081
You could take this part right over...

00:07:58.081 --> 00:07:58.738
Which part?

00:07:58.738 --> 00:07:59.920
Yeah, this part right over here

00:07:59.920 --> 00:08:02.062
and you could add those two middle terms,

00:08:02.062 --> 00:08:05.126
and then factor in a non-grouping way,

00:08:05.126 --> 00:08:07.170
and I encourage you to do that.

00:08:07.170 --> 00:08:08.864
But just to see that this makes sense

00:08:08.864 --> 00:08:11.453
that zeros really are the x-intercepts.

00:08:11.453 --> 00:08:16.313
I went to Wolfram|Alpha and
I graphed this polynomial

00:08:16.313 --> 00:08:19.357
and this is what I got.

00:08:19.357 --> 00:08:23.481
So, this is what I got, right over here.

00:08:23.981 --> 00:08:25.591
If you see a fifth-degree polynomial,

00:08:25.591 --> 00:08:29.120
say, it'll have as many
as five real zeros.

00:08:29.120 --> 00:08:30.316
But, if it has some imaginary zeros,

00:08:30.316 --> 00:08:32.197
it won't have five real zeros.

00:08:32.197 --> 00:08:34.194
Instead, this one has three.

00:08:34.194 --> 00:08:35.854
And that's because the imaginary zeros,

00:08:35.854 --> 00:08:37.549
which we'll talk more about in the future,

00:08:37.549 --> 00:08:39.638
they come in these conjugate pairs.

00:08:39.638 --> 00:08:42.192
So, if you don't have five real roots,

00:08:42.192 --> 00:08:44.366
the next possibility is
that you're going to have

00:08:44.366 --> 00:08:45.234
three real roots.

00:08:45.234 --> 00:08:46.557
And, if you don't have three real roots,

00:08:46.557 --> 00:08:48.811
the next possibility is you're
gonna have one real root.

00:08:48.811 --> 00:08:50.818
So, that's an interesting
thing to think about.

00:08:50.818 --> 00:08:54.978
And so, here you see,
your three real roots.

00:08:55.438 --> 00:08:57.412
You see your three real roots

00:08:57.412 --> 00:08:59.791
which correspond to the x-values

00:08:59.791 --> 00:09:02.589
at which the function is equal to zero,

00:09:02.589 --> 00:09:07.431
which is where we have our x-intercepts.


Finding zeros of polynomials (2 of 2)

WEBVTT
Kind: captions
Language: en

00:00:00.363 --> 00:00:03.103
- [Voiceover] In the last video,
we factored this polynomial

00:00:03.103 --> 00:00:04.660
in order to find the real roots.

00:00:04.660 --> 00:00:06.388
We factored it by grouping,

00:00:06.388 --> 00:00:08.606
which essentially doing
the distributive property

00:00:08.606 --> 00:00:10.139
in reverse twice.

00:00:10.139 --> 00:00:11.984
And I mentioned that there's
two ways you could do it.

00:00:11.984 --> 00:00:13.772
You could actually, from the get go,

00:00:13.772 --> 00:00:16.534
add these two middle degree terms,

00:00:16.534 --> 00:00:19.192
and then, think about it from there.

00:00:19.192 --> 00:00:20.888
So, what I thought I'd
do is just a quick video

00:00:20.888 --> 00:00:22.908
on that alternative.

00:00:22.908 --> 00:00:26.387
So, if we add, instead of grouping,

00:00:26.387 --> 00:00:29.011
if we add these middle two terms.

00:00:29.011 --> 00:00:30.787
Actually, I'll just focus

00:00:30.787 --> 00:00:32.436
on the fourth degree polynomial here.

00:00:32.436 --> 00:00:34.247
We know that we have an x out front.

00:00:36.591 --> 00:00:41.050
This fourth degree polynomial
is going to simplify to

00:00:42.120 --> 00:00:44.017
x to the fourth

00:00:44.017 --> 00:00:46.886
plus seven x squared

00:00:50.005 --> 00:00:51.306
minus 18.

00:00:54.664 --> 00:00:58.331
If we wanna factor this, we
could recognize a pattern here.

00:00:59.323 --> 00:01:00.762
You probably remember.

00:01:00.762 --> 00:01:01.691
Hopefully you remember.

00:01:01.691 --> 00:01:03.109
If you don't, then you might wanna review

00:01:03.109 --> 00:01:04.108
your factoring polynomials.

00:01:04.108 --> 00:01:06.209
But if you have x plus a

00:01:06.209 --> 00:01:08.624
times x plus b,

00:01:08.624 --> 00:01:10.288
that's going to be equal to,

00:01:10.288 --> 00:01:14.548
x is going to be equal to x squared plus

00:01:14.548 --> 00:01:16.951
the sum of those two numbers, a and b,

00:01:16.951 --> 00:01:18.751
as being the coefficient of the x term

00:01:18.751 --> 00:01:20.411
plus the product of those two numbers.

00:01:20.411 --> 00:01:22.628
If you just multiply this out,

00:01:22.628 --> 00:01:24.126
this is what you would get.

00:01:24.126 --> 00:01:28.827
But if this was x squared
plus a times x squared plus b,

00:01:28.827 --> 00:01:30.119
instead of this being x squared,

00:01:30.119 --> 00:01:31.701
this would be x to the fourth.

00:01:31.701 --> 00:01:34.232
Instead of this being x,
this would be x squared,

00:01:34.232 --> 00:01:36.913
which is exactly the pattern we have here.

00:01:36.913 --> 00:01:39.978
So, what two a's and b's
that if I add them up,

00:01:39.978 --> 00:01:41.894
I would get seven,

00:01:41.894 --> 00:01:43.241
and if I were to take their product,

00:01:44.633 --> 00:01:47.844
I get negative 18?

00:01:48.314 --> 00:01:49.977
Well, since their product
is negative, we know

00:01:49.977 --> 00:01:51.519
that they are of different signs.

00:01:51.519 --> 00:01:53.655
One will be positive,
one will be negative.

00:01:53.655 --> 00:01:55.664
And since their sum is positive, we know

00:01:55.664 --> 00:01:57.823
that the larger of the two numbers

00:01:57.823 --> 00:02:00.238
is going to be positive.

00:02:00.238 --> 00:02:03.953
So, what jumps out at me is nine times

00:02:03.953 --> 00:02:05.102
negative two.

00:02:05.102 --> 00:02:06.774
You multiply those, you get negative 18.

00:02:06.774 --> 00:02:08.933
You take their sum, you get seven.

00:02:08.933 --> 00:02:12.334
So, we can rewrite this, just
looking at this pattern here

00:02:12.334 --> 00:02:15.956
as x squared plus nine

00:02:15.956 --> 00:02:19.624
times x squared minus two.

00:02:19.624 --> 00:02:21.203
I could say plus negative two.

00:02:21.203 --> 00:02:23.199
That's the same thing
as x squared minus two.

00:02:23.199 --> 00:02:24.999
And then, that's exactly what we got

00:02:26.332 --> 00:02:28.226
right over here.

00:02:28.226 --> 00:02:29.759
Of course, you have this x out front

00:02:29.759 --> 00:02:31.546
that I didn't consider right over here.

00:02:31.546 --> 00:02:33.798
And then, this, as we did
in the previous video,

00:02:33.798 --> 00:02:37.246
you could recognize as
a difference of squares

00:02:37.246 --> 00:02:40.356
and then, factor it further
to actually find the roots.

00:02:40.356 --> 00:02:42.539
But I just wanted to show
that you could solve this

00:02:42.539 --> 00:02:46.672
by regrouping, or you can solve this by,

00:02:46.672 --> 00:02:49.678
I guess you could say, more
traditional factoring means.

00:02:49.678 --> 00:02:52.534
And notice this nine and negative two,

00:02:52.534 --> 00:02:54.450
this is what was already broken up for us,

00:02:54.450 --> 00:02:57.446
so we could factor by regrouping.


Determining if a function is invertible

WEBVTT
Kind: captions
Language: en

00:00:10.355 --> 00:00:12.038
- [Voiceover] So if x is equal to a

00:00:12.038 --> 00:00:14.696
then, so if we input a into our function

00:00:14.696 --> 00:00:16.415
then we output -6.

00:00:16.415 --> 00:00:17.959
f of a is -6.

00:00:17.959 --> 00:00:20.582
We input b we get three,
we input c we get -6,

00:00:20.582 --> 00:00:23.988
we input d we get two,
we input e we get -6.

00:00:36.146 --> 00:00:38.782
Alright, so let's see
what's going on over here.

00:00:38.782 --> 00:00:40.488
Let me scroll down a little bit more.

00:00:40.488 --> 00:00:44.587
So in this purple oval, this
is representing the domain

00:00:44.587 --> 00:00:46.723
of our function f and this is the range.

00:00:46.723 --> 00:00:47.930
So the function is going to,

00:00:47.930 --> 00:00:49.822
if you give it a member of the domain

00:00:49.822 --> 00:00:51.888
it's going to map from
that member of domain

00:00:51.888 --> 00:00:53.282
to a member of the range.

00:00:53.282 --> 00:00:56.682
So, for example, you
input a into the function

00:00:56.682 --> 00:00:58.284
it goes to -6.

00:00:58.284 --> 00:01:01.721
So a goes to -6, so I drag
that right over there.

00:01:01.721 --> 00:01:03.643
b goes to three,

00:01:03.643 --> 00:01:07.218
b goes to three.

00:01:07.218 --> 00:01:09.679
c goes to -6, so it's already interesting

00:01:09.679 --> 00:01:12.871
that we have multiple
values that point to -6.

00:01:12.871 --> 00:01:15.658
So this is okay for f to be a function

00:01:15.658 --> 00:01:17.712
but we'll see it might
make it a little bit tricky

00:01:17.712 --> 00:01:19.315
for f to be invertible.

00:01:19.315 --> 00:01:23.680
So let's see, d is points
to two, or maps to two.

00:01:23.680 --> 00:01:26.871
So you input d into our
function you're gonna output two

00:01:26.871 --> 00:01:31.040
and then finally e maps to -6 as well.

00:01:31.040 --> 00:01:33.513
e maps to -6 as well.

00:01:33.513 --> 00:01:36.809
So, that's a visualization
of how this function f maps

00:01:36.809 --> 00:01:40.256
from a through e to members of the range

00:01:40.256 --> 00:01:43.252
but also ask ourselves is
this function invertible?

00:01:43.252 --> 00:01:45.180
And I already hinted at it a little bit.

00:01:45.180 --> 00:01:49.057
Well in order for it to
be invertible you need a,

00:01:49.057 --> 00:01:51.878
you need a function that could take

00:01:51.878 --> 00:01:54.350
go from each of these points to,

00:01:54.350 --> 00:01:56.555
they can do the inverse mapping.

00:01:56.555 --> 00:01:58.530
But it has to be a function.

00:01:58.530 --> 00:02:01.756
So, if you input three
into this inverse function

00:02:01.756 --> 00:02:03.115
it should give you b.

00:02:03.115 --> 00:02:05.472
If you input two into
this inverse function

00:02:05.472 --> 00:02:07.713
it should output d.

00:02:07.713 --> 00:02:12.217
If you input -6 into
this inverse function,

00:02:12.217 --> 00:02:14.910
well this hypothetical inverse function.

00:02:14.910 --> 00:02:16.477
what should it do?

00:02:16.477 --> 00:02:19.031
Well you can't have a function
that if you input one,

00:02:19.031 --> 00:02:22.619
if you input a number it could
have three possible values,

00:02:22.619 --> 00:02:26.531
a, c, or e, you can only map to one value.

00:02:26.531 --> 00:02:28.530
So there isn't, you actually can't set up

00:02:28.530 --> 00:02:30.168
an inverse function that does this

00:02:30.168 --> 00:02:31.072
'cause it wouldn't be a function.

00:02:31.072 --> 00:02:33.986
You can't go from input -6
into that inverse function

00:02:33.986 --> 00:02:35.785
and get three different values.

00:02:35.785 --> 00:02:38.606
So this is not invertible.

00:02:38.606 --> 00:02:40.731
Let's do another example.

00:02:40.731 --> 00:02:43.865
So here, so this is the same drill.

00:02:43.865 --> 00:02:46.570
We have our members of our
domain, members of our range.

00:02:46.570 --> 00:02:48.545
We can build our mapping diagram.

00:02:48.545 --> 00:02:51.202
a maps to -36,

00:02:51.202 --> 00:02:54.189
b maps to nine,

00:02:54.189 --> 00:02:57.438
c maps to -4,

00:02:57.438 --> 00:03:00.550
d maps to 49,

00:03:00.550 --> 00:03:04.996
and then finally e maps to 25.

00:03:04.996 --> 00:03:08.352
e maps to 25.

00:03:08.352 --> 00:03:11.230
Now is this function invertible?

00:03:11.230 --> 00:03:12.334
Well let's think about it.

00:03:12.334 --> 00:03:15.996
The inverse, woops, the,
was it d maps to 49.

00:03:15.996 --> 00:03:17.740
So, let's think about what the inverse,

00:03:17.740 --> 00:03:19.981
this hypothetical inverse
function would have to do.

00:03:19.981 --> 00:03:22.081
It would have to take each
of these members of the range

00:03:22.081 --> 00:03:23.846
and do the inverse mapping.

00:03:23.846 --> 00:03:27.421
So if you input 49 into
our inverse function

00:03:27.421 --> 00:03:28.362
it should give you d.

00:03:28.362 --> 00:03:30.265
Input 25 it should give you e.

00:03:30.265 --> 00:03:32.297
Input nine it gives you b.

00:03:32.297 --> 00:03:34.596
You input -4 it inputs c.

00:03:34.596 --> 00:03:37.464
You input -36 it gives you a.

00:03:37.464 --> 00:03:40.505
So you could easily construct
an inverse function here.

00:03:40.505 --> 00:03:45.322
So this is very much, this
is very much invertible.

00:03:45.322 --> 00:03:47.901
One way to think about it is these are a,

00:03:47.901 --> 00:03:51.394
this is a one to one mapping.

00:03:51.394 --> 00:03:53.716
Each of the members of the domain

00:03:53.716 --> 00:03:56.095
correspond to a unique
member of the range.

00:03:56.095 --> 00:03:57.860
You don't have two members of the domain

00:03:57.860 --> 00:04:01.019
pointing to the same member of the range.

00:04:01.019 --> 00:04:03.068
Anyway, hopefully you
found that interesting.


Verifying inverse functions by composition: not inverse

WEBVTT
Kind: captions
Language: en

00:00:00.310 --> 00:00:04.596
- [Voiceover] Let's say that
f of x is equal to two x

00:00:04.596 --> 00:00:07.853
minus three, and g of x,

00:00:08.461 --> 00:00:12.849
g of x is equal to 1/2 x

00:00:13.310 --> 00:00:14.590
plus three.

00:00:14.590 --> 00:00:17.008
What I wanna do in this video is evaluate

00:00:17.008 --> 00:00:19.405
what f of g of x is,

00:00:20.975 --> 00:00:23.528
and then I wanna evaluate
what g of f of x is.

00:00:23.528 --> 00:00:26.797
So first, I wanna evaluate f of g of x,

00:00:27.522 --> 00:00:29.188
and then I'm gonna evaluate
the other way around.

00:00:29.188 --> 00:00:30.873
I'm gonna evaluate g of f of x.

00:00:30.873 --> 00:00:32.580
But let's evaluate f of g of x first.

00:00:32.580 --> 00:00:34.734
And I, like always, encourage
you to pause the video

00:00:34.734 --> 00:00:36.868
and see if you can work through it.

00:00:36.868 --> 00:00:38.596
This is going to be equal to,

00:00:38.596 --> 00:00:41.309
f of g of x is going to be equal to,

00:00:41.309 --> 00:00:44.499
wherever we see the x in
our definition for f of x,

00:00:44.499 --> 00:00:47.994
the input now is g of x, so
we'd replace it with the g of x.

00:00:47.994 --> 00:00:50.714
It's gonna be two times g of x.

00:00:51.256 --> 00:00:54.981
Two times g of x minus three.

00:00:57.555 --> 00:01:00.521
And this is going to
be equal to two times,

00:01:00.521 --> 00:01:02.756
well, g of x is all of that business,

00:01:02.756 --> 00:01:07.415
two times 1/2 x plus three,

00:01:07.998 --> 00:01:10.137
and then we have the minus three.

00:01:11.655 --> 00:01:14.160
And now we can distribute this two,

00:01:14.804 --> 00:01:19.218
two times 1/2 x is just
going to be equal to x.

00:01:19.226 --> 00:01:21.724
Two times three is going to be six.

00:01:21.724 --> 00:01:25.422
So x plus six minus three.

00:01:27.882 --> 00:01:31.688
This is going to equal x plus three.

00:01:31.695 --> 00:01:34.255
X plus three, all right, interesting.

00:01:34.255 --> 00:01:36.064
That's f of g of x.

00:01:36.064 --> 00:01:39.396
Now let's think about what
g of f of x is going to be.

00:01:39.396 --> 00:01:42.183
So g of,

00:01:42.183 --> 00:01:44.572
our input, instead of being,
instead of calling our input x,

00:01:44.576 --> 00:01:46.737
we're gonna call our input f of x.

00:01:47.014 --> 00:01:51.707
So g of f of x is going to be equal to

00:01:54.728 --> 00:01:59.728
1/2 times our input, which
in this case is f of x.

00:02:00.357 --> 00:02:03.511
1/2 time f of x plus three.

00:02:05.477 --> 00:02:08.198
You can view the x up
here as the placeholder

00:02:08.198 --> 00:02:10.373
for whatever our input happens to be.

00:02:10.373 --> 00:02:12.770
And now our input is going to be f of x.

00:02:12.770 --> 00:02:16.224
And so, this is going to
be equal to 1/2 times,

00:02:16.224 --> 00:02:17.422
what is f of x?

00:02:17.422 --> 00:02:19.533
It is two x minus three.

00:02:20.177 --> 00:02:23.570
So, two times x minus three,

00:02:23.570 --> 00:02:26.217
and we have a plus three.

00:02:26.536 --> 00:02:27.969
And now we can distribute the 1/2.

00:02:27.969 --> 00:02:31.057
1/2 times two x is going to be x.

00:02:31.057 --> 00:02:36.057
1/2 times negative three is negative 3/2s.

00:02:36.705 --> 00:02:39.590
And then we have a plus three.

00:02:39.590 --> 00:02:41.907
So let's see, three is
the same thing as 6/2s.

00:02:41.907 --> 00:02:44.934
So 6/2s minus 3/2s is going to be 3/2s.

00:02:44.934 --> 00:02:49.517
So this is going to be
equal to x plus 3/2s.

00:02:49.517 --> 00:02:52.402
So notice, we definitely
got different things

00:02:52.402 --> 00:02:54.637
for f of g of x and g of f of x.

00:02:54.637 --> 00:02:56.242
And we also didn't do a round trip.

00:02:56.242 --> 00:02:58.050
We didn't go back to x.

00:02:58.050 --> 00:03:00.854
So we know that these are
not inverses of each other.

00:03:00.854 --> 00:03:03.292
In fact, we just have to
do either this or that

00:03:03.292 --> 00:03:06.055
to know that they're not
inverses of each other.

00:03:06.055 --> 00:03:07.957
These are not inverses.

00:03:07.957 --> 00:03:09.765
So we write it this way.

00:03:09.765 --> 00:03:13.502
F of x does not equal

00:03:14.772 --> 00:03:17.360
the inverse of g of x.

00:03:20.117 --> 00:03:23.019
And g of x does not equal

00:03:23.019 --> 00:03:25.646
the inverse of f of x.

00:03:28.505 --> 00:03:30.902
In order for them to be inverses,

00:03:30.902 --> 00:03:33.869
if you have an x value right over here,

00:03:33.869 --> 00:03:37.694
and if you apply g to it,
if you input it into g,

00:03:38.155 --> 00:03:40.979
and then that takes you to g of x,

00:03:40.979 --> 00:03:42.869
so that takes you to g
of x right over here,

00:03:42.869 --> 00:03:44.189
so that's the function g,

00:03:44.189 --> 00:03:46.221
and then you apply f to it,

00:03:46.221 --> 00:03:48.456
you would have to get
back to the same place.

00:03:48.456 --> 00:03:51.971
So g inverse would get us back

00:03:51.971 --> 00:03:53.901
to the same place.

00:03:53.901 --> 00:03:55.892
And clearly, we did not
get back to the same place.

00:03:55.892 --> 00:03:58.655
We didn't get back to x, we
got back to x plus three.

00:03:58.655 --> 00:04:00.037
Same thing over here.

00:04:00.037 --> 00:04:02.001
We see that we did not get,

00:04:02.001 --> 00:04:04.199
we did not go get back to x,

00:04:04.843 --> 00:04:06.956
we got to x plus 3/2s.

00:04:06.956 --> 00:04:10.152
So they're definitely not
inverses of each other.


Verifying inverse functions by composition

WEBVTT
Kind: captions
Language: en

00:00:00.378 --> 00:00:04.488
- [Voiceover] Let's say
that f of x is equal to

00:00:04.488 --> 00:00:09.262
x plus 7 to the third power, minus one.

00:00:09.262 --> 00:00:12.177
And let's say that g of x

00:00:12.177 --> 00:00:15.307
g of x is equal to

00:00:15.307 --> 00:00:19.879
the cube root of x plus one

00:00:19.879 --> 00:00:23.340
the cube root of x plus one, minus seven.

00:00:23.340 --> 00:00:26.970
Now, what I wanna do now is evaluate

00:00:26.970 --> 00:00:31.910
f of g of x

00:00:31.910 --> 00:00:34.819
I wanna evaluate f of g of x

00:00:34.819 --> 00:00:39.226
and I also wanna evaluate g of f of x

00:00:40.146 --> 00:00:43.415
g of f of x, and see what I get

00:00:43.415 --> 00:00:48.222
I encourage you, like always,
pause the video and try it out

00:00:48.222 --> 00:00:50.816
Let's first evaluate f of g of x.

00:00:50.816 --> 00:00:54.032
That means, g of x, this
expression is going to be our input

00:00:54.032 --> 00:00:56.889
So, everywhere we see an x
in the definition for f of x,

00:00:56.889 --> 00:00:59.506
we would replace it with all of g of x

00:00:59.506 --> 00:01:03.094
so, f of g of x is going to be equal to

00:01:03.094 --> 00:01:04.797
so it's going to be equal to,

00:01:04.797 --> 00:01:06.207
well I see an x right over there

00:01:06.207 --> 00:01:08.162
so I'll write all of g of x there

00:01:08.162 --> 00:01:12.453
so that's the cube root of

00:01:12.453 --> 00:01:15.253
x plus one

00:01:15.253 --> 00:01:19.620
minus seven, and then I have plus seven

00:01:19.620 --> 00:01:24.587
plus seven to the third power, minus one

00:01:24.587 --> 00:01:29.305
notice, whenever I saw the x,
since I'm taking f of g of x,

00:01:29.305 --> 00:01:31.662
I replace it with what g of x is,

00:01:31.662 --> 00:01:35.284
so, that is, the cube root
of x plus one minus seven.

00:01:35.284 --> 00:01:37.350
Alright, I'll see if we can simplify this.

00:01:37.350 --> 00:01:39.544
Well, we have a minus seven plus seven

00:01:39.544 --> 00:01:41.170
so that simplifies nicely.

00:01:41.170 --> 00:01:44.339
So, this just becomes, this is equal to,

00:01:44.339 --> 00:01:46.145
I can do a neutral color now,

00:01:46.145 --> 00:01:47.705
this is equal to

00:01:47.705 --> 00:01:51.707
the cube root of x plus one

00:01:51.707 --> 00:01:56.142
to the third power, minus one.

00:01:56.142 --> 00:01:57.858
Well, if I take the
cube root of x plus one

00:01:57.858 --> 00:01:59.475
and then I raise it to the third power,

00:01:59.475 --> 00:02:01.861
well, that's just gonna
give me x plus one.

00:02:01.861 --> 00:02:04.265
So, this part

00:02:04.265 --> 00:02:07.685
this part just simplifies to x plus one,

00:02:07.685 --> 00:02:10.080
and then I subtract one,

00:02:10.080 --> 00:02:14.538
so it all simplified out
to just being equal to x.

00:02:14.538 --> 00:02:16.662
So we're just left with an x.

00:02:16.662 --> 00:02:19.797
So, f of g of x is just x.

00:02:19.797 --> 00:02:23.477
So now, let's try what g of f of x is.

00:02:23.477 --> 00:02:26.635
So, g of f of x is going to be equal to,

00:02:26.635 --> 00:02:28.039
I'll do it right over here,

00:02:28.039 --> 00:02:30.361
this is going to be equal to

00:02:30.361 --> 00:02:33.983
the cube root of

00:02:33.983 --> 00:02:35.457
actually, let me write it out,

00:02:35.457 --> 00:02:38.128
wherever I see an x, I
can write f of x instead,

00:02:38.128 --> 00:02:40.223
I didn't do it that last
time, I went directly

00:02:40.223 --> 00:02:42.042
and replaced with the definition of f of x

00:02:42.042 --> 00:02:43.489
but just to make it clear what I'm doing

00:02:43.489 --> 00:02:47.083
everywhere I'm seeing an x,
I replace it with an f of x.

00:02:47.083 --> 00:02:51.725
So, the cube root of f of x plus one,

00:02:51.725 --> 00:02:53.755
minus seven.

00:02:53.755 --> 00:02:57.335
Well, that's going to be
equal to the cube root of

00:02:57.335 --> 00:02:59.701
cube root of f of x,

00:02:59.701 --> 00:03:03.051
which is all of this business over here

00:03:03.051 --> 00:03:07.956
so that is x plus seven
to the third power,

00:03:08.586 --> 00:03:12.377
minus one, and then we add one

00:03:12.377 --> 00:03:16.436
and we add one, and then
we subtract the seven

00:03:16.436 --> 00:03:19.538
lucky for us, subtracting
one and adding one,

00:03:19.538 --> 00:03:20.590
those cancel out.

00:03:20.590 --> 00:03:21.461
Next, we're gonna take the

00:03:21.461 --> 00:03:24.514
cube root of x plus
seven to the third power.

00:03:24.514 --> 00:03:27.068
Well, the cube root of x
plus seven to the third power

00:03:27.068 --> 00:03:29.148
is just going to be x plus seven

00:03:29.148 --> 00:03:30.628
so, this is going to be

00:03:30.628 --> 00:03:32.787
x plus seven,

00:03:32.787 --> 00:03:35.945
for all of this business
simplifies to x plus seven,

00:03:35.945 --> 00:03:39.021
and then we do subtract seven

00:03:39.021 --> 00:03:42.248
and these two cancel out,
or they negate each other

00:03:42.248 --> 00:03:44.444
and we are just left with x.

00:03:44.444 --> 00:03:46.296
So, we see something very interesting.

00:03:46.296 --> 00:03:49.105
f of g of x is just x

00:03:49.105 --> 00:03:51.264
and g of f of x is x.

00:03:51.264 --> 00:03:54.967
So, in this case, if we start with an x

00:03:54.967 --> 00:03:59.298
if we start with an x, we
input it into the function g

00:03:59.298 --> 00:04:01.918
and we get g of x

00:04:01.918 --> 00:04:04.128
we get g of x

00:04:04.128 --> 00:04:07.085
and then we input that
into the function f,

00:04:07.085 --> 00:04:09.970
then we input that into the function f,

00:04:09.970 --> 00:04:13.441
f of g of x gets us back to x.

00:04:13.441 --> 00:04:15.844
It gets us back to x.

00:04:15.844 --> 00:04:17.701
So we kind of did a round-trip.

00:04:17.701 --> 00:04:19.585
And the same thing is happening over here.

00:04:19.585 --> 00:04:22.867
If I put x into f of x...

00:04:25.097 --> 00:04:27.542
I'm sorry, if I put x into the function f,

00:04:27.542 --> 00:04:31.175
and I get f of x, the output is f of x,

00:04:31.175 --> 00:04:35.494
and then I input that into
g, into the function g

00:04:35.494 --> 00:04:38.220
into the function g, once
again I do this round-trip

00:04:38.220 --> 00:04:39.743
and I get back to x.

00:04:39.743 --> 00:04:43.275
Another way to think about it,

00:04:45.005 --> 00:04:47.298
these are both composite functions,

00:04:47.298 --> 00:04:49.098
one way to think about it is,

00:04:49.098 --> 00:04:51.350
if these are the set
of all possible inputs

00:04:51.350 --> 00:04:53.591
into either of these composite functions,

00:04:53.591 --> 00:04:56.110
and then, these are the outputs,

00:04:56.110 --> 00:04:59.627
so you're starting with an x,

00:04:59.627 --> 00:05:01.450
I'll do this case first,

00:05:01.450 --> 00:05:03.830
so, g is a mapping,

00:05:03.830 --> 00:05:05.571
let me write down,

00:05:05.571 --> 00:05:07.675
so, g is going to be a mapping

00:05:07.675 --> 00:05:11.185
from x to g of x

00:05:11.185 --> 00:05:13.594
so, this is what g is doing

00:05:13.594 --> 00:05:18.180
so, the function g maps
from x to some value, g of x

00:05:19.520 --> 00:05:23.905
and then if you'd apply f to
this value right over here

00:05:23.905 --> 00:05:27.208
if you apply f to this value, the g of x,

00:05:27.208 --> 00:05:30.128
you get all the way back to x.

00:05:30.128 --> 00:05:33.552
So, that is f of g of x.

00:05:33.552 --> 00:05:35.254
And vice versa.

00:05:35.254 --> 00:05:39.631
If you start with x
and apply f of x first,

00:05:39.631 --> 00:05:42.896
so, if you start with f,
if you apply f of x first,

00:05:42.896 --> 00:05:44.252
let me do that,

00:05:44.252 --> 00:05:47.488
so, if you apply f of x first,
you see you get to this value

00:05:47.488 --> 00:05:51.643
so, that is f of x, so
you applied the function f

00:05:51.643 --> 00:05:54.460
when you apply the function g to that,

00:05:54.460 --> 00:05:57.727
you apply the function
g to that, you get back.

00:05:57.727 --> 00:06:02.405
So this g of f of x,

00:06:02.405 --> 00:06:06.393
I should say, or g of f,

00:06:06.393 --> 00:06:10.665
we're applying the function
g to the value f of x

00:06:10.665 --> 00:06:14.212
and so, since we get a
round-trip either way, we know

00:06:14.212 --> 00:06:17.912
that the functions g and f
are inverses of each other

00:06:17.912 --> 00:06:20.598
in fact, we can write that

00:06:20.598 --> 00:06:25.251
f of x is equal to the inverse of g of x,

00:06:26.131 --> 00:06:29.096
inverse of g of x, and vice versa,

00:06:29.096 --> 00:06:33.645
g of x is equal to the inverse of f of x

00:06:33.645 --> 00:06:37.326
inverse of f of x.

00:06:37.326 --> 00:06:38.976
Hope you enjoyed that.


Evaluating composite functions: using graphs

WEBVTT
Kind: captions
Language: en

00:00:00.380 --> 00:00:02.516
- So we have the graphs
of two functions here.

00:00:02.516 --> 00:00:04.629
We have the graph y equals f of x

00:00:04.629 --> 00:00:07.554
and we have the graph
y is equal to g of x.

00:00:07.554 --> 00:00:09.482
And what I wanna do in this video

00:00:09.482 --> 00:00:14.482
is evaluate what g of, f of,

00:00:16.799 --> 00:00:19.187
let me do the f of it another color,

00:00:19.187 --> 00:00:24.088
f of negative five is,
f of negative five is.

00:00:24.088 --> 00:00:26.082
And it can sometimes
seem a little daunting

00:00:26.082 --> 00:00:28.010
when you see these composite functions.

00:00:28.010 --> 00:00:32.747
You're evaluating the function
g at f of negative five.

00:00:32.747 --> 00:00:33.884
What does all this mean?

00:00:33.884 --> 00:00:35.045
We just have to remind ourselves

00:00:35.045 --> 00:00:36.160
what functions are all about.

00:00:36.160 --> 00:00:38.505
They take an input and
they give you an output.

00:00:38.505 --> 00:00:39.921
So really, what we're doing is

00:00:39.921 --> 00:00:41.129
we're going to take,

00:00:41.129 --> 00:00:43.219
we have the function f.

00:00:43.219 --> 00:00:45.843
We have the function f.

00:00:45.843 --> 00:00:48.838
We're going to input negative
five into that function.

00:00:48.838 --> 00:00:51.089
We're going to input negative five

00:00:51.089 --> 00:00:53.876
into that function and
it's going to output

00:00:53.876 --> 00:00:55.595
f of negative five.

00:00:55.595 --> 00:00:58.961
It's going to output f of negative five

00:00:58.961 --> 00:01:00.354
and we can figure what that is.

00:01:00.354 --> 00:01:01.747
And then that's going to be the input

00:01:01.747 --> 00:01:03.883
into the function g.

00:01:03.883 --> 00:01:08.040
So that's going to be the
input into the function g

00:01:08.040 --> 00:01:10.014
and so we're going to,

00:01:10.014 --> 00:01:13.913
and then the output is going
to be g of f of negative five,

00:01:13.913 --> 00:01:18.581
g of f of negative five.

00:01:18.581 --> 00:01:19.718
Let's just do it step by step.

00:01:19.718 --> 00:01:21.646
So the first thing we wanna figure out is

00:01:21.646 --> 00:01:25.245
what is the function f when
x is equal to negative five?

00:01:25.245 --> 00:01:27.381
What is f of negative five?

00:01:27.381 --> 00:01:31.816
Well we just have to see when
x is equal to negative five.

00:01:31.816 --> 00:01:33.279
When x is equal to negative five,

00:01:33.279 --> 00:01:35.739
the function is right over here.

00:01:35.739 --> 00:01:38.293
Let's see, let me see if I
can draw a straight line.

00:01:38.293 --> 00:01:41.475
So then x is equal to negative five.

00:01:41.475 --> 00:01:43.635
The function is right over here.

00:01:43.635 --> 00:01:48.580
It looks like f of negative
five is equal to negative two.

00:01:48.580 --> 00:01:51.320
It's equal to negative two.

00:01:51.320 --> 00:01:52.852
You see that right over there.

00:01:52.852 --> 00:01:57.852
So, f of negative five is negative two.

00:01:58.704 --> 00:02:00.119
And so we can now think of this

00:02:00.119 --> 00:02:02.489
instead of saying g of f of negative five,

00:02:02.489 --> 00:02:06.272
we could say well f of negative
five is just negative two,

00:02:06.272 --> 00:02:08.385
is just negative two.

00:02:08.385 --> 00:02:12.170
So this is going to be
equivalent to g of negative two,

00:02:12.170 --> 00:02:17.170
g of negative two, g of negative two.

00:02:17.790 --> 00:02:20.042
We're gonna take negative two into g

00:02:20.042 --> 00:02:23.548
and we're gonna output g of negative two.

00:02:23.548 --> 00:02:25.172
So we're taking that output, negative two

00:02:25.172 --> 00:02:26.451
and we're inputting into g.

00:02:26.451 --> 00:02:28.796
So when x is negative two,

00:02:28.796 --> 00:02:32.556
when x is negative two, what is g?

00:02:32.556 --> 00:02:35.761
So we see, when x is negative two,

00:02:35.761 --> 00:02:38.223
g, the graph is right over there,

00:02:38.223 --> 00:02:42.587
g of negative two is one.

00:02:42.587 --> 00:02:44.956
So this is going to be one.

00:02:44.956 --> 00:02:49.552
So g of f of negative five
sounds really complicated,

00:02:49.552 --> 00:02:50.922
we were able to figure out is one

00:02:50.922 --> 00:02:52.967
'cause you input negative five into f,

00:02:52.967 --> 00:02:54.521
it outputs negative two.

00:02:54.521 --> 00:02:58.261
And then you input negative
two into g, it outputs one

00:02:58.261 --> 00:03:00.089
and we're all done.


Evaluating composite functions: using tables

WEBVTT
Kind: captions
Language: en

00:00:00.516 --> 00:00:02.223
- [Voiceover] So we have some tables here

00:00:02.223 --> 00:00:06.773
that give us what the
functions f and g are

00:00:06.773 --> 00:00:08.735
when you give it certain inputs.

00:00:08.735 --> 00:00:11.000
So, when you input negative four,

00:00:11.000 --> 00:00:13.797
f of negative four is 29.

00:00:13.797 --> 00:00:15.876
That's going to be the
output of that function.

00:00:15.876 --> 00:00:18.476
So we have that for both f and g,

00:00:18.476 --> 00:00:21.402
and what I want to do is
evaluate two composite functions.

00:00:21.402 --> 00:00:25.499
I want to evaluate f of g of zero,

00:00:26.169 --> 00:00:30.540
and I want to evaluate g of f of zero.

00:00:30.540 --> 00:00:32.398
So like always, pause the video

00:00:32.398 --> 00:00:34.731
and see if you can figure it out.

00:00:35.241 --> 00:00:38.369
Let's first think about f of g of zero.

00:00:38.449 --> 00:00:41.556
F of g of zero.

00:00:41.556 --> 00:00:42.670
What is this all about?

00:00:42.670 --> 00:00:45.712
Actually let me use multiple colors here.

00:00:45.712 --> 00:00:49.370
F of g of zero.

00:00:50.020 --> 00:00:51.797
Well, this means that we're going to

00:00:51.797 --> 00:00:54.347
evaluate g at zero,

00:00:54.347 --> 00:00:57.277
so we're gonna input zero into g.

00:00:58.497 --> 00:00:59.467
Do it in that.

00:00:59.467 --> 00:01:03.304
So we're gonna input
zero into our function g,

00:01:03.304 --> 00:01:05.300
and we're going to output,

00:01:05.300 --> 00:01:07.681
whatever we output is
going to be g of zero.

00:01:07.681 --> 00:01:09.202
I'll write it right over here,

00:01:09.202 --> 00:01:12.139
and then we're going to input
that into our function f.

00:01:12.139 --> 00:01:15.378
We're going to input
that into our function f,

00:01:15.378 --> 00:01:17.177
and whatever I output then is going to be

00:01:17.177 --> 00:01:18.942
f of g of zero.

00:01:18.942 --> 00:01:21.829
F of g of zero.

00:01:22.731 --> 00:01:26.325
F of g of zero.

00:01:26.325 --> 00:01:27.509
I wrote these small here so we have space

00:01:27.509 --> 00:01:28.914
for the actual values.

00:01:28.914 --> 00:01:30.446
So first let's just evaluate,

00:01:30.446 --> 00:01:32.595
and if you are now inspired,

00:01:32.595 --> 00:01:35.194
pause the video again and
see if you can solve it.

00:01:35.194 --> 00:01:36.309
Although, if you solved it the first time,

00:01:36.309 --> 00:01:37.527
you don't have to do that now.

00:01:37.527 --> 00:01:39.653
What's g of zero?

00:01:39.653 --> 00:01:43.203
Well, when we input x equals zero,

00:01:43.203 --> 00:01:46.931
we get g of zero is equal to five.

00:01:47.291 --> 00:01:49.334
So g of zero is five.

00:01:49.334 --> 00:01:51.331
So that is five.

00:01:51.331 --> 00:01:53.246
So we're now going to input five

00:01:53.246 --> 00:01:54.650
into our function f.

00:01:54.650 --> 00:01:56.926
We're essentially going
to evaluate f of five.

00:01:56.926 --> 00:01:59.747
So when you input five into our function.

00:02:00.277 --> 00:02:02.725
I'm gonna do it in this brown color.

00:02:02.725 --> 00:02:05.417
When you input x equals five into f,

00:02:05.417 --> 00:02:09.341
you get the function f
of five is equal to 11.

00:02:09.341 --> 00:02:11.838
So this is going to be 11.

00:02:11.838 --> 00:02:15.541
So, f of g of zero is equal to 11.

00:02:16.081 --> 00:02:19.100
Now, let's do g of f of zero.

00:02:19.580 --> 00:02:21.269
So now let's evaluate.

00:02:21.269 --> 00:02:23.138
I'll do this is different colors.

00:02:23.138 --> 00:02:26.238
G, maybe I'll use those
same two colors actually.

00:02:26.238 --> 00:02:30.127
So now we're going to
evaluate g of f of zero.

00:02:31.667 --> 00:02:34.789
G of f of zero, and the key realization is

00:02:34.789 --> 00:02:36.437
you wanna go within the parenthesis.

00:02:36.437 --> 00:02:39.317
Evaluate that first so
then you can evaluate

00:02:39.317 --> 00:02:42.370
the function that's
kind of on the outside.

00:02:42.370 --> 00:02:45.167
So here we're going to
take zero as an input

00:02:45.167 --> 00:02:47.071
into the function f,

00:02:47.071 --> 00:02:49.779
and then whatever that is, that f of zero,

00:02:49.779 --> 00:02:52.995
we're going to input into our function g.

00:02:52.995 --> 00:02:54.826
We're going to input into our function g,

00:02:54.826 --> 00:02:56.115
and what we're going to be,

00:02:56.115 --> 00:02:59.934
and then the output of that
is going to be g of f of zero.

00:03:00.554 --> 00:03:03.157
So, let's see, what is f of zero?

00:03:03.157 --> 00:03:06.965
You see over here when our input is zero,

00:03:06.965 --> 00:03:10.761
this table tells us that
f of zero is equal to one.

00:03:11.101 --> 00:03:13.526
So f of zero is equal to one.

00:03:13.526 --> 00:03:15.462
F of zero is equal to one.

00:03:15.462 --> 00:03:17.576
So now we use one as an input into g.

00:03:17.576 --> 00:03:20.141
We're now evaluating g of one,

00:03:20.141 --> 00:03:21.696
or I can just write this.

00:03:21.696 --> 00:03:26.496
This is the same thing as g of one.

00:03:27.636 --> 00:03:29.249
G of one.

00:03:29.249 --> 00:03:30.294
Once again, why was that?

00:03:30.294 --> 00:03:33.035
'Cause f of zero is equal to,

00:03:33.035 --> 00:03:35.031
f of zero is equal to one.

00:03:35.031 --> 00:03:36.315
And let me, I wrote those parenthesis

00:03:36.315 --> 00:03:37.683
too far away from the g.

00:03:37.683 --> 00:03:40.122
This is the same thing as

00:03:40.122 --> 00:03:43.057
g of one.

00:03:43.057 --> 00:03:45.260
Because once again f of zero is one.

00:03:45.260 --> 00:03:46.643
Now what is g of one?

00:03:46.643 --> 00:03:51.077
Well, when I input one
into our function g,

00:03:51.897 --> 00:03:55.615
I get g of one is equal to eight.

00:03:55.615 --> 00:03:57.008
So this is going to be equal,

00:03:57.008 --> 00:03:58.993
this is equal to eight,

00:03:58.993 --> 00:03:59.789
and we're done.

00:03:59.789 --> 00:04:01.078
And notice these are different values,

00:04:01.078 --> 00:04:02.729
because these are different
composite functions.

00:04:02.729 --> 00:04:07.456
F of g of zero is 11, and
g of f of zero is eight.


Manipulating expressions using structure (example 2)

WEBVTT
Kind: captions
Language: en

00:00:00.551 --> 00:00:01.329
- [Voiceover] We're told,

00:00:01.329 --> 00:00:05.412
suppose a+b=2a.

00:00:05.412 --> 00:00:10.320
Which of these expressions equals b-a?

00:00:10.326 --> 00:00:12.078
Alright, I encourage
you to pause the video

00:00:12.078 --> 00:00:13.514
and see if you can figure that out.

00:00:13.514 --> 00:00:14.888
Which of these expressions

00:00:14.888 --> 00:00:17.618
would be equal to b-a

00:00:17.618 --> 00:00:21.453
and it's going to just involve
some algebraic manipulation.

00:00:22.000 --> 00:00:23.548
Alright, let's work through this together.

00:00:23.548 --> 00:00:25.296
So we are told,

00:00:25.382 --> 00:00:30.322
we are told that a+b=2a

00:00:30.968 --> 00:00:32.475
so the first thing I would want to do is

00:00:32.475 --> 00:00:34.943
get all my a's in one place

00:00:34.983 --> 00:00:36.353
and one way I could do that is

00:00:36.353 --> 00:00:39.122
I could subract a from both sides.

00:00:39.369 --> 00:00:43.684
So if I subtract a from both sides

00:00:43.736 --> 00:00:47.438
I'm going to be left with
just a b on the left-hand side

00:00:47.438 --> 00:00:49.192
and on the right-hand side
I'm going to be left with

00:00:49.192 --> 00:00:51.793
2a-a, well that's just going to be a.

00:00:51.793 --> 00:00:53.755
If I have two of something
and I subtract one of them,

00:00:53.755 --> 00:00:54.997
take away one of them I'm going to have

00:00:54.997 --> 00:00:57.546
just one of those somethings equal to 1a.

00:00:57.980 --> 00:01:00.285
So we want to figure out what b-a is.

00:01:00.627 --> 00:01:02.287
Well luckily I can figure that out

00:01:02.287 --> 00:01:04.464
if I subtract a from both sides.

00:01:04.679 --> 00:01:08.759
So if I subtract a from both sides

00:01:09.079 --> 00:01:11.250
well then I'm going to
get on the left-hand side

00:01:11.250 --> 00:01:14.228
b-a, which is what we want to figure out,

00:01:14.279 --> 00:01:17.304
is equal to a-a=0.

00:01:17.518 --> 00:01:22.518
So b-a=0 which is not one of the choices.

00:01:25.796 --> 00:01:29.336
Alright, so let's see if we can figure out

00:01:29.336 --> 00:01:32.076
some other things over here.

00:01:32.076 --> 00:01:37.076
So b-a=0 but that is
not one of the choices.

00:01:39.065 --> 00:01:40.736
Alright, so let's see is there

00:01:40.736 --> 00:01:43.627
any other way to manipulate this?

00:01:43.627 --> 00:01:46.631
No, b minu- I could just go straight ahead

00:01:46.631 --> 00:01:48.325
and subtract 2a from both sides

00:01:48.325 --> 00:01:50.805
and I would get b-a=0.

00:01:51.368 --> 00:01:53.565
Oh, this is interesting,
this is a tricky one.

00:01:53.921 --> 00:01:56.772
So b-a=0.

00:01:57.237 --> 00:02:00.318
Well, if b-a=0 then if we take

00:02:00.318 --> 00:02:02.530
the negative of both sides of this,

00:02:03.093 --> 00:02:04.973
if we take the negative of both sides,

00:02:04.973 --> 00:02:07.394
if we multiply both sides by -1.

00:02:08.003 --> 00:02:12.263
So x, I should write the times,
I should write like this,

00:02:12.263 --> 00:02:13.285
alright, because we don't want to

00:02:13.285 --> 00:02:14.608
confuse it with the variable x,

00:02:14.608 --> 00:02:16.663
so if we multiply both sides by -1

00:02:16.663 --> 00:02:17.511
what do we get?

00:02:17.511 --> 00:02:20.250
Well on the left-hand side we get a-b

00:02:20.250 --> 00:02:23.678
and on the right-hand side we still get 0.

00:02:23.695 --> 00:02:25.788
If b-a=0 then the negative of it,

00:02:25.788 --> 00:02:28.589
which is a-b is also
going to be equal to 0.

00:02:28.850 --> 00:02:30.021
And that's this choice.

00:02:30.844 --> 00:02:33.707
That is, let me do that
in a little darker color,

00:02:33.808 --> 00:02:36.833
that is this choice right over there.

00:02:37.047 --> 00:02:38.493
That was a good one.


Manipulating expressions using structure

WEBVTT
Kind: captions
Language: en

00:00:00.283 --> 00:00:02.384
- [Voiceover] So, we're
sold that suppose a plus b

00:00:02.384 --> 00:00:03.661
is equal to zero.

00:00:03.661 --> 00:00:06.253
Which of these expressions
equal a times b?

00:00:06.253 --> 00:00:07.418
And like always, pause the video,

00:00:07.418 --> 00:00:08.707
and see if you can figure it out.

00:00:08.707 --> 00:00:11.263
These are actually pretty fun problems.

00:00:11.882 --> 00:00:12.951
Alright, so, let's see if we can

00:00:12.951 --> 00:00:14.310
do a little bit of manipulation.

00:00:14.310 --> 00:00:18.001
So, we're told that a
plus b is equal to zero.

00:00:18.552 --> 00:00:21.705
So, if we subtract b from both sides,

00:00:21.705 --> 00:00:23.262
we would get.

00:00:23.262 --> 00:00:26.082
So, if I subtract b from both sides,

00:00:26.082 --> 00:00:29.809
I would get a is equal to negative b.

00:00:29.809 --> 00:00:32.520
So, what is a time b equal to?

00:00:32.520 --> 00:00:34.181
So, a times b.

00:00:34.181 --> 00:00:35.655
Well, I could write this a few ways.

00:00:35.655 --> 00:00:39.452
I could substitute the
a with a negative b,

00:00:39.452 --> 00:00:41.980
since we know a is equal to negative b.

00:00:43.410 --> 00:00:48.030
In which case, a times b
would be negative b times b,

00:00:48.030 --> 00:00:50.846
which is equal to negative b squared.

00:00:50.846 --> 00:00:53.668
Another option is I could

00:00:54.645 --> 00:00:56.642
instead of saying a is
equal to negative b,

00:00:56.642 --> 00:01:00.044
I could say that b is equal to negative a

00:01:00.044 --> 00:01:02.945
if we multiply both
sides times negative one.

00:01:02.945 --> 00:01:05.199
So, b is equal to negative a.

00:01:05.199 --> 00:01:08.867
And so, instead of substituting
a, I could substitute b

00:01:08.867 --> 00:01:11.537
with negative a.

00:01:11.537 --> 00:01:14.347
And so, this expression
would be a times negative a,

00:01:15.419 --> 00:01:17.974
a times negative a,

00:01:17.974 --> 00:01:20.575
which is equal to negative a squared.

00:01:20.575 --> 00:01:23.036
So, let's see, which of
these choices are there?

00:01:23.036 --> 00:01:25.473
There's a b squared, by
not negative b squared.

00:01:25.473 --> 00:01:27.678
So, we see the negative a
squared right over there.

00:01:27.678 --> 00:01:29.073
So, I'm feeling good.

00:01:29.073 --> 00:01:31.969
I'm feeling good about that choice.


Analyzing graphs of exponential functions: negative initial value

WEBVTT
Kind: captions
Language: en

00:00:00.550 --> 00:00:02.644
- [Voiceover] So we have a
graph here of the function

00:00:02.644 --> 00:00:05.869
f of x, and I'm telling
you right now that f of x

00:00:05.869 --> 00:00:08.031
is going to be an exponential function.

00:00:08.031 --> 00:00:09.852
It looks like one, but
it's even nicer when

00:00:09.852 --> 00:00:11.522
someone tells you that.

00:00:11.522 --> 00:00:16.016
And our goal in this video is
to figure out at what x value,

00:00:16.295 --> 00:00:21.017
so when does f of x?

00:00:22.468 --> 00:00:24.385
At what x value is f of x

00:00:24.385 --> 00:00:27.416
going to be equal to negative 1/25.

00:00:27.820 --> 00:00:30.026
And you might be tempted to
just eyeball it over here,

00:00:30.026 --> 00:00:34.294
but when f of x is
negative 1/25, that's like

00:00:34.622 --> 00:00:36.411
right below the x-axis.

00:00:37.541 --> 00:00:41.315
If I try to eyeball it, it
would be very difficult.

00:00:41.669 --> 00:00:44.245
It's very difficult to
tell what value that is.

00:00:44.245 --> 00:00:46.573
It might be at three, it might be at four.

00:00:46.938 --> 00:00:48.632
I am not sure.

00:00:50.842 --> 00:00:53.633
Well, I don't want to just
eyeball it, just guess it.

00:00:53.709 --> 00:00:55.872
Instead, I'm gonna
actually find an expression

00:00:55.872 --> 00:00:57.755
that defines f of x because

00:00:57.755 --> 00:00:59.351
they've given us some information here.

00:00:59.351 --> 00:01:02.864
And then I can just solve
for x, so let's do that.

00:01:03.388 --> 00:01:05.753
Well, since we know that f of
x is an exponential function,

00:01:05.753 --> 00:01:07.194
we know it's going to take the form

00:01:07.194 --> 00:01:11.047
f of x is equal to our initial value a

00:01:11.586 --> 00:01:16.529
times our common ratio r to the xth power.

00:01:17.319 --> 00:01:19.359
Well, the initial value
is straightforward enough.

00:01:19.359 --> 00:01:21.849
That's going to be the value
that the function takes on

00:01:21.849 --> 00:01:23.919
when x is equal to zero.

00:01:23.919 --> 00:01:25.734
And you can even see it
here, if x is equal to zero,

00:01:25.734 --> 00:01:27.574
the r to the x would just be one.

00:01:27.574 --> 00:01:30.357
And so f of zero will just be equal to a.

00:01:30.579 --> 00:01:32.790
And so what is f of zero?

00:01:33.559 --> 00:01:35.631
Well, when x is equal to zero,

00:01:35.631 --> 00:01:37.654
which essentially we're saying,

00:01:38.017 --> 00:01:40.712
where does it intersect the y-axis?

00:01:40.712 --> 00:01:44.433
We see f of zero is negative 25.

00:01:45.079 --> 00:01:48.391
So a is going to be negative 25.

00:01:50.545 --> 00:01:53.265
When x is zero, the r
to the x is just one.

00:01:53.507 --> 00:01:55.880
So f of zero is going to be negative 25.

00:01:55.880 --> 00:01:57.805
We see that right over there.

00:01:57.922 --> 00:02:00.110
Now to figure out the common ratio,

00:02:00.110 --> 00:02:01.579
there's a couple of ways
you could think about it.

00:02:01.579 --> 00:02:03.756
The common ratio is the ratio between

00:02:06.736 --> 00:02:09.110
two successive values
that are separated by one.

00:02:09.110 --> 00:02:10.397
What do I mean by that?

00:02:10.397 --> 00:02:13.086
Well, you could view
it as the ratio between

00:02:13.086 --> 00:02:16.317
f of one and f of zero.

00:02:16.434 --> 00:02:18.145
That's going to be the common ratio.

00:02:18.145 --> 00:02:20.670
Or the ratio between f of two

00:02:20.670 --> 00:02:22.387
and f of one.

00:02:22.387 --> 00:02:24.393
That is going to be the common ratio.

00:02:24.715 --> 00:02:29.247
Well, lucky for us, we know
f of zero is negative 25.

00:02:29.649 --> 00:02:34.496
And we know that f of one,
when x is equal to one,

00:02:35.100 --> 00:02:38.241
y, or f of x, or f of one
is equal to negative five.

00:02:40.325 --> 00:02:41.990
And so just like that
we're able to figure out

00:02:41.990 --> 00:02:46.294
that our common ratio r is
negative five over negative 25,

00:02:46.535 --> 00:02:49.810
which is the same thing as 1/5.

00:02:49.977 --> 00:02:51.597
Divide a negative by a
negative, you get a positive.

00:02:51.597 --> 00:02:54.495
So you get a five over 25, which is 1/5.

00:02:58.230 --> 00:03:01.012
We can write an expression
that defines f of x.

00:03:01.232 --> 00:03:03.560
f of x is going to be equal to

00:03:04.043 --> 00:03:07.227
negative 25 times

00:03:09.459 --> 00:03:13.135
1/5 to the x power.

00:03:13.741 --> 00:03:15.323
And so let's go back to our question.

00:03:15.323 --> 00:03:19.543
When is this going to be
equal to negative 1/25?

00:03:20.032 --> 00:03:21.789
So when is this equal to negative 1/25?

00:03:21.789 --> 00:03:23.712
Well, let's just set
them equal to each other.

00:03:24.419 --> 00:03:25.908
There's a siren outside.

00:03:25.908 --> 00:03:27.336
I don't know if you hear it.

00:03:28.657 --> 00:03:30.363
I'll power through.

00:03:31.643 --> 00:03:34.790
So let's see at what x
value does this expression

00:03:34.790 --> 00:03:37.325
equal negative 1/25?

00:03:38.055 --> 00:03:40.637
Let's see, we can multiply.

00:03:41.159 --> 00:03:43.032
Well, actually, we want to solve for x.

00:03:43.032 --> 00:03:47.501
So let's see, let's divide
both sides by negative 25.

00:03:47.957 --> 00:03:51.679
And so we are going to get 1/5

00:03:51.679 --> 00:03:53.879
to the xth power

00:03:54.002 --> 00:03:57.632
is equal to, if we divide
both sides by negative 25,

00:03:57.632 --> 00:03:59.530
this negative 25 is gonna go away.

00:03:59.530 --> 00:04:01.332
And on the right-hand
side, we're going to have,

00:04:01.332 --> 00:04:02.902
divide a negative by a negative
is going to be positive.

00:04:02.902 --> 00:04:05.399
It's going to be 1/625.

00:04:10.954 --> 00:04:12.783
And 1/5 to the xth power.

00:04:12.783 --> 00:04:15.851
This is the same thing
as one to the xth power

00:04:16.898 --> 00:04:21.068
over five to the xth power is equal to

00:04:22.938 --> 00:04:25.407
1/625.

00:04:28.816 --> 00:04:31.278
Well, one to the xth power

00:04:34.428 --> 00:04:36.116
is just going to be equal to one.

00:04:37.273 --> 00:04:38.856
It doesn't matter that we have this

00:04:38.856 --> 00:04:40.669
to the xth power over here.

00:04:41.649 --> 00:04:43.783
I thought I was erasing
that with a black color.

00:04:43.783 --> 00:04:45.988
There you go, that's a black
color right over there.

00:04:46.185 --> 00:04:48.141
So we can see that five to the xth power

00:04:48.141 --> 00:04:50.355
needs to be equal to 625.

00:04:51.535 --> 00:04:53.062
So let me write that over here.

00:04:53.562 --> 00:04:55.289
Whoops, didn't change my color.

00:04:55.408 --> 00:04:58.836
Five to the xth power needs to be equal to

00:04:59.238 --> 00:05:03.156
625.

00:05:03.515 --> 00:05:05.599
Now the best way I could
think of doing this

00:05:05.599 --> 00:05:07.685
is let's just think
about our powers of five.

00:05:08.129 --> 00:05:10.665
So five to the first power is five.

00:05:10.903 --> 00:05:13.401
Five squared is 25.

00:05:13.803 --> 00:05:17.359
Five to the third is 125.

00:05:17.600 --> 00:05:20.575
Five to the fourth, we'll
multiply that by five,

00:05:20.575 --> 00:05:23.038
you're going to get 625.

00:05:23.555 --> 00:05:26.135
So x is going to be 4

00:05:27.722 --> 00:05:30.509
cause five to the fourth power is 625.

00:05:30.584 --> 00:05:33.802
So we can now say that f of four

00:05:36.964 --> 00:05:38.974
is equal to

00:05:40.471 --> 00:05:42.996
negative 1/25.

00:05:46.132 --> 00:05:47.978
And once again, you can verify that.

00:05:48.138 --> 00:05:50.224
You can verify that right over here.

00:05:51.984 --> 00:05:56.057
1/5 to the fourth power is gonna be 1/625.

00:05:57.193 --> 00:06:02.017
Negative 25 over positive 625
is going to be negative 1/25.

00:06:02.574 --> 00:06:05.412
So hopefully that clears
things up a little bit.


Analyzing graphs of exponential functions

WEBVTT
Kind: captions
Language: en

00:00:00.480 --> 00:00:01.779
- [Voiceover] So we have the graph of an

00:00:01.779 --> 00:00:06.779
exponential function here,
and the function is m of x.

00:00:06.829 --> 00:00:08.950
And what I want to do
is figure out what is

00:00:08.950 --> 00:00:12.990
m of six going to be equal to?

00:00:12.990 --> 00:00:14.930
And like always, pause the video,

00:00:14.930 --> 00:00:17.090
and see if you can work it out.

00:00:17.090 --> 00:00:19.710
Well, as I mentioned, this
is an exponential function,

00:00:19.710 --> 00:00:22.030
so m is going to take the form.

00:00:22.030 --> 00:00:22.880
Let me write it this way.

00:00:22.880 --> 00:00:26.069
M of x is going to take the form

00:00:26.069 --> 00:00:30.409
a times r to the xth power,

00:00:30.409 --> 00:00:34.969
where a is our initial power,
and r is our common ratio.

00:00:34.969 --> 00:00:37.209
Well the initial value is
pretty straightforward.

00:00:37.209 --> 00:00:39.348
It's just going to be what m of zero is.

00:00:39.348 --> 00:00:42.668
So a is going to be equal to m of zero,

00:00:42.668 --> 00:00:44.149
and we can just look at this graph.

00:00:44.149 --> 00:00:48.369
When x is equal to zero, the
function is equal to nine.

00:00:48.369 --> 00:00:51.909
So it's equal to nine, and now we need to

00:00:51.909 --> 00:00:53.720
figure out our common ratio.

00:00:53.720 --> 00:00:55.720
So let me set up a little
bit of a table here,

00:00:55.720 --> 00:00:57.320
just to help us with this.

00:00:57.320 --> 00:01:01.900
So let me draw some straight lines.

00:01:01.900 --> 00:01:06.420
And so this is x and m of x.

00:01:06.420 --> 00:01:08.220
We already know that when x is zero,

00:01:08.220 --> 00:01:10.720
m of x is equal to nine.

00:01:10.720 --> 00:01:14.680
We also know when x is, let's see.

00:01:14.680 --> 00:01:19.680
When x is one, when x
is one, m of x is three.

00:01:20.280 --> 00:01:22.340
M of x is three.

00:01:22.340 --> 00:01:25.860
So when we increase our x by one,

00:01:25.860 --> 00:01:28.260
what happened to our m of x?

00:01:28.260 --> 00:01:30.340
Well, what did we have to multiply it by?

00:01:30.340 --> 00:01:32.120
Well, to go from nine to three,

00:01:32.120 --> 00:01:34.960
you multiplied by 1/3.

00:01:34.960 --> 00:01:36.780
So that's going to be our common ratio.

00:01:36.780 --> 00:01:37.787
And in fact, if we wanted to care what

00:01:37.787 --> 00:01:42.787
m of two is going to be, we
would multiply by 1/3 again.

00:01:43.977 --> 00:01:47.127
And m of two should be equal to one,

00:01:47.127 --> 00:01:48.967
and we see that right over here.

00:01:48.967 --> 00:01:51.647
M of two is, indeed, equal to one.

00:01:51.647 --> 00:01:54.877
So our common ratio, our common ratio

00:01:54.877 --> 00:01:57.797
right over here is equal to 1/3.

00:01:57.797 --> 00:02:01.437
So m of x, we can write it as, m of x

00:02:01.437 --> 00:02:06.437
is going to be equal to
our initial value, a,

00:02:06.577 --> 00:02:11.237
which we already figured it out, as a is,

00:02:11.237 --> 00:02:13.815
a is equal to nine, so
it's going to be nine,

00:02:13.815 --> 00:02:18.395
times our common ratio,
times our common ratio,

00:02:18.395 --> 00:02:21.495
1/3 to the xth power.

00:02:21.495 --> 00:02:25.535
So I was able to figure out
the formula for our definition

00:02:25.535 --> 00:02:28.355
for m of x, but that's not what I wanted.

00:02:28.355 --> 00:02:30.835
I just wanted to figure out
what m of six is going to be.

00:02:30.835 --> 00:02:33.975
So we can write down that m of six,

00:02:33.975 --> 00:02:38.095
m of six is going to be nine times

00:02:38.095 --> 00:02:41.975
one over three to the sixth power.

00:02:41.975 --> 00:02:45.055
Let's see, that is going to be equal to,

00:02:45.055 --> 00:02:47.135
that's the same thing as nine times,

00:02:47.135 --> 00:02:49.215
well one to the six is just one.

00:02:49.215 --> 00:02:50.695
That's just going to be one to the six,

00:02:50.695 --> 00:02:54.015
which is just one, over
three to the sixth power.

00:02:54.015 --> 00:02:56.335
Now what is three to the sixth power?

00:02:56.335 --> 00:02:58.475
In fact, I can even simplify
this a little bit more.

00:02:58.475 --> 00:03:01.735
I can recognize that
nine is three squared,

00:03:01.735 --> 00:03:03.512
so I could say this is going to be

00:03:03.512 --> 00:03:05.412
three squared over three to the sixth.

00:03:05.412 --> 00:03:09.585
Three squared over three to the sixth,

00:03:09.585 --> 00:03:12.085
and then, I could tackle
this a couple of ways.

00:03:12.085 --> 00:03:13.884
I could just divide the
numerator into the denominator

00:03:13.884 --> 00:03:16.164
by three squared, in
which case, I would get

00:03:16.164 --> 00:03:20.225
one over three to the fourth power.

00:03:20.225 --> 00:03:21.904
Or another way to think
about it, this should

00:03:21.904 --> 00:03:25.220
be the same thing as three
to the two minus sixth power,

00:03:25.220 --> 00:03:29.020
which is the same thing as three
to the negative four power,

00:03:29.020 --> 00:03:30.340
which is, of course, the same thing as

00:03:30.340 --> 00:03:31.840
one over three to the fourth.

00:03:31.840 --> 00:03:33.580
So what's three to the fourth?

00:03:33.580 --> 00:03:34.960
So three squared is nine.

00:03:34.960 --> 00:03:36.679
Three to the third is 27.

00:03:36.679 --> 00:03:38.600
Three to the fourth is 81.

00:03:38.600 --> 00:03:42.120
So this is going to be
equal to one over 81.

00:03:42.120 --> 00:03:44.740
M of six is equal to one over 81.

00:03:44.740 --> 00:03:46.040
We could also, we could've done that

00:03:46.040 --> 00:03:47.830
if we kept going by our table.

00:03:47.830 --> 00:03:50.270
M of three, multiply it by 1/3,

00:03:50.270 --> 00:03:52.650
is going to be 1/3.

00:03:52.650 --> 00:03:55.190
M of four, multiply by 1/3 again,

00:03:55.190 --> 00:03:57.510
is going to be 1/9.

00:03:57.510 --> 00:03:59.071
Then we could, m of five is going to be

00:03:59.071 --> 00:04:03.883
1/27, and m of six is going to be 181st.


Analyzing tables of exponential functions

WEBVTT
Kind: captions
Language: en

00:00:00.330 --> 00:00:01.886
- [Voiceover] Let's say
that we have an exponential

00:00:01.886 --> 00:00:03.116
function, h of n,

00:00:03.116 --> 00:00:04.625
and since it's an exponential function

00:00:04.625 --> 00:00:07.830
it's going to be in the
form a times r to the n,

00:00:07.830 --> 00:00:09.339
where a is our initial value

00:00:09.339 --> 00:00:10.648
and r is our common ratio,

00:00:10.648 --> 00:00:13.147
and we're going to assume
that r is greater than zero.

00:00:13.147 --> 00:00:15.631
And they've given us some
information on h of n.

00:00:15.631 --> 00:00:17.373
We know that when n is equal to 2,

00:00:17.373 --> 00:00:19.393
h of 2 is 144,

00:00:19.393 --> 00:00:21.599
that h of 4 is 324,

00:00:21.599 --> 00:00:24.710
that h of 6 is 729.

00:00:24.710 --> 00:00:26.498
So, based on the information here,

00:00:26.498 --> 00:00:27.984
let's see if we can actually figure out

00:00:27.984 --> 00:00:31.142
what a and r are going to be.

00:00:31.142 --> 00:00:34.833
And like always, pause the
video and try to give it a go.

00:00:34.833 --> 00:00:36.737
All right, now let's do this together.

00:00:36.737 --> 00:00:39.384
So I'm going to focus on r first.

00:00:39.384 --> 00:00:41.706
The common ratio

00:00:41.706 --> 00:00:44.910
and if we had successive n's.

00:00:44.910 --> 00:00:47.836
If we had h of 3, then
we could just find the

00:00:47.836 --> 00:00:50.460
ratio between h of 3 and h of 2

00:00:50.460 --> 00:00:52.294
and r would just come out of that,

00:00:52.294 --> 00:00:55.150
or if the ratio between h of 4 and h of 3,

00:00:55.150 --> 00:00:56.938
we could solve explicitly for r,

00:00:56.938 --> 00:00:58.517
but we can get pretty close to that.

00:00:58.517 --> 00:01:00.630
We can just find the ratio between h of 4

00:01:00.630 --> 00:01:01.907
and h of 2.

00:01:01.907 --> 00:01:04.229
So, h of 4,

00:01:04.229 --> 00:01:05.854
the ratio between h of 4

00:01:05.854 --> 00:01:07.595
and h of 2

00:01:07.595 --> 00:01:09.615
is going to be equal to,

00:01:09.615 --> 00:01:13.075
well, we know h of 4 is 324,

00:01:13.075 --> 00:01:15.977
and h of 2 is 144,

00:01:15.977 --> 00:01:18.578
and we could simplify this a little bit.

00:01:18.578 --> 00:01:20.691
Let's see, if we,

00:01:20.691 --> 00:01:23.245
if we simplify this, we would get,

00:01:23.245 --> 00:01:24.499
they're both divisible by 2.

00:01:24.499 --> 00:01:25.985
If we divide them both by 2,

00:01:25.985 --> 00:01:27.308
this one in the numerator,

00:01:27.308 --> 00:01:30.721
324 divided by 2 is 162.

00:01:30.721 --> 00:01:33.624
144 divided by 2 is 72.

00:01:33.624 --> 00:01:35.621
Let's see, we can divide by 2 again.

00:01:35.621 --> 00:01:38.430
81 over 36.

00:01:38.430 --> 00:01:39.289
Divide by 2,

00:01:39.289 --> 00:01:41.100
actually, no, we can't
divide by 2 anymore,

00:01:41.100 --> 00:01:42.865
but we can divide by 9 now,

00:01:42.865 --> 00:01:46.000
so 81 divided by 9 is 9,

00:01:46.000 --> 00:01:48.113
and 36 divided by 9 is 4.

00:01:48.113 --> 00:01:51.897
So this thing can be rewritten as 9/4,

00:01:51.897 --> 00:01:54.405
but we could also rewrite this ratio

00:01:54.405 --> 00:01:55.519
by using,

00:01:55.519 --> 00:01:58.700
using this form of an
exponential function,

00:01:58.700 --> 00:02:01.254
so we could also say that
this is going to be equal to,

00:02:01.254 --> 00:02:03.367
h of 4 is going to be a

00:02:03.367 --> 00:02:04.993
times r.

00:02:04.993 --> 00:02:07.616
Now n is 4, so r to the 4th power,

00:02:07.616 --> 00:02:11.842
and h of 2 is going to be
a times r to the 2nd power.

00:02:11.842 --> 00:02:15.070
A times r to the 2nd power,

00:02:15.070 --> 00:02:16.695
and this simplifies nicely,

00:02:16.695 --> 00:02:18.274
a divided by a,

00:02:18.274 --> 00:02:19.690
cancels to just 1,

00:02:19.690 --> 00:02:22.500
and r to the 4th divided by r-squared,

00:02:22.500 --> 00:02:24.613
well that's going to be
r to the 4 minus 2 power,

00:02:24.613 --> 00:02:25.402
or r,

00:02:25.402 --> 00:02:26.981
it's going to be r to the 2nd power,

00:02:26.981 --> 00:02:29.024
it's going to be r-squared,

00:02:29.024 --> 00:02:30.812
and so we have a nice
little equation set up.

00:02:30.812 --> 00:02:35.061
R-squared needs to be equal to 9/4.

00:02:35.061 --> 00:02:36.083
So let me write that down.

00:02:36.083 --> 00:02:38.382
R-squared is equal to 9/4,

00:02:38.382 --> 00:02:40.587
and r needs to be greater than zero,

00:02:40.587 --> 00:02:43.536
so we could just say r is
going to be the principal root

00:02:43.536 --> 00:02:45.719
of 9/4,

00:02:45.719 --> 00:02:48.528
which is equal to 3/2.

00:02:48.528 --> 00:02:51.315
So we were able to figure out r.

00:02:51.315 --> 00:02:53.056
So now how do we figure out,

00:02:53.056 --> 00:02:55.378
how do we figure out a?

00:02:55.378 --> 00:02:56.980
Well, there's a couple of ways to do it.

00:02:56.980 --> 00:03:00.533
You can think about a is
going to be what h of zero

00:03:00.533 --> 00:03:01.392
is equal to,

00:03:01.392 --> 00:03:02.274
so we could do,

00:03:02.274 --> 00:03:04.758
I guess we could call it a tabular method,

00:03:04.758 --> 00:03:06.709
where, let me set up

00:03:06.709 --> 00:03:08.984
a little table here.

00:03:08.984 --> 00:03:10.540
So, a little table,

00:03:10.540 --> 00:03:14.510
so if this is n and this is h of n,

00:03:14.510 --> 00:03:15.857
so n is zero,

00:03:15.857 --> 00:03:17.529
we don't know what h of zero is just yet,

00:03:17.529 --> 00:03:18.574
it's going to be a.

00:03:18.574 --> 00:03:22.149
We don't know what h of 1 is yet.

00:03:22.149 --> 00:03:26.352
We do know that h of 2 is 144.

00:03:26.352 --> 00:03:29.162
And we do know, since
the common ratio is 3/2,

00:03:29.162 --> 00:03:33.179
if we take h of 1 and
we multiply it by 3/2,

00:03:33.179 --> 00:03:35.175
we're going to get h of 2,

00:03:35.175 --> 00:03:38.240
and if we take h of zero
and multiply it by 3/2,

00:03:38.240 --> 00:03:40.701
we're going to get h of 1.

00:03:40.701 --> 00:03:45.136
So, h of 1 is going to
be 144 divided by 3/2.

00:03:45.136 --> 00:03:45.949
So let's write that down.

00:03:45.949 --> 00:03:49.246
H of 1 is equal to 144

00:03:49.246 --> 00:03:52.450
divided by 3/2,

00:03:52.450 --> 00:03:56.165
which is going to be 144,

00:03:56.165 --> 00:04:00.530
144 times 2 over 3,

00:04:00.530 --> 00:04:01.297
and let's see,

00:04:01.297 --> 00:04:05.639
144 divided by 3 is

00:04:05.639 --> 00:04:08.170
going to be equal to,

00:04:08.170 --> 00:04:09.330
is it,

00:04:09.330 --> 00:04:09.841
let's see,

00:04:09.841 --> 00:04:11.699
one 3 goes into,

00:04:11.699 --> 00:04:12.535
I'll just do this by hand,

00:04:12.535 --> 00:04:14.276
my brain doesn't work that well

00:04:14.276 --> 00:04:15.460
while I'm making videos.

00:04:15.460 --> 00:04:18.502
3 goes into 14 four times,

00:04:18.502 --> 00:04:19.756
4 times 3 is 12,

00:04:19.756 --> 00:04:20.731
subtract,

00:04:20.731 --> 00:04:22.147
so it's going to be,

00:04:22.147 --> 00:04:24.372
I see, it's going to be 48.

00:04:24.372 --> 00:04:26.113
3 goes into 24 eight times.

00:04:26.113 --> 00:04:27.274
8 times 3 is 24,

00:04:27.274 --> 00:04:28.899
you have no remainder,

00:04:28.899 --> 00:04:30.455
so,

00:04:30.455 --> 00:04:34.751
this is going to be 48 times 2,

00:04:34.751 --> 00:04:37.189
which is going to be equal to 96

00:04:37.189 --> 00:04:38.953
so this is going to be 96,

00:04:38.953 --> 00:04:40.602
and so if we want to figure out h of zero,

00:04:40.602 --> 00:04:42.831
we just divide by 3/2 again,

00:04:42.831 --> 00:04:46.778
so, h of zero is 96

00:04:46.778 --> 00:04:49.588
divided by 3/2

00:04:49.588 --> 00:04:54.046
which is equal to 96 times 2 over 3.

00:04:54.046 --> 00:04:56.507
96 divided by 3, let's see,

00:04:56.507 --> 00:04:57.900
it's going to be 32,

00:04:57.900 --> 00:05:01.290
so this is going to be 32

00:05:01.290 --> 00:05:03.124
times, did I do that right?

00:05:03.124 --> 00:05:04.680
Yup, 32 times 2

00:05:04.680 --> 00:05:07.513
which is equal to 64.

00:05:07.513 --> 00:05:08.975
64.

00:05:08.975 --> 00:05:10.740
And so, just like that,
we've figured out that a

00:05:10.740 --> 00:05:13.108
is equal to 64,

00:05:13.108 --> 00:05:15.059
and r,

00:05:15.059 --> 00:05:18.054
and r is equal to 3/2.

00:05:18.054 --> 00:05:20.097
So we can write h of n,

00:05:20.097 --> 00:05:24.648
we can say that h of n is equal to 64,

00:05:24.648 --> 00:05:28.061
is equal to 64

00:05:28.061 --> 00:05:30.982
times 3/2,

00:05:31.660 --> 00:05:33.657
times 3/2,

00:05:33.657 --> 00:05:34.934
our common ratio,

00:05:34.934 --> 00:05:37.117
to the nth power.

00:05:37.117 --> 00:05:38.858
Now, there's another way that
we could have tackled this

00:05:38.858 --> 00:05:40.484
instead of this tabular method.

00:05:40.484 --> 00:05:42.991
We could have just solved for a now,

00:05:42.991 --> 00:05:43.966
since we know r.

00:05:43.966 --> 00:05:45.777
We know, for example,

00:05:45.777 --> 00:05:49.237
we know for example that h of 2

00:05:49.237 --> 00:05:53.300
which is going to be equal to a times,

00:05:53.300 --> 00:05:55.088
we know what our common ratio is,

00:05:55.088 --> 00:05:56.969
it's 3/2

00:05:56.969 --> 00:05:59.361
to the 2nd power,

00:05:59.361 --> 00:06:03.424
is going to be equal to 144.

00:06:03.424 --> 00:06:07.603
And so, we could say a times 9 over 4

00:06:07.603 --> 00:06:10.297
is equal to 144,

00:06:10.297 --> 00:06:12.479
and so we can multiply both
sides times the reciprocal

00:06:12.479 --> 00:06:13.362
of 9 over 4,

00:06:13.362 --> 00:06:16.682
so we multiply both sides times 4 over 9,

00:06:16.682 --> 00:06:19.445
times 4 over 9,

00:06:19.445 --> 00:06:20.327
and that cancels with that,

00:06:20.327 --> 00:06:21.744
that cancels with that,

00:06:21.744 --> 00:06:23.671
and we are left with

00:06:23.671 --> 00:06:25.993
a is equal to, well let's see,

00:06:25.993 --> 00:06:29.174
144 divided by 9

00:06:29.174 --> 00:06:31.356
is going to be,

00:06:31.356 --> 00:06:35.559
is going to be, I want
to say it's equal to 16?

00:06:35.559 --> 00:06:37.997
Is that right?

00:06:37.997 --> 00:06:39.181
I think that is right.

00:06:39.181 --> 00:06:40.574
Yup, and then 16 times 4,

00:06:40.574 --> 00:06:43.453
so this is going to be 16 and 1,

00:06:43.453 --> 00:06:46.495
16 times 4 is 64,

00:06:46.495 --> 00:06:48.492
which is exactly what we got before,

00:06:48.492 --> 00:06:49.560
and so this actually,

00:06:49.560 --> 00:06:50.698
this method, now that I look at it,

00:06:50.698 --> 00:06:53.298
is actually a good bit easier,

00:06:53.298 --> 00:06:53.972
but they're equivalent,

00:06:53.972 --> 00:06:55.829
and I like this one because
you get to see the common

00:06:55.829 --> 00:06:58.499
ratio in action, and you get
to see that the initial value

00:06:58.499 --> 00:06:59.776
is the initial value.

00:06:59.776 --> 00:07:01.541
It is h of zero,

00:07:01.541 --> 00:07:02.470
but either way,

00:07:02.470 --> 00:07:03.445
once you figure out r,

00:07:03.445 --> 00:07:05.860
and you know one of the

00:07:05.860 --> 00:07:06.765
h of,

00:07:06.765 --> 00:07:08.855
you know what the function is at a value,

00:07:08.855 --> 00:07:10.271
you can solve for a,

00:07:10.271 --> 00:07:11.711
and frankly, if you
knew a and you knew what

00:07:11.711 --> 00:07:13.104
the function was for a given n,

00:07:13.104 --> 00:07:15.983
you could likewise solve for r.

00:07:15.983 --> 00:07:18.253
So hopefully you enjoyed that.


Initial value & common ratio of exponential functions

WEBVTT
Kind: captions
Language: en

00:00:00.396 --> 00:00:02.340
- [Voiceover] So let's
think about a function.

00:00:02.340 --> 00:00:03.117
I'll just give an example.

00:00:03.117 --> 00:00:04.530
Let's say, h of n

00:00:04.530 --> 00:00:06.643
is equal to

00:00:06.643 --> 00:00:08.161
one-fourth

00:00:08.161 --> 00:00:09.740
times

00:00:09.740 --> 00:00:10.555
two

00:00:10.555 --> 00:00:12.060
to the n.

00:00:12.060 --> 00:00:13.015
So, first of all,

00:00:13.015 --> 00:00:15.059
you might notice something
interesting here.

00:00:15.059 --> 00:00:18.704
We have the variable, the
input into our function.

00:00:18.704 --> 00:00:20.402
It's in the exponent.

00:00:20.402 --> 00:00:24.474
And a function like this is
called an exponential function.

00:00:24.474 --> 00:00:27.221
So this is an exponential.

00:00:27.221 --> 00:00:30.641
Ex-po-nen-tial.

00:00:30.641 --> 00:00:33.751
Exponential function, and that's because

00:00:33.751 --> 00:00:36.376
the variable, the input into our function,

00:00:36.376 --> 00:00:38.447
is sitting in its definition of

00:00:38.447 --> 00:00:40.396
what is the output of
that function going to be.

00:00:40.396 --> 00:00:41.942
The input is in the exponent.

00:00:41.942 --> 00:00:44.682
I could write another
exponential function.

00:00:44.682 --> 00:00:45.428
I could write,

00:00:45.428 --> 00:00:46.154
f of,

00:00:46.154 --> 00:00:48.581
let's say the input is a variable, t,

00:00:48.581 --> 00:00:50.659
is equal to

00:00:50.659 --> 00:00:53.213
is equal to five

00:00:53.213 --> 00:00:54.943
times

00:00:54.943 --> 00:00:56.521
times three

00:00:56.521 --> 00:00:57.792
to the t.

00:00:57.792 --> 00:00:58.755
Once again,

00:00:58.755 --> 00:01:02.110
this is an exponential function.

00:01:02.110 --> 00:01:04.293
Now there's a couple of
interesting things to think about

00:01:04.293 --> 00:01:05.168
in exponential function.

00:01:05.168 --> 00:01:07.710
In fact, we'll explore many of them,

00:01:07.710 --> 00:01:09.271
but I'll get a little
used to the terminology,

00:01:09.271 --> 00:01:11.604
so one thing that you might see

00:01:11.604 --> 00:01:12.687
is a notion of

00:01:12.687 --> 00:01:14.040
an initial value.

00:01:14.040 --> 00:01:16.373
In-i-tial

00:01:16.373 --> 00:01:18.196
Intitial value.

00:01:18.196 --> 00:01:20.344
And this is essentially
the value of the function

00:01:20.344 --> 00:01:22.758
when the input is zero.

00:01:22.758 --> 00:01:24.348
So, for in these cases,

00:01:24.348 --> 00:01:26.647
the initial value for the function, h,

00:01:26.647 --> 00:01:29.026
is going to be, h of zero.

00:01:29.026 --> 00:01:30.245
And when we evaluate that,

00:01:30.245 --> 00:01:31.846
that's going to be one-fourth

00:01:31.846 --> 00:01:34.245
times two to the zero.

00:01:34.245 --> 00:01:36.288
Well, two to the zero power, is just one.

00:01:36.288 --> 00:01:37.437
So it's equal to

00:01:37.437 --> 00:01:38.505
one-fourth.

00:01:38.505 --> 00:01:40.305
So the initial value,
at least in this case,

00:01:40.305 --> 00:01:41.883
it seems to just be that number

00:01:41.883 --> 00:01:42.987
that sits out here.

00:01:42.987 --> 00:01:44.867
We have the initial value times

00:01:44.867 --> 00:01:46.724
some number

00:01:46.724 --> 00:01:48.164
to this exponent.

00:01:48.164 --> 00:01:49.988
And we'll come up with
the name for this number.

00:01:49.988 --> 00:01:52.900
Well let's see if this was
true over here for, f of t.

00:01:52.900 --> 00:01:55.083
So, if we look at its intial value,

00:01:55.083 --> 00:01:56.280
f of zero

00:01:56.280 --> 00:01:57.070
is going to be

00:01:57.070 --> 00:01:58.673
five times

00:01:58.673 --> 00:02:00.703
three to the zero power

00:02:00.703 --> 00:02:02.074
and, the same thing again.

00:02:02.074 --> 00:02:03.376
Three to the zero is just one.

00:02:03.376 --> 00:02:05.419
Five times one is just five.

00:02:05.419 --> 00:02:07.799
So the initial value is once again, that.

00:02:07.799 --> 00:02:08.375
So if you have

00:02:08.375 --> 00:02:10.555
exponential functions of
this form, it makes sense.

00:02:10.555 --> 00:02:11.996
Your initial value,

00:02:11.996 --> 00:02:14.712
well if you put a zero
in for the exponent,

00:02:14.712 --> 00:02:16.405
then the number raised

00:02:16.405 --> 00:02:18.029
to the exponent is just going to be one,

00:02:18.029 --> 00:02:19.277
and you're just going to be left with

00:02:19.277 --> 00:02:20.777
that thing that you're
multiplying by that.

00:02:20.777 --> 00:02:21.857
Hopefully that makes sense,

00:02:21.857 --> 00:02:22.937
but since you're looking at it,

00:02:22.937 --> 00:02:25.061
hopefully it does make a little bit.

00:02:25.061 --> 00:02:26.118
Now, you might be saying,

00:02:26.118 --> 00:02:28.033
well what do we call this number?

00:02:28.033 --> 00:02:29.832
What do we call that number there?

00:02:29.832 --> 00:02:31.261
Or that number there?

00:02:31.261 --> 00:02:33.868
And that's called the common ratio.

00:02:33.868 --> 00:02:35.029
The

00:02:35.029 --> 00:02:36.909
common

00:02:36.909 --> 00:02:38.718
common ratio.

00:02:38.718 --> 00:02:39.703
And in my brain,

00:02:39.703 --> 00:02:42.257
we say well why is it
called a common ratio?

00:02:42.257 --> 00:02:45.044
Well, if you thought about
integer inputs into this,

00:02:45.044 --> 00:02:48.282
especially sequential
integer inputs into it,

00:02:48.282 --> 00:02:50.001
you would see a pattern.

00:02:50.001 --> 00:02:51.401
For example,

00:02:51.401 --> 00:02:52.433
h of,

00:02:52.433 --> 00:02:54.169
let me do this in that green color,

00:02:54.169 --> 00:02:56.305
h of zero

00:02:56.305 --> 00:02:59.155
is equal to, we already
established one-fourth.

00:02:59.155 --> 00:03:00.081
Now, what is h of one

00:03:00.081 --> 00:03:01.682
going to be equal to?

00:03:01.682 --> 00:03:03.408
It's going to be one-fourth times

00:03:03.408 --> 00:03:04.707
two to the first power.

00:03:04.707 --> 00:03:06.564
So it's going to be one-fourth

00:03:06.564 --> 00:03:06.894
times

00:03:06.894 --> 00:03:07.826
two.

00:03:07.826 --> 00:03:09.420
What is h of two

00:03:09.420 --> 00:03:11.220
going to be equal to?

00:03:11.220 --> 00:03:12.461
Well, it's going to be

00:03:12.461 --> 00:03:14.331
one-fourth

00:03:14.331 --> 00:03:15.294
times

00:03:15.294 --> 00:03:16.018
two squared,

00:03:16.018 --> 00:03:18.924
so it's going to be times two times two.

00:03:18.924 --> 00:03:21.745
Or, we could just view this as

00:03:21.745 --> 00:03:23.988
this is going to be two times

00:03:23.988 --> 00:03:25.430
h of one.

00:03:25.430 --> 00:03:26.634
And actually I should have done this

00:03:26.634 --> 00:03:27.640
when I wrote this one out,

00:03:27.640 --> 00:03:28.801
but this we can write as

00:03:28.801 --> 00:03:29.835
two times

00:03:29.835 --> 00:03:31.899
h of zero.

00:03:31.899 --> 00:03:34.665
So notice, if we were to
take the ratio between

00:03:34.665 --> 00:03:35.698
h of two

00:03:35.698 --> 00:03:36.549
and h of one,

00:03:36.549 --> 00:03:37.690
it would be two.

00:03:37.690 --> 00:03:38.645
If we were to take the ratio between

00:03:38.645 --> 00:03:39.312
h of one

00:03:39.312 --> 00:03:39.887
and h of zero,

00:03:39.887 --> 00:03:41.023
it would be two.

00:03:41.023 --> 00:03:42.544
That is the common ratio between

00:03:42.544 --> 00:03:46.185
successive whole number
inputs into our function.

00:03:46.185 --> 00:03:47.563
So, h of

00:03:47.563 --> 00:03:49.037
I could say

00:03:49.037 --> 00:03:49.731
h of n

00:03:49.731 --> 00:03:50.792
plus one

00:03:50.792 --> 00:03:51.952
over

00:03:51.952 --> 00:03:53.442
h of n

00:03:53.442 --> 00:03:55.682
is going to be equal to

00:03:55.682 --> 00:03:57.354
is going to be equal to

00:03:57.354 --> 00:03:58.782
actually I can work it out mathematically.

00:03:58.782 --> 00:03:59.528
One-fourth

00:03:59.528 --> 00:04:00.689
times two

00:04:00.689 --> 00:04:02.568
to the n plus one

00:04:02.568 --> 00:04:03.088
over

00:04:03.088 --> 00:04:04.040
one-fourth

00:04:04.040 --> 00:04:04.725
times

00:04:04.725 --> 00:04:06.129
two to the n.

00:04:06.129 --> 00:04:06.898
That cancels.

00:04:06.898 --> 00:04:08.566
Two to the n plus one,
divided by two to the n

00:04:08.566 --> 00:04:10.609
is just going to be equal to two.

00:04:10.609 --> 00:04:12.988
That is your common ratio.

00:04:12.988 --> 00:04:14.997
So for the function h.

00:04:14.997 --> 00:04:15.995
For the function f,

00:04:15.995 --> 00:04:17.438
our common ratio is

00:04:17.438 --> 00:04:18.576
three.

00:04:18.576 --> 00:04:20.178
If we were to go the other way around,

00:04:20.178 --> 00:04:22.674
if someone said, hey, I have some function

00:04:22.674 --> 00:04:24.252
whose initial value,

00:04:24.252 --> 00:04:25.228
so let's say,

00:04:25.228 --> 00:04:27.046
I have some function, I'll
do this in a new color,

00:04:27.046 --> 00:04:29.426
I have some function, g,

00:04:29.426 --> 00:04:31.627
and we know that its initial

00:04:31.627 --> 00:04:32.927
initial

00:04:32.927 --> 00:04:34.529
value

00:04:34.529 --> 00:04:35.388
is

00:04:35.388 --> 00:04:36.851
five.

00:04:36.851 --> 00:04:37.826
And someone were to say

00:04:37.826 --> 00:04:39.789
its common ratio

00:04:39.789 --> 00:04:42.621
its common

00:04:42.621 --> 00:04:44.327
ratio

00:04:44.327 --> 00:04:45.627
is six,

00:04:45.627 --> 00:04:47.566
what would this exponential
function look like?

00:04:47.566 --> 00:04:50.201
And they're telling you this
is an exponential function.

00:04:50.201 --> 00:04:51.850
Well, g of

00:04:51.850 --> 00:04:53.824
let's say x is the input,

00:04:53.824 --> 00:04:55.205
is going to be equal to our

00:04:55.205 --> 00:04:57.909
initial value, which is five.

00:04:57.909 --> 00:05:00.126
That's not a negative sign there,

00:05:00.126 --> 00:05:01.591
Our initial value is five.

00:05:01.591 --> 00:05:03.557
I'll write equals to make that clear.

00:05:03.557 --> 00:05:04.623
And then times

00:05:04.623 --> 00:05:06.177
our common ratio

00:05:06.177 --> 00:05:08.441
to the x power.

00:05:08.441 --> 00:05:09.335
So once again,

00:05:09.335 --> 00:05:12.484
initial value, right over
there, that's the five.

00:05:12.484 --> 00:05:14.514
And then our common ratio

00:05:14.514 --> 00:05:15.385
is

00:05:15.385 --> 00:05:16.474
the six,

00:05:16.474 --> 00:05:17.773
right over there.

00:05:17.773 --> 00:05:19.382
So hopefully that gets
you a little bit familiar

00:05:19.382 --> 00:05:20.984
with some of the parts

00:05:20.984 --> 00:05:22.099
of an exponential function,

00:05:22.099 --> 00:05:24.492
why they are called what they are called.


Solving exponent equation using exponent properties

WEBVTT
Kind: captions
Language: en

00:00:00.080 --> 00:00:01.763
- [Voiceover] So I have an
interesting equation here.

00:00:01.763 --> 00:00:04.874
It says V to the negative six fifths power

00:00:04.874 --> 00:00:08.160
times the fifth root of V

00:00:08.490 --> 00:00:10.728
is equal to V to the K power,

00:00:10.728 --> 00:00:14.110
for V being greater than/equal to zero.

00:00:14.324 --> 00:00:15.579
And what I wanna do

00:00:15.789 --> 00:00:18.236
is try to figure out what K needs to be.

00:00:18.236 --> 00:00:19.972
So what is...

00:00:20.314 --> 00:00:22.857
what is K going to be equal to?

00:00:22.857 --> 00:00:25.113
So pause the video and see
if you can figure out K,

00:00:25.443 --> 00:00:27.419
and I'll give you a hint,
you just have to leverage

00:00:27.419 --> 00:00:29.352
some of your exponent properties.

00:00:30.135 --> 00:00:32.057
Alright, let's work this out together.

00:00:32.237 --> 00:00:33.702
So the first thing I'd want to do is

00:00:34.022 --> 00:00:35.342
being a little bit consistent in

00:00:35.592 --> 00:00:36.881
how I write my exponents.

00:00:36.881 --> 00:00:39.081
So here I've written it as
negative six fifths power,

00:00:39.551 --> 00:00:41.095
and here I've written it as a fifth root,

00:00:41.095 --> 00:00:43.050
but we know that the
fifth root of something...

00:00:43.780 --> 00:00:46.272
we know that the fifth root...

00:00:47.192 --> 00:00:48.814
the fifth root of V,

00:00:49.044 --> 00:00:50.173
that's the same way,

00:00:50.173 --> 00:00:53.690
that's the same thing as saying
V to the one fifth power,

00:00:53.690 --> 00:00:55.422
and the reason I want
to say that is because

00:00:55.422 --> 00:00:58.158
then I'm multiplying two
different powers of the same base,

00:00:58.158 --> 00:00:59.861
two different powers of V.

00:00:59.971 --> 00:01:02.029
And so we can use our
exponent properties there.

00:01:02.029 --> 00:01:04.441
So, this is gonna be the same thing as

00:01:04.441 --> 00:01:07.486
V to the negative six fifths

00:01:08.136 --> 00:01:10.651
times, instead of saying
the fifth root of V

00:01:10.651 --> 00:01:14.894
I can say times V to the one fifth power

00:01:15.294 --> 00:01:17.756
is going to be equal to V to the K.

00:01:18.126 --> 00:01:20.606
It's gonna be equal to V to the K power.

00:01:21.121 --> 00:01:23.909
Now, if I'm multiplying V to some power

00:01:23.909 --> 00:01:25.682
times V to some other power,

00:01:25.962 --> 00:01:28.105
we know what the exponent
properties would tell us,

00:01:28.297 --> 00:01:30.346
and I could remind us.

00:01:30.346 --> 00:01:31.166
I'll do it over here.

00:01:31.166 --> 00:01:33.289
If I have X to the A

00:01:33.609 --> 00:01:35.993
times X to the B,

00:01:36.273 --> 00:01:40.626
that's going to be X
to the A plus B power.

00:01:41.266 --> 00:01:43.806
So here, I have the same base, V.

00:01:44.236 --> 00:01:46.407
So this is going to be V to the,

00:01:46.407 --> 00:01:47.789
and I could just add the exponents.

00:01:48.119 --> 00:01:51.631
V to the negative six fifths power

00:01:52.001 --> 00:01:53.534
plus one fifth power,

00:01:53.534 --> 00:01:57.911
or V to the negative six
fifths plus one fifth power

00:01:58.241 --> 00:02:00.024
is going to be equal to V to the K.

00:02:01.024 --> 00:02:03.727
Is equal to V to the K.

00:02:03.727 --> 00:02:05.503
I think you might see where
this is all going now.

00:02:06.313 --> 00:02:09.834
So this is going to be equal to V.

00:02:10.214 --> 00:02:12.585
So negative six fifths plus one fifth

00:02:12.995 --> 00:02:16.311
is going to be negative
five fifths or negative one.

00:02:17.761 --> 00:02:19.527
So all of this

00:02:19.777 --> 00:02:22.592
is going to be equal to negative one,

00:02:23.042 --> 00:02:26.447
and that's going to be
equal to V to the K.

00:02:27.047 --> 00:02:31.538
So K must be equal to
negative one, and we're done.

00:02:31.908 --> 00:02:36.581
K is equal to negative one.