|Start||Ok, so we are done with trigonometry for a bit.
And, now we are going to talk about functions.
So, yeah it is sort of the theory of functions.
Where is some chalk.
So, first of all, this is the theory part of the lecture.
What is a function?
Input results in an output... Sort of.
Not a great definition, but I can tell where you are going. Other ideas?
Ooooo. That sounds like somebody who took math before.
For every input, there is only one output. That is almost technically a good definition. But a function is something that you do. Okay?
So, yes that is pretty good.
|1:03||You guys are on the right track.
Don't make me do it.
So, for every input there is only one output is part of the definition of a function. Yeah?
Not quite a system of equations. Yeah.
That is what functions are.
Functions pass the vertical line test.
Well, log is a type of function.
These are all things that are true about functions. When you graph them the x values do not repeat. Yes?
(Student) Describes a line?
Describes a line, sure.
Functions can describe a line. Yeah.
|2:02||It tells me how to find y? Pretty close.
Is it like a relationship?
Sure. Where you like thinking Kayne? Or... like a healthy relationship? (laughter) Don't tell them I said it. Don't tell them.
I will bring Jay-Z in it and it will be all over.
Any other ideas? No body on this side of the room has answered. Go ahead...
You are all saying things that are true about functions.
You all learned this at some point in high school or a previous college course.
So, a function is doing something, when you take something from the input and put it into the output. So, it is an operation of some kind.
So, but a function is is just a way of matching.
You do not have to write down this definition if you do not want to.
What happens is you have
|3:00||something which you call the domain.
Then, something else which you call range.
Then you have elements in your domain.
And, it maps them.
Puts them onto elements in your range.
So, there is a way of taking an element in the domain, the input, and creating something in the output. Now it gets a little tricky because what do we mean by domain
|3:30||and what do we mean by range
and all sorts of things
so a function can be
you each have a Stony Brook ID number.
So, a function could be take Lane you give him a Stony Brook ID number.
And then, take Lamar and get a different Stony Brook ID number.
So, part of what you want a function, is you what to decide can it be the same output?
Because if you guys were the same, he would get into a fight over it.
|4:01||Okay. You would hate that.
You know, part of what you want to decide is for the domain is there a single answer in the range?
So, you think about Stony Brook ID numbers they are all nine digits. They start with one and end in zero.
Well, not you guys. You start with 1 and end in 1.
So, it can't be all nine digit numbers.
That you can get out because you don't have Stony Brook ID numbers that start with 9. We may some day.
A long time from now.
You know that they are counting the students.
So, you would have to have 800 million or so people go through Stony Brook.
But, however For each student, there is only one Stony Brook ID number.
Okay, um. Or you could think about email address. However, email address you may not get 1 to 1 because there may be more than one David Kahn.
There is another David Kahn, however, he is Khan.
|5:00||That is actually a different email address. I will tell you
what happens if you email me and you
use his email address because you do not know how to spell my name.
So, spell it correctly. But, there are definitely people in this school who have the same name.
Okay? So, for example, if you have the last name Kim, there are a lot of people here whose last name is Kim. Okay, so, you are going to get you have to figure out how you want to map those.
So, that they work. So that you get single emails that match.
The input and the output.
Um, you could do all sorts of, so a function doesn't have to be mathematical. Okay?
A function can be as simple as a user mailbox.
Each person has a single mailbox, so you are matched to a mail box.
But, the idea is every element in the domain Is only one element in the range.
So, we will write that down.
|6:01||What somebody said before
about not repeating x values.
What does that mean: For every element in the domain, there is only one element in the range.
Well, if you think of it as x and y, it means that when you have an x, there is only one y. So, There is only one Stony Brook ID number for each of you. Right, you do not get 2 Stony Brook ID numbers.
Alright, so 2 people I'm sorry, backwards.
2 people cannot have the same Stony Brook ID number.
|6:31||So for every x,
a person, there is only one Stony Brook ID
You know that there's no second person with the same Stony Brook ID number.
Okay? In fact they go 1 to 1. Because you can go in the other direction as well.
So the idea is if you think about it.
Sort of to jump around.
You have 2 different x values.
x 1 and x 2.
For every x, there is only one y.
|7:01||Which means you can't have a y here
and a y there.
That is why there is what we call the vertical line test.
So, if you had a graph of a function And you could do a vertical line test to determine if it is a function.
The vertical line test says that there is no place on the graph where if you draw a vertical line, it cuts through the graph more than one place. I will say that again.
There is no place on the graph
|7:30||Where if you draw a vertical line
it touches the graph more than once.
But that doesn't mean that it always has to cut through the graph.
But, if it does, it doesn't intersect twice.
So, if you have a curve that goes like this Then, if you think about drawing vertical
|8:00||lines, you are never going to cut through that
more than once. So, for
every x you are only going to hit it once.
If you had a graph that looked like that... Now There are places where you can cut it with a vertical line and you could intersect like so.
So, that would not be a function of x. Okay?
So far so good?
|8:31||This is what you mean when you talk about functions.
So, functions really is just a way to relate from what is going in to what is going out. So, you already saw, with trigonometric functions.
Now we can do exponential functions.
Which is logarithm functions.We can really do all sorts of fun functions How are we doing so far?
You learned this before?
You did not get it last time either?
You just do the vertical line test.
And, that is a good test. Except
|9:01||however, when they call it as a function of x.
You can have a function theta.
y=sin theta. So, it doesn't have to actually be x and y.
But, what we call the x is what we mean by the domain.
The domain is what you are allowed to plug in to the function. Range, I am not going to spend a lot of time on this, range is what you get out.
Okay? So, we did sine and cosine. Can you take the sine of anything?
What are things you can't take the sine of?
|9:31||Think about it.
Is there anything that you are not able to take the sine of?
Not that you don't know it. You could use your calculators.
Could you take the sine of 1 billion?
What do you think?
Can you take the sine of a billion?
I don't know. How about I take out my calculator and find out?
sine of a billion
|10:00||Yeah, I get an answer. So, could I take the sine of
a negative a billion? Sure.
Why not? You can take sine of anything that you want.
I heard someone say no.
So, I could take the sine and cosine of anything I want. So, the domain of sine and cosine is anything, all reals. Okay?
Range is not, but the domain is.
So, one of the things we are going to work on for a few minutes.
f of x which you have seen before, f of x stands for function of x. Okay?
|10:31||If I had f of theta,
That would be a function of theta.
If I had f of Stony Brook Student, That would be the function of Stony Brook Student.
So, whatever is in the parentheses is the domain. Okay?
So, if you put in anything you want, then you get out something else.
Remember you get out what you put in.
So, what you put in you get out the same thing, that is a function.
Okay, then you want to figure out what is the domain.
Well, lots of functions you can put in anything you want. Okay, however, half of Stony Brook students, well we need them to be Stony Brook students. So, we can't put in a student from Harvard.
You can only put in a Stony Brook student for input, anything you get out you get for output. Right?
Medical school could be the output.
You put in a Stony Brook student and you get out a doctor.
|11:30||So, the domain is what you
are allowed to put into
the function. So, let us make this fact a
If I had f of x equals x squared, what is the domain?
Well, x is the letter. Okay?
What is the domain? In other words, what are we allowed to put in. What are we allowed to do the square to?
Anything. There isn't anything that we are not allowed to square.
Right. So, it belongs to the x axis.
When you square, x could be anything.
So, the domain of this, all reals. By the way, This is as theoretical as we are going to get here. Okay?
What are real numbers? Well, let me put it another way. First you have counting numbers. So, those are numbers you could use your fingers.
Right? So, 1,2,3
|12:31||to a bazillion. Okay?
I should say 1, 2, 3, with dots. 4,5... Okay?
Um, okay, So, the integers included any of the counting numbers and then there is 0.
So, if you say that the domain is the integers, then it could be any whole number positive, negative, or 0.
Positive if you start with 1, negative if you start with negative 1.
|13:01||And, so on. So, the integers are 0, 1, 2,
3, -1, -2, -3,
out to infinity.
Okay? Rational numbers.
You guys in the front?
What else are rationals?
Decimals, fractions, kind of the same thing. Yeah.
Anything we can take a fraction of.
Is Pi over 3? It is a fraction
Is Pi over 3 a rational number?
How do you know Pi is irrational?
Ah, you are getting very close. Okay.
This is mathematicians being very annoying with definitions.
A ending, non-repeating decimal.
Sure, but a fraction is a good idea.
That is not a rational number.
But, a fraction, wait you have to go a fraction of what? So, what is in the numerator in the fraction?
Could i go in the numerator in a fraction?
No. So, what is in the numerator in a fraction? An integer.
What is in the denominator of a fraction?
|14:31||An integer other than 0.
Okay? So, a rational number, is a number that can be represented as a ratio, a fraction of an integer and another integer.
Or as they say in math speak, It is sort of P over Q where P and Q are integers.
Q not equal to 0.
If 0 is in the bottom it is undefined.
You have to learn how to deal with that.
Okay? So, that is a rational number.
And when you do these, you get integers, that is why a fraction wasn't complete.
4 over 1 is a rational number, right? 1 over 4 is a rational number. So, You get integers. You can get fractions, right?
You can get decimals that
end. So, finite decimals.
You can get infinite repeating decimals.
You can get an infinite non-repeating decimal. Well, if you take this, and divide by this, you wil not get an infinite, non-repeating decimal.
So, a irrational number is a number that is not rational.
If you look at your friend, you say, you are being irrational right now.
Okay? It means you are not being rational.
Okay. So, all
|16:00||reals are both rationals
and irrationals. Okay?
Or, if you remember your imaginary numbers.
not allowed to be rational.
You can get Pi and some things like that.
We could look at trigonometry with rational and irrational numbers.
But, we are not doing that.
When we say the domain is all reals, as far as we are concerned you can plug in anything that you want.
You can plug in fractions, you can plug in
|16:31||integers. You can plug in Pi.
You can plug in 0. You can plug in a billion.
Negative a billion. It doesn't matter.
What is the domain?
Can you take the square root of anything you want?
Well, could it be the integers?
Positive numbers. That word numbers needs to be defined.
|17:00||Positive real numbers. Almost.
Want to give it a shot?
Come on, you just left out one little thing.
The positive reals.
Almost right, forgot something.
No, no, not just integers, real numbers is good.
Can you take the square root of a negative number?
You really can't hear yourself think.
|17:31||You guys are getting so close.
Positive real numbers was almost correct. You just forgot something.
Zero? You forgot 0. Okay? So, positive real numbers or 0. Because you can take the square root of 0. What is the square root of 0?
So, the domain would be written in a couple of different ways.
Okay? You would say x is greater than or equal to 0.
|18:01||You could say x
So, non-negative means positive or 0. Okay? So, if I wanted to say positive, positive means greater than 0.
Negative means less than 0.
So, not negative means, take away all the numbers less than 0.
Or, one other thing
|18:31||you could represent the domains
with interval notation.
Did you guys learn interval notation before?
In high school?
Something like that? We could do that in just a minute.
We will use the side of this board for review.
Okay, we use a square bracket and parentheses. Okay?
Square bracket means that we include the number/ Parentheses mean it does not include the number.
Okay it is the difference between greater than
|19:01||and greater than or equal to.
Or, less than or less than or equal to.
That is what the square bracket is for.
You can signify a range by using two elements with a comma between them.
So, If I add this, that means all the numbers from 4 to 6.
Including 4, but not including 6. So another way to write this.
Is something like that.
So, if I have, That is all the numbers from 0 to 10. Including 0, including 10.
If I have this, looks like something like that.
|20:01||That means all the numbers from 0 to
to 10 not including
0 or 10.
Webassign uses interval notation.
And, of course, you should learn it.
It is very compact. One thing about mathematicians, they are always looking for the shortest or easiest way to write something.
So, if I had, 0 to 10 this way. Then, that would be x including 0.
|20:30||Not including 10.
You can guess what is coming.
That would be you don't include 0 and do include 10.
So far so good?
Now, questions, do we include infinity?
Do include infinity.
Do not include infinity. Which is it?
Who wants to go with no you cannot?
Good. The majority.
|21:01||You never actually get to infinity.
So, if you want to represent infinity.
So, say you wanted to to do this.
That means all real numbers greater than or equal to 0 out to infinity.
That is saying that x is greater than or equal to zero. So, infinity or negative infinity.
You never get a square bracket.
That would be all real numbers
|21:31||less than or equal to zero.
Okay? They always get, infinity, always get parentheses.
So, if you can think about the number line.
Negative infinity down at this end.
That is infinity down at this end.
0 in the middle. Okay?
So, since you can't actually get to infinity. It gets parentheses.
You should have all seen this stuff before.
What is that to represent something like all reals except x=5.
Well, I would start at negative infinity.
And, go up to 5.
But, I don't include 5.
Then, I go back to 5.
and go up to infinity.
And you need a union symbol. Not that union.
The union going together.
So, that would be anything except 5. So, when you are doing this on Webassign. Pay attention to this.
|23:01||you guys get something wrong
because you did not include a union sign.
Okay? So, negative infinity to 5, 5 to infinity.
So, where does that show up?
So, take a second and find the domain of that.
|23:30||x - 3 could be anything greater than
or equal to 0.
So, binomials you cannot square root.
You take what is underneath the square root, which is called the radicand.
There is a radical and the radicand is what is inside the radical.
And you set it greater than or equal to 0.
The domain Will be x is greater than or equal to 3.
So, in interval notation,
|24:00||Looks like that.
Right, well, what you are doing is when you want to find the domain, of a square root, you take what is inside the square root, and set it greater than or equal to zero.
So, if we had, You would take 5 minus 4x and set it greater than or equal to zero.
You get x is less than or equal to
5 over 4. Do not make that 4 over 5.
Another common mistake.
You guys see that over there?
This would be anything less than 5 over 4.
So you go from -infinity
|25:02||to 5 over 4.
Square bracket at the 5 over 4.
Parentheses on the negative infinity.
So, what if instead of square root of x-3, cube root of x-3?
|25:31||What if it is cube root instead of square root?
Now comes the fun part.
Can you take a cube root of a negative number?
What is the cube root of negative 8?
Cube root of 8 is 2. What is the cube root of -8? -2.
You can take the cube root of a negative number.
Can you take the cube root of a positive number?
Sure. Can you take the cube root of 0?
Sure. Can you take the cube root of Pi?
Sure. I mean it is some number. Right?
It is not that easy to do, but it certainly
|26:01||something times something times something will give you Pi.
That is how you get the cube root of Pi.
Multiply it three times to get Pi.
So, domain is all reals.
Or in interval notation, Everything from minus infinity to positive infinity.
So, watch out.
|26:31||When you have a radical, if it is
an odd radical,
cubed root, 5th root, 7th root, 9th root
it is all reals.
If it is an even radical, square root, 4th root, 6th root, etc. cannot.
Okay. So, we only take an even radical of a positive number or 0. If we have an odd radical it is all real numbers.
Should I repeat that?
|27:01||Look at the root. We will write it out.
We will be slightly mathy.
We will use math terms.
If f(x) is the nth root of x, then, if n is even, positive even
it doesn't really make sense.
You never really see the negative root of something.
or 1/3, it doesn't really make any sense.
But, if positive and even then the domain is from 0 to infinity cannot take the even root of a negative number.
|28:01||is positive odd
domain is all reals.
All these years and I am not so good at the infinity symbols.
But, I am trying.
What is the domain?
So, the domain
x does not equal 2.
If I have all reals that is that funny R.
For that compact notation.
Or, we could do this in interval notation.
Minus infinity to 2 union 2 to infinity.
So, those who did not hear that,
that is probably all of you, okay.
What do we know? Well, first you can't have 0 in the denominator. Right, because if it is 0 in the denominator of a fraction, you have never really seen a fraction like this have you?
If I remember, no, no, pretty much.
You can't have 0 in the denominator.
And, you can't take the square root of a negative number.
|30:01||So, you just have
the square root of x- 2.
The domain would be you take x-2 and you set it greater or equal to 0.
Here x-2 has to be just greater than 0.
Because You can't have negative values of square root and you can't have 0.
Because you can't have 0 in the denominator.
Here the domain would be x is greater than 2.
If I had square root of x-2 over 5.
Well, now you could have 0 in the numerator.
So here, the domain would be x-2 is greater than or equal to 0.
x is greater than or equal to 2.
Or, 2 to infinity.
Go ahead and take a minute.
See if you can figure that one out.
Okay, the numerator can be anything that you want.
The numerator can be anything that you want, but the denominator has to be, can't be 0 and you can't take the square root of a negative number.
So, x-3 has to be greater than 0. The domain is x is greater than 3. Or you
|31:31||could write it
3 to infinity. Either of
those notations is fine. It is
kind of traditional
to put curly braces
around that, but you do not have to.
You can't have 3.
Parentheses take care of that.
It does not matter which notation that you use on the test.
You could use either one. You could
|32:01||even write the domain is
all reals where x is greater than 3.
It takes a little work. But, you could just write it out like that.
Interval notation will be useful for Webassign.
Um. The reason you can't pick 3 have 3 in the denominator, in the domain, it is because if you plug in 3 it will make 0 in the denominator.
Where as normally you can take the square root of 0 and get 0.
But, here you take the square root of 0 and have
|32:31||a problem. Suppose I change the function.
I move it around.
And, I do this.
Right?! So here this is where they get you.
What is the problem now.
The top you want to go for it?
|33:01||you could put 0 in the top.
Because you could, numerator, because you could take the square root of 0.
So, if we just had the numerator.
We would say the domain is x is greater than or equal to 3. Because, we plug in 3 or any number greater than 3 that's fine. But, We can't plug in 5. Because 5 would give us a 0 in the denominator.
Okay? So, x can be
|33:31||any number greater than or equal to 3 except 5.
You can write the word except. Okay? You could say x is greater than or equal to 3 except 5. You can do any indication on the exam that you know what you are doing.
Perfectly fine. other than writing, what the person next to me has.
If you write that than we will be a little suspicious. Okay?
In interval notation, Well we would write it off to infinity,
|34:00||but we have a problem. 5 is in the way.
So, go up to 5 don't even include it, and we start at 5 again and go up to infinity.
Got the idea?
You have to get this.
The domain question on the exam is the domain question on Webassin.
It would be a good idea. And remember, what this is really about is in a function, okay, domain is what will allow
|34:31||to begin. So, domains
are very important when you are looking at functions
in general. Because you want to say
to yourself, in math in general
you want to have very specific definitions. You know when I am
asking questions and you don't have that
exactly right. You want to know exactly
what you mean. So, domain
tells you exactly what is allowed
to go into the machine, into the function.
Then, once you know what goes in, then you know what comes out.
So, as we learned already the sines and cosines the domain is anything. It is not true of tangent.
Okay. You cannot
|35:01||take the tangent of anything. You can take the sine and cosine.
Okay. Let's have you practice another of these.
So, one of the things that you should feel comfortable with so, most of the kinds of functions we are going to give you at this stage, if you see a fraction bar, then the bottom cannot be 0. So, one of the things you look for in domains. The denominator can't be 0. So, tangent is undefined where cosine of x is 0.
Remember we had that little chart.
of x cannot
be 0. That would be the domain.
Now, where's cosine of x equal to 0?
Well, let us not do it with degrees.
So, x cannot equal Pi over 2, 3 Pi over 2, what about 5 Pi over 2?
Can't be 5 Pi over 2 either.
7 Pi over 2, can't be any of those.
|36:00||So, it can't be the negatives of those, so,
plus or minus.
Then you can write dot, dot, dot, once you get the pattern.
We also write Pi over 2 plus, n Pi, where n is an integer.
|36:30||Pi over 2 plus Pi, is
really 3 Pi over 2. Pi over 2 plus 2 Pi is really
5 Pi over 2. Pi over 2 plus negative
Pi is negative Pi over 2
and so on. So, there
is a bunch of different ways to write this.
Okay? So, domain of tangent is x cannot be any odd multiple of Pi over 2.
You could write this in
interval notation and it would get messy.
Okay? Because you would have -Pi over 2 to Pi over 2 union Pi over 2 3 Pi over 2 union 3 Pi over 2, 5 Pi over 2.
And you would have to go the other way too.
So it would be very hard to do.
So, that is not the kind of thing you want to write.
So, tangents is
sine over cosine
tangent is undefined any place where
the denominator is 0. So, we see
functions, as soon as you see a denominator,
you have to say to yourself
the domain can't be, the thing can't be 0.
Ooooo. That is low.
I put in Pi.
Pi is a great way to get 10 percent of the class in questions.
What is the domain?
Well, if I did not have Pi, if I had a number, I would take x minus that number and set it greater than or equal to 0.
So, we take x-Pi and set it greater than or equal to 0.
So next, It is greater than or equal to Pi.
You are right.
Erase that. Because can't have 0 in the denominator. Thank you.
x has to be greater than Pi.
Everytime I say Pi, I get hungry, just so you know.
|39:00||Okay, just like everybody
What if I have Okay.
So how about this one?
x, we look at the top and we say well, x has to be greater than or equal to 0.
And, in the bottom, and you say x cannot equal plus or minus 3.
However, ooo, get it?
got him now.
I was in class this morning and there was a spider.
A spider web and there was no spider.
Which is good.
Spider web, okay, spider, not so good.
x is greater than or equal to 0.
x cannot be plus or minus. We actually don't care about
|40:00||the minus 3 in this one. Because it already has
to be greater than 0
it already cannot be negative 3, that won't
matter because we are restricted
by the fact that x is not equal to 0.
Okay? So, this would be x is greater than or equal to 0 and x cannot be 3. So, it would be 0 to 3 3 to infinity.
Well, you take negative 3 and you
|40:30||square root it what do you get?
When you take negative 3 and you square it what do you get? 9-9=0.
-3 squared gives you 9.
However, we actually don't care in this particular one about the negative 3 because it is already eliminated by the fact that x has to be greater than or equal to 0. But,
If I had this,
f(x)= x over x squared minus 9
Now, the domain is x cannot
equal plus or minus 3.
Because now the numerator can be anything it wants.
The denominator can't be 0.
The denominator cannot be 3 or negative 3 in interval notation form,
Okay, you start at negative infinity
you go up to negative 3, then you stop.
You go to negative 3 and you go to positive 3 and stop.
Then you go 3 to infinity on your graph.
Along this number line.
You can do something like that.
Those are really entertaining to enter into Webassign.
Okay, there is one other
This would be if I am really feeling nasty at exam time.
How are we doing on this? You all get it copied down?
You do have it copied down.
I am going to cover it.
It will be available
on video again as of
tomorrow at some point.
Assuming that we are videoing this. Technically, one class last semester we videoed where we forgot about the sound.
That was not my most popular video.
Find the domain of that one.
This is something that you did in MAP 103. Did the denominator of this and you did not know why.
Now you do.
Okay, now for the fun part.
You kind of know what you needed to do.
Right? You say x squared minus 9 has to be greater than 0.
So, if this were a test question, you would get something to put in that to demonstrate you know what you are doing. And, the tricky part is how do you find where x squared minus 9 is greater than 0. Which you do not do is add 9 to both sides and square root it.
What you do is you factor this.
He needs x + 3 and x - 3.
Why can't you do that though?
Why can't it equal 9?
Why can't it equal 0?
Because we can't have 0 in the denominator.
If this weren't a fraction, and you just had the square root of x squared minus 9, then you could just set it equal to 0.
|44:30||But, because it is in the denominator,
you can't set it equal to 0. Okay?
You get 0 in the bottom of a fraction and that is bad.
You sure that we got that?
This is the place where people trip up.
So, now you say great. But, how do I know where (x+3)(x-3) is greater than 0?
The simple trick is to draw your number line.
Put the zeros in the number line.
This is called doing a sign test.
You take a number in each of the 3 regions.
|45:01||The region less than negative 3.
The region between negative 3 and 3.
The region greater than 3.
Plug it into here and see where you get a positive answer.
You take a number less than -3. What is a number less then -3?
-4. We all guess negative 4, right.
Plug in -4 and you get -4+3=-1 -4- -3=-7.
Negative times a negative is a positive.
So, this function is positive
|45:31||the left of negative 3. So, again,
take a number less than negative 3:
-4, -5 or -billion.
Plug it in, and you will get a negative times a negative is a positive.
So, when x is less than negative 3 this will come out positive.
Now pick a number between negative 3 and 3. Like 0, 0 is always a good choice.
You plug in 0 to get 3 times -3. And a positive
|46:00||times a negative is negative.
Okay? Positive and a negative is negative.
So it is negative in that zone.
Now you pick a number greater than 3, like 5, sure, or 4 or 5 or a trillion.
It doesn't matter.
5+3 is a positive number.
5-3 is a positive number.
Positive times a positive is a positive.
So, it is positive here again.
So, therefore, the zone will be
x is less than
x is greater than positive 3.
And in interval notation.
You get that.
So, it is these regions from negative, oh this is negative.
I am sorry. From negative infinity to -3.
|47:00||Combined with the region from
3 to infinity.
Say that again.
Say I put and I put it in?
Right? So, -2 is not in that region, but -4 is.
Okay. So remember, less than negative 3 is going to the left
|47:30||on the number line.
Okay? There is still a lot of time left.
Everyone is so restless. You want to rush out.
Okay, that is enough for today.