Welcome to my presentation

on domain of a function. So what’s is the domain? The domain of a function,

you’ll often hear it combined with domain and range. But the domain of a function is

just what values can I put into a function and get

a valid output. So let’s start with

something examples. Let’s say I had f of x is equal

to, let’s say, x squared. So let me ask you a question. What values of x can I put

in here so I get a valid answer for x squared? Well, I can really put anything

in here, any real number. So here I’ll say that the

domain is the set of x’s such that x is a member

of the real numbers. So this is just a fancy way of

saying that OK, this r with this kind of double backbone

here, that just means real numbers, and I think you’re

familiar with real numbers now. That’s pretty much every number

outside of the complex numbers. And if you don’t know

what complex numbers are, that’s fine. You probably won’t need

to know it right now. The real numbers are every

number that most people are familiar with, including

irrational numbers, including transcendental numbers,

including fractions — every number is a real number. So the domain here is x —

x just has to be a member of the real numbers. And this little backwards

looking e or something, this just means x is a member

of the real numbers. So let’s do another one

in a slight variation. So let’s say I had f of x is

equal to 1 over x squared. So is this same thing now? Can I still put any x

value in here and get a reasonable answer? Well what’s f of 0? f of zero is equal to 1 over 0. And what’s 1 over 0? I don’t know what it is,

so this is undefined. No one ever took the trouble to

define what 1 over 0 should be. And they probably didn’t do, so

some people probably thought about what should be, but they

probably couldn’t find out with a good definition for 1 over

0 that’s consistent with the rest of mathematics. So 1 over 0 stays undefined. So f of 0 is undefined. So we can’t put 0 in and get

a valid answer for f of 0. So here we say the domain is

equal to — do little brackets, that shows kind of the

set of what x’s apply. That’s those little curly

brackets, I didn’t draw it that well. x is a member of the real

numbers still, such that x does not equal 0. So here I just made a slight

variation on what I had before. Before we said when f of x is

equal to x squared that x is just any real number. Now we’re saying that x is any

real number except for 0. This is just a fancy way of

saying it, and then these curly brackets just mean a set. Let’s do a couple more ones. Let’s say f of x is equal to

the square root of x minus 3. So up here we said, well this

function isn’t defined when we get a 0 in the denominator. But what’s interesting

about this function? Can we take a square root

of a negative number? Well until we learn about

imaginary and complex numbers we can’t. So here we say well, any x is

valid here except for the x’s that make this expression under

the radical sign negative. So we have to say that x minus

3 has to be greater than or equal to 0, right, because you

could have the square to 0, that’s fine, it’s just 0. So x minus 3 has to be greater

than or equal to 0, so x has to be greater than or equal to 3. So here our domain is x is a

member of the real numbers, such that x is greater

than or equal to 3. Let’s do a slightly

more difficult one. What if I said f of x is equal

to the square root of the absolute value of x minus 3. So now it’s getting a little

bit more complicated. Well just like this time

around, this expression of the radical still has to be

greater than or equal to 0. So you can just say that the

absolute value of x minus 3 is greater than or equal to 0. So we have the absolute value

of x has to be greater than or equal to 3. And if order for the absolute

value of something to be greater than or equal to

something, then that means that x has to be less than or equal

to negative 3, or x has to be greater than or equal to 3. It makes sense because x

can’t be negative 2, right? Because negative 2 has an

absolute value less than 3. So x has to be less

than negative 3. It has to be further in the

negative direction than negative 3, or it has to be

further in the positive direction than positive 3. So, once again, x has to be

less than negative 3 or x has to be greater than 3,

so we have our domain. So we have it as x is

a member of the reals — I always forget. Is that the line? I forget, it’s either

a colon or a line. I’m rusty, it’s been

years since I’ve done this kind of stuff. But anyway, I think

you get the point. It could be any real number

here, as long as x is less than negative 3, less than or

equal to negative 3, or x is greater than or equal to 3. Let me ask a question now. What if instead of this it was

— that was the denominator, this is all a separate

problem up here. So now we have 1 over the

square root of the absolute value of x minus 3. So now how does this

change the situation? So not only does this

expression in the denominator, not only does this have to be

greater than or equal to 0, can it be 0 anymore? Well no, because then you would

get the square root of 0, which is 0 and you would get a

0 in the denominator. So it’s kind of like

this problem plus this problem combined. So now when you have 1 over the

square root of the absolute value of x minus 3, now it’s no

longer greater than or equal to 0, it’s just a greater

than 0, right? it’s just greater than 0. Because we can’t have a 0

here in the denominator. So if it’s greater than 0, then

we just say greater than 3. And essentially just get rid of

the equal signs right here. Let me erase it properly. It’s a slightly different

color, but maybe you won’t notice. So there you go. Actually, we should do another

example since we have time. Let me erase this. OK. Now let’s say that f of x is

equal to 2, if x is even, and 1 over x minus 2 times

x minus 1, if x is odd. So what’s the domain here? What is a valid x I

can put in here. So immediately we

have two clauses. If x is even we use this

clause, so f of 4 — well, that’s just equal to 2 because

we used this clause here. But this clause applies

when x is odd. Just like we did in the last

example, what are the situations where this

kind of breaks down? Well, when the

denominator is 0. Well the denominator is 0

when x is equal to 2, or x is equal to 1, right? But this clause only

applies when x is odd. So x is equal to 2 won’t

apply to this clause. So only x is equal to 1

would apply to this clause. So the domain is x is a member

of the reals, such that x does not equal 1. I think that’s all the

time I have for now. Have fun practicing

these domain problems.

## 100 Comments

## Patrick Ransbottom

Very informative, thank you for the video.

## Norah Aldrak

Natural numbers are from 1 and more, which means the second example you can also say Domain= { x Belongs to N }

## vladikuss1

I would like to know what program you use for you video. (not for video recording but for blackboard)

## sunjanium

i have a d in algerbra 2 because i dont understand the way my teacher teaches me so now i am going to try and watch ur videos for functions to try and salvage my grade wish me luck friends

## IRAQYsniper95

my teacher fucking sucks at teaching ur the fucking best man. fuck my math teacher.

## Butt Ball

@sunjanium yeah, this dude is awesome! im only in algebra 1 but i had a test and i had no idea what to do because i didnt understand how my teacher taught it either so i searched ''how to do ….'' and i totally understand now!

## rafay qazi

ahhhh something happened to my pen! … Lol!

## NoseNoseNose

My "Math II" Proff sucks.

Ur the math dude!!

## 444zane3

@Rensyl math 2? you mean like algebra 2 or something?

## NoseNoseNose

@444zane3 I'm from Argentina and im studying economics.

We have MAthematics 1, 2, 3 and 4.

We don´t call it calculus, algebra or etc etc.

Greetings.

## RemoteHelper

Thanks! 🙂

## zodiac832

Thank You for the video!! i looked at it before i went to take my final, really helped me out

## Moe Baller

The very last 4 minutes confused me. Make another video to explain them please.

## PortugalFootballer

I LOVE YOU, you are saving me for my diploma exam this week.

## DemonikOu

3:00 That can be also written as {x∈R*}!

R* means all R except 0 😉

## L King

Y'know I was just thinking, shouldn't 1 divided by 0 = 1? Because technically, you aren't dividing by anything, so 1 would remain the same right?

## MrUnladenswallow

think of it this way, how many times can you throw nothing into a bucket.( lets say the bucket never decayed and you could live forever and never run out of energy) . you could keep throwing nothing into the bucket forever and it will never fill up. ( bad analogy but i hope that makes sense) 🙂

## L King

Ahh, but then what was already in the bucket remains unchanged doesn't it? (I feel like this is edging towards philosophical thinking haha)

## MrUnladenswallow

Now you have succeeded in confusing me aswell , i hope you are happy XD (kidding)

## L King

haha, whoops! XD

## Hulyo Geroleo

Sir Sal what I just want to ask what happened in the algebra playlist yesterday I started with its history I played almost 20 videos and now Im confused of what happened now it started in domain of functions and the sequence of videos is re-ordered.

## jbassmesser183

lol. good one! maybe in a logical or reasonable world. math follows pointless logic, in my opinion. nothing you will ever encounter in the real world. i surely haven't in my 38 years. but, 1/0 is an impossible situation. I guess that's a better way of thinking. In other words, you literally can't divide any number by zero. Thus, it is undefined.

## jbassmesser183

I just want to know why the index of 6 to the radicand 5-x is (in interval notation) (-infinite, 5], yet the index of 7 to the radicand 3x+5 is (in interval notation) (-infinite, infinite). Why wouldn't it be [-5/3,infinite) as I have concluded in my calculation?? Does it have to do with the odd number index? Thus, all odd number indexes would (-infinite, infinite) in interval notation? I'm utterly confused.

## L King

haha, that is very true! touché my friend

## Aaron Brown

You cannot Divide by zero.

:L

## L King

Lol I know it's mathematically impossibly but philosophically anything is possible XD

## Nasar Shah

HOW THE [email protected]#$ DO you remember this lol. "Its been years since I done this"

## Jihoon Choe Music

Dude. Anything divided by 0 is undefined! So you can't divide by zero.

## Mohyuddin Nasir

Did I Hear Curly Brackets 3:21! lol

## boblee666

what is love?

## Dorian P.

has this guy written a book yet. il love to read how how he studied, like his schedule he is a man to imitate for greatness in school.

## TheJuleslMeister

The answer to life is 42, therefore we can write {xl 1/0=42 and x€R} Hope that helped took seconds of research

## theT0MMYNATOR

i guess it technically isn't doing anything…. but you could also say it goes into 1 an infinite number of times while also never being able to equal 1……..i guess thats why they say its undefined

## L King

Ahhh I see what you're saying, interesting point

## Diana Dai

why i can place all number in x square?

## ManDudeYeah

That's what my South Korean cal teacher taught us today. She really tries to drill that in our heads. Admittedly, it makes me a bit uncomfortable.

## ManDudeYeah

I think that's one of the biggest problems. Even good books don't teach nearly as well as a clear demonstration with some one's voice guiding you. That's why Khan is so badass….

## antiflx

KHAAAAN!

## Arturo Verboonen

The reason division by 0 is undefined is because we have this thing called properties of real numbers. One of this properties says that every number has another number (its inverse) such as when you multiply them, you get 1. The thing with 0 is that there is no such number by which you can multiply it to get 1, and given that division is nothing more than multiplication by the inverse, division by 0 is undefined.

I hope this helped.

## Mohamed Tabaa

djis was here!!!!

## Jazen Valencia

Came here to learn how to kill Kirk ended up learning how to kill 9:59

## BIGBase Entertanment

thanks for your help i enjoy watching and listing to you teaching,it awesome but i would like to ask a question which is "i thought whenever the minus is moving to the other side that the sign will change as well" so i think it suppose to be x is less than and equal to positive 3,please i will be waiting for your reply once again thanks for making out time to educate us

## shoopdeedoop

thanks Sal

## L King

ahh that really does make sense! I discussed your comment with my brother and I understand now, thanks!

## permanganate89

thanks khan maahnnn

## Chris Cullen

Actually my teacher is great, I'm just here because I've jumped into a maths course without having the full prerequisites and i need to catch up in my own time 🙂

## Bryce Hoffman

I fucking hate math…like this is going to help me in life…

"Oh yes you want personal trainer position at Gold's Gym in California? SHOW ME YOU KNOW HOW TO FIND THE DOMAIN OF THIS FUNCTION!"

## ayush modi

good

## RyleZor

It won't help for that.. For people that want to have a real job it will. For example; engineers, scientists, doctors. They will all use mathematics quite extensively. Don't be so ignorant, and stop doing maths at school if you have no goals in life.

## RyleZor

Lol, accounting is mathematics you learn in year 8. I'm already over qualified for that shit.

## ben neale

hi

## Ivan Jolly

3:29 f(x) = my penis lol

## Luis Arias

him saying he hasnt done this in years kinda proves we dont need this -______-

## RAGE Dark

wow, personal trainer thats pathetic. that just shows how little and ignorant you are. your life sounds amazing. lol.

## Bryce Hoffman

You dumbfucks don't understand an example.

## NotoriousPhD

You're a very ignorant individual. Many more important jobs require high levels of understanding of mathematics, jobs that help sustain the life you live. Where would we be if we didn't have physics or biology? We would be neanderthals/cavemen. Very primitive.

## NotoriousPhD

Not everybody, but some people do. Sal was saying he doesn't remember the peculiarities of this subject. Don't have such an irrational fear of math.

## Bryce Hoffman

Without money you wouldn't have internet and therefore would not have been able to type that comment. Point made.

## NotoriousPhD

Yes, in the type of society we live in that is how it works, but there are alternative and more effective and peaceful economic systems. If you look at the venus project, that would create a better world and more advanced society.

## Bryce Hoffman

Without Googlinge venus project, I'm just gonna assume this is something similar to Communism, and we all know how that turned out.

## Brenda

he kept on fucking up this entire video oh my god

## Samiur Rahman Sefat

thank you 🙂

## eurofilmkid

shit sal, get your pen together. also, isn't the answer to the last question x is a member of the real numbers such that x is not equal to 0? X can be equal to 1, right? 1/(x-2)(x-1) is a valid input….

## Samuel Gitonga

thank you Sal,,owe u big time

## Gabriel Shadwick

having a sound issue

## Barbara Lopez

Very helpful

## Rainier7777

THANK YOU, THANK YOU, THANK YOU!

## Tadayen

(X<3)(X>3) xcannot be equal to 3 because that creates the 0 you divide by…..

## charles bai

This guy sounds like etho from minecraft lel

## Akshay Appan

Wait- why did you take three? in the squared root of x and why did you divide x by 1??

## blingz

im still having problems to understand whats domain and whats undefined and all that stuff

## Minjarez96

Your name should be Mathew

## Merana Summers

I couldn't understand what you did in the absolute value of x part.

## Raining Starz

Your me money for tuition thank so much

## Chris Matthew

Can I specify domain values as "x not equal to 6" ( or some number.. ) or do I have to use {…. : …..} ?

## Smit Patel

hey there, can anyone please explain me why in the end of video, there's a popup saying x beongs to integers… why we ain't considering the real ones ?

## ahmad abdolsaheb

this was uploaded in 2007 everyone 🙂

## Cathy B

Ummm… What is the name of this sign? ===> Є

It looks like E. But… What is it called in Math?

## Katelin Morrison

How come you were putting a negative domain as the absolute value of x when you said in the previous equation there cannot be negatives under a radical? Please help 🙂

## Kevin Lee

so I can say the domain is equal to any real number except 0?

And what do you mean by x is odd or even?

## thebutter flyNL

wtf 240p quality

## VWLZ

is there such thing as f(x, y) in math?

## yert koop

somebody buy this guy a pen

## AME MA

Drinking game: take a shot for every time he says "my pen"

## clouden the kid

#mypen

## Ime Ekpo

cool

## RS

i didnt get the last problem.. somebody explain plz

## American Adventures

he uploaded this when youtube was a joke!

## Skimtar

Stumbled across this video literally an hour past the 10 year mark. Creepy.

## Jerferson de Matos

For a second I thought the yellow dots were ads 😂

## Arun Pandey

How do I recognize whether the domain of a particular function is of integer type or real number type??

## mario napolitano

Such that is a line

## ashi mohta

whats,√x^2

## Aditya Nagesh

You are an awesome math teacher.

## Alex K

Ant man and the Wasp was fun

## Justin S

1,000,000th view!

## Christal Horton

This boi still helpful in uni kids

## إياد حصني

That is very very nice

## Renaldo

can you do a video on business functions such as c(X) where c is cost etc

## Coleton Messmer

240p lol