Let’s have a little bit of a review of what

a function is before we talk about what it means that what the domain of a function means.

So function we can view as something — so I put a function in this box

here and it takes inputs, and for a given input, it’s going to

produce an output which we call f of x. So, for example, let’s say that

we have the function — let’s say we have the function f of x is equal to 2 over x. So in this case if — let me see — that’s my function f. If I were to input the number 3. Well, f of 3 that we’re going to output — we have, we

know how to figure that out. We’ve defined it right over here. It’s going to be

equal to 2 over 3. It’s going to be equal to 2 over 3. So we’re able, for that input, we’re

able to find an output. If our input was pi, then we input into our function and then

f of pi — when x is pi, we’re going to output

f of pi, which is equal to 2 over pi. So we

could write this as 2 over pi. We’re able to find the output

pretty easily. But I want to do something interesting. Let’s attempt to input 0 into the function. If we input 0 then

the function tells us what we need to output. Does this

definition tell us what we need to output? So if I attempt to put x equal 0, then this

definition would say f of 0 be 2 over 0, but 2 over 0 is

undefined. Rewrite this — 2 over 0. This is undefined.

This function definition does not tell us what to actually do with 0. It gives us an

undefined answer. So this function is not defined here.

It gives a question mark. So this gets to the essence of what domain is. Domain is the set of all inputs over

which the function is defined. So the domain of this

function f would be all real numbers except for x equals 0. So we write down these, these big ideas.

This is the domain — the domain of a function — Actually let me write that out. The domain of a function A domain of a function is the set of all inputs — inputs over which the function is defined — over which the function is defined,

or the function has defined outputs over which the function has defined outputs. So the domain for this f

in particular — so the domain for this one — if I want

to say its domain, I could say, look, it’s going to

be the set of these curly brackets. These are kind of typical mathy set notation. I said OK

, it could be the set of — I gonna put curly brackets like that. Well, x can be a

member So this little symbol means a

member of the real numbers. But it can’t be any real number. It could be most of the real numbers except it

cannot be 0 because we don’t know — this definition is undefined when

you put the input as 0 So x is a member of the real numbers,

and we write real numbers — we write it with this kind of double stroke right over here.

That’s the set of all real numbers such that — we have to put

the exception. 0 is not a — x equals to 0 is not a member of that

domain — such that x does not — does not equal 0. Now let’s make this a little bit

more concrete by do some more examples So more examples we do, hopefully the clearer this will become.

So let’s say we have another function. Just be clear, we don’t

always have to use f’s and x’s. We could say, let’s say we

have g of y is equal to the square root of y minus 6. So what is the domain here? What is the set of all inputs over which this function g

is defined? So here we are in putting a y it to function g and we’re gonna output g of y. Well it’s going to be defined

as long as whatever we have under the radical right over here is non-negative. If this becomes a

negative, our traditional principal root operator here is not defined. We need something that —

if this was a negative number, how would you take the principal

root of a negative number? We just think this is kind of the the

traditional principal root operator. So y minus 6, y minus 6 needs to be greater than or equal to 0, in order for, in order for g

to be defined for that input y. Or you could say add six to both sides.

y is to be greater than or equal 6. Or you could say g is defined for any inputs

y that are greater than or equal to 6. So you could say the domain here, we

could say the domain here is the set of all y’s that are members of

the real numbers such that y, such that they’re also

greater than or equal, such that they’re also greater than or equal to 6. So hopefully

this is starting to make some sense — You’re all used to a function that is

defined this way. You could even see functions that are divided fairly exotic ways. You could see a

function — let me say h of x — h of x could be defined as — it literally

could be defined as, well h of x is gonna be 1 if x is equal to pi and it’s equal to 0 if, if, x is equal to 3. Now what’s the domain

here? And I encourage you to pause the video and think about it. Well, this function is actually only

defined for two input. If you, we know h of — we know h of pi —

if you input pi into it we know you’re gonna output 1, and

we know that if you input 3 into it h of 3, when x equals 3, you’re going to

— you’re going to — put some commas here. You’re gonna get 0.

But if you input anything else, what’s h of 4 going to be? Well, it hasn’t defined. It’s undefined.

What’s h of negative 1 going to be? It hasn’t defined. So the domain, the domain here, the domain of h is literally — it’s just literally

going to be the the two valid inputs that x can be are 3 and pi. These are the only valid inputs.

These are the only two numbers over which this function is actually defined.

So this hopefully starts to give you a flavor of why we care about to the domain.

It’s not all functions are defined over all real numbers. Some are defined

for only a small subset of real numbers, or for some other

thing, or only whole numbers, or natural numbers, or positive numbers, and negative

numbers. So they have exceptions. So we’ll see that as we do

more and more examples.

## 30 Comments

## Ben B

First

## Luke Joseph

First to comment

## Suhaib Alkhoory

my college literally uses khan academy instead of actual classes 😀

## Chibi Prussia

Oh sure, you upload this AFTER I failed a quiz miserably. Several weeks ago.

## TEAMiANDu

Ive bern going through notes trying to remember this and you post it the day BEFORE my math final . God bless you ?

## Crud O'Matic

float func(float x) {

float ret;

return ret = (x) ? 2.0/x : 0.0;

}

## CHINMAYACADEMY

Nice video & explanation !!!

## Fady Hameed

THANK YOU SO MUCH , IT WAS VERY HELPFUL

## Tajender Singh

thanks

## Maritza Soto

for your example g(y)= the square roots of y-6. would interval notation look like [6, infinity) ?

## Rafa

5:04 hopefully

## goodcleanfun

3:14, does someone in the background laugh at "double stroke"? Obviously asking the important questions here.

## Riccardo

You're the man, very clear explanation

## Azure Phoenix

I am thoroughly impressed by his mouse handwriting. lol

## Yunirang

Khan academy, could u guys make more vids like this? It's great for studying for ap

## Vinícius Horta

isnt square root of 2 undefined, so that the in the domain of g(y) y is different than 8?

## NCARalph

I'm sorry, as much as I like Khan Academy, this video sucks. Just jumping into set notation doesn't seem to have any purpose other than convincing students they will never understand math.

## Choosy teens

Why can't domain have negative no.s

## Tropixs

Ok…I don't know if I finally lost it or not BUT Is there any audio in this video or not?

## Curious Squared

Thank you

## NASIR MAHMOOD

what will be the domain of f ( x ) = 1/ sqrt ( x^2 – 1 )

## SHAFAQUE SHIREEN

Thnx alot sir.. Realy it helps me alot to understood and find the domain of a function… Tysm sr.. ❤❤

## Hebeli Haguna

Good math ?️?️?

## YoBoi21 XD

Sir i do no speak smart math language

## YoBoi21 XD

Can you speak normal math language for dumb people

## Coco L.

Shouldn’t the domain for the blue one b greater than or equal to7, bc otherwise u still get a negative number?

## Miguel Martinez

I still don't understand this. I don't know what's wrong with me for not getting it. 🙁

## JoyCon Boyz Forever

This looked really cryptic but it’s actually easy.

## Ali

شكراً يا أستاذ خان ?

## Everything Is Love

How do u Wright with the mouse?? Dude I get cramps at school when I do that…