Hi guys, I’m Nancy. And I’m going to show

you how to find the domain of any function. The domain is kind of an annoying question

that you’ll get about a function, but it can be very simple. And all it means is all possible

x values that you can have in your function. So all allowable x values that you can have

in your function. To do this you should already know how to solve a quadratic by factoring.

And how to solve an inequality. Okay so these are the five types of functions you might

have to find the domain for. I’m going to go through all of them. If you want to skip

ahead you can. Just look below in the description for the time to skip to. The first case is

really easy. If you have a polynomial – just an x expression like x squared plus 3x plus

1, no square roots, no fractions, your domain answer is just all real numbers. And you can

write that phrase “all real numbers” or you can write interval notation: parenthesis negative

infinity comma positive infinity, end parenthesis. Now if you don’t have just a polynomial and

you have something more complicated like a fraction or a square root or possibly both,

it’s going to be one of these four cases. it’s either just a fraction, it’s a square

on its own, it could be fraction with a square root in the bottom, or a fraction with a square

root in the top (the numerator). And I’m going to explain how to find the domain for all

of those cases. Okay, say you have just a fraction. To find the domain all you have

to do is take the bottom, whatever that may be, and set it not equal to zero. So in this

example we have x squared plus 5x plus 6 “not equal to zero. Notice that it doesn’t matter

what the top is, just take the bottom and set it not equal to zero. And you can solve

this just as if it were a normal equality. And that involves factoring. If you need help

with factoring or solving by factoring you can look at one of my other videos. So in

this equation we get two restrictions two solutions: x cannot equal negative 3 and x

cannot equal negative 2. So when we want to write the domain, the final answer for domain,

you can write it as all real numbers but x cannot equal negative 3 and cannot equal negative

2. Or you can use interval notation and write it as everything from negative infinity, all

the way up to negative 3, not including negative 3, combined with everything from negative

3 to negative 2, not inclusive of either number, combined with everything from negative 2 all

the way up to infinity. So either of these notations is fine – they mean the same thing.

Let’s look at another example. What if you’re given something like this? Where the denominator

is x squared plus 3. Same thing: take the bottom, set it not equal to zero because we

cannot have the denominator equal zero. It’s not defined when it is. Then when you solve

this equation, you end up with x squared cannot equal negative 3. But x squared can never

equal negative 3, so we just can’t have that ever. There’s no x value that would ever make

that happen, so you don’t need to worry about that restriction. All you need to write in

that case would just be your normal all real numbers domain. So you can say all real numbers.

You could say capital R symbol for all real numbers. Or you could say, with interval notation,

everything from negative infinity to positive infinity, with parentheses on either side.

So that would be your three ways of writing the domain for this fraction. Now let me show

you what to do if you have just a square root in your function. I’m going to focus on a

square root, but this is true for any even root – fourth root, sixth root, etc. It’s

not true for odd roots like third root. If you have a square root, or an even root, you

take the expression underneath the radical sign, which is the radicand (everything that’s

underneath the root symbol), so in this case x plus 1, and you need it to be greater than

or equal to zero, because we can’t have a negative number underneath a root. You can

only have a positive number or zero. So what’s underneath must be greater than or equal to

zero. Now you’re going to solve that normally, as you would solve any inequality, and you

get the restriction x is greater than or equal to negative 1. So your domain would be all

real numbers, x greater than or equal to negative 1. Or you can write it in interval notation

as everything from negative 1, all the way up to infinity. Notice that I used a square

bracket for negative 1 because you’re including negative 1, and everything all the way up

to infinity, parenthesis. So either of these ways is fine, as notation for the domain.

Okay let me show you quickly also an example with a quadratic expression under the root.

So here we have x squared plus 5x plus 6 underneath the square root symbol. You still do just

take the expression underneath the root symbol and require that it be greater than or equal

to zero. Okay you need to solve this inequality. It’s a quadratic inequality, which is more

complicated than the last example. If you don’t know how to solve that, you can look

it up somewhere else. Once you have solved this quadratic inequality, you will get that

x can be less than or equal to negative 3, or x can be greater than or equal to negative

2. So your domain you can write either as all real numbers but x less than or equal

to negative 3, x greater than or equal to negative 2. If you were going to write this

in interval notation, it would look like everything from negative infinity all the way up to negative

3, including negative 3, and everything from negative 2 all the way to positive infinity.

So either of those ways are fine to write your domain answer. Okay, if you have a square

root in the bottom of your fraction, all you need to do is take the expression under the

root, x plus 1, and set it greater than zero. Now usually with a square root, we would set

it greater than or equal to zero, but we can’t have that the denominator equal zero, so we

leave that part out. So if you see the square root in the bottom, all you need to do is

take the expression under the root and set it greater than zero, and solve. So in this

case, you subtract one from each side, and you would get that x is greater than negative

1, so your domain would be

all real numbers, x greater than negative 1, or if you wanted to use interval

notation, you would write parenthesis negative 1, everything from negative 1, not including

negative 1, all the way up to infinity. So either of these ways is fine to write your

domain answer. Okay, if your square root is in the top of your fraction, you’ll have two

restrictions. The first one will be that what’s under the square root has to be greater than

or equal to zero. So x plus 1 greater than or equal to zero. And when you solve that

by subtracting 1 from each side, minus 1, minus 1, you get x greater than or equal to

negative 1. The other restriction will come from the bottom. That the bottom can’t be

zero. So we have x squared minus 4 cannot equal zero. Now to solve that as before, you

would have to factor, so you have x plus 2, times x minus 2, can’t be zero. So I’ve re-written

that as a product of two factors, and when you solve each of them separately, set each

of them not equal to zero, you get x cannot be negative 2, and x cannot be 2. Okay so

we have this restriction from the root, and this restriction from the bottom. What you

have to do in this case is take the intersection of the two. So if you think about this, x

has to be greater than or equal to negative 1, which already takes care of negative 2

– if x is greater than or equal to negative 1, we can’t have negative 2 anyway, so you

can ignore that one, throw it away. All you need to write for your domain: x greater than

or equal to negative 1, except it cannot be 2, positive 2, and in interval notation that

would be that you can have everything from negative 1, including negative 1, up to 2

but not including 2, and then everything from 2, not including 2, up to infinity. So either

of these notations is fine, and that’s your answer for the domain. So I hope that was

super fun, finding the domain of your function. It’s okay if you didn’t have fun. You don’t

have to like math. But you can like my video, so if you did, please click like below.

I haven't understand any thing

Hottest maths teacher

Best video I have found on Youtube on Domain!

Wait she is writing in mirror?????

I don't understand on the FRACTION ONLY why was the domain all real numbers. it can't equal -3 but then it can't equal -2 or -1 either? confusedddd

Thanks

Hi Nancy… 👍

Thanks, like, sincerely. Damn this is so useful. This makes more sense than my class

Can you do more better than that because i knew all these function . i think you have to add more function like log,ln,modulus

Can u solve this question domain and also range g(x) =|x_3|

Can you please explain with pseudo quadratic as well

If she didn't make her math videos, i would fail my math class. she's a saint.

I like ya

What about logarithm function and it’s domain

Fantastically organized and straight to the point video, thank you

Is your hand okay?

Find the domain and range of f(X)=X|X|

Beauty helps me study. That is why I study using Nancy's video.. And I don't watch any other math video who a guy is teaching because Math guys teachers are assholes! and difficult to understand!!

Does the domain only focus on denominators???

Maths has never been so beautiful 😍😍

THANK YOU!!

Thank you. Simple and made me actually pay attention. Kudos to you!!

there's a mistake, anyone noticed?

Outstanding Nancy, thanks so much. -Mary

Why is it that all nancy are beautiful? nancypi, nancymcdoney, and nancy binay.

Why are you so nervous

I watched kahn academy 5 times and nothing helped this makes so much more sense thanks

To find the domain of the function: look at the A in the symbol f : A -> B

How do you write backwards?

Thank you!!!!

she's too hot I cant focus…

How to find range please ? huhu… best teacher in youtube <3

You don't have to like math … But you can like my video.

Me: i like you

@5:03 why is the domain negative infinite to positive infinity, wouldn't it be negative infinity to negative 3??NEED HELP UNDERSTANDING WHY ITS TO POSITIVE INFINITY

THANK YOU SO MUCH. nowhere else in any textbook or internet explained finding the domain as well as you have.

You did a great. It's so benefit to me.

Great video, thanks a lot, this really helped me with my functions!

Thank you so much!! I get it now lol

thanks for the video…really helped

Wow what a great video 😍.

Everything in just small video superb🤗.

daaaaaaaang she fine affff… dat voice tho geeez

How to find range make video on that also fast🏃⛷️

Nancy your too attractive for me to actually learn anything from what your saying.

Thank you for making my life easier! Lol teachers really tend to make it complicated -_- smh. Btw i love your voice❤

Can you do a video like this but on the range

Give yourself a hug cause you rule!

hello, thank you for the service i would like you to help me with sketching the absolute values of functions

holy FUCK this girl is hot

I love when random people on YouTube are better teachers than people with >masters degrees. I bet it was great back in the day when teaching was a revered career.

Your beauty disturbed me ;p

Thankkkkkk youuuuu

on of the best explanation ever and i just love you

When would I know to put a bracket or a parenthesis

THANK YOU SO MUCH!

Hi

Just wanted to let you know that this was very helpful 🙂

Brilliant

She made a mistake in square root of X^2+5x+6 coz inequality flip in value of 3.

the coolest finishing statement,dont have to like math….

Her backwards handwriting is better than my handwriting at 1 wpm

Pre-calc teacher showed me this in class…. couldnt focus… SHE WRITES BACKWARDS

I’ve had TWO different teachers explain this to me and nothing, I tried khan academy and nothing. But you is a blessing because I understand every. Single. Word.

Hi I’m gonna need help more Thank grateful for helping but need more help

very difficult to focus on question

how r u writing backwards

You are sexy as fuck

Wow I have never enjoyed math this much. You are so calm

thankyou

also thanks for the description of video

Why is she kinda mad in this video?🤔

Beautiful video and gorgeous face

What if you have a root inside the root

Excuse me ! May i go to Washroom 🖐🏻

Maam are you writing on screen of the cam 😵🤦🏻♂️

You are a great teacher ❤️ thanks for your effort

yo thanks for the explanation, I have finals this Friday this really gonna help me out

You are a life saver❤️

Ad: WHAT? STILL SEARCHING YOUTUBE FOR MATH HELP?

Me: Yes. I am indeed. Because there are some really good teachers on youtube who deserve respect.

This video was a great help & now problems’ solved

You remind me of Violet from the Incredibles.

OMG Im so happy she made a video on this!

Thank you so much. God bless you

Be careful with the numbers included and not included accordingly apply the brackets

How the hell can a 10 minute video on Youtube literally be more educational than a 3 hour math class, taught by someone who has done it for years. We need to replace teachers with youtubers.

Please make a video on "how to find range of any function". I absolutely need this!

And thank you for the video! You are an excellent teacher.

Liked, Subscribed!

Dear Mam, Tomorrow morning is my math exam (first term). I took my math teacher consultant hour also 3 extra class about domain and range but more than 60% student did not understand this term. after watching this video i totally understand how to find Domain. but i still do not understand how to find Range.

Would you please make second video like this about How to Find the Range?

Honestly, you are really saving my non-mathematical bottom part. Thank you! 😀

🌿🍀🌿🍀🌿🍀🌿🍀🌿🍀🌿🍀🌿

Barely noticed that if you write on anything transparent like her glass or plastic the reflection on a mirror or flipped video comes out the same because it’s the back side of the writing, pretty cool… and something about the domain of fractions I think?

Why couldn't my teacher just explain it like this?! Thank you 🙂

Her videos were very useful for me and now I can help other people with their algebra

What about absolute value?

I searched for lie derivative—but this is much better

You are absolutely gorgeous

Does this works for inverse function and composite function？HELPPP！

Girl this does not help at all. You people come on here calling yourself helping but you aren't be clear about how or why you are making the solutions. I need to know why you got the (-3) and so on.

You didn't have to have like maths but can like my video 😂👍🏿

Make on range too plz

why cant i still understand it!!! how my teachers explains are different from this.the examples they give is so damn hard. but the way she teaches makes it easy!!!! but i still cant understand it!! im so stupid.

Good content