>>Natalie Denny: Welcome back. Now we’re going to look at writing domain in

interval notation. Let’s take a look at f of x is

equal to the square root of x. So in the last video, we

talked about the domain of this function are all numbers that are greater

than or equal to 0. So interval notation

looks like this. We want things that are

greater than or equal to 0. So, actually, I want to

start with this bracket. I’ll explain why in a moment. That means I’m going

to start at 0. It could be equal to 0,

and I can go all the way up to any number that’s greater

than 0 forever, so infinity. So that’s what interval

notation looks like. If this is a review for you,

then you might be able to go through this video

kind of quickly. If not, then you might need

to watch this a couple times or pause every now and then. So if this is new for you,

I usually start to think — until I get used to writing

in interval notation, I think about a number line. And I’m just going to put 0 here

at the center and, oh, you know, 5, 10, okay, negative

5, negative 10. And I want to think

about this statement — “all numbers that are

greater than or equal to 0.” So that certainly mean that

0’s valid and 5 is valid, and here in the middle at 2 1/2. Any of these numbers, all

of these numbers are valid. So because it could be equal to

0, I’m going to use a bracket. Okay? So special notation. We need specifically a bracket, and then I would shade all the

way to the right because all of these things are valid. So this bracket is

— that we put on the number line is how we

start that interval notation. This literally reads start at 0. It can be equal to 0. That’s what that bracket means. And then comma. So we separate this interval. This is where we

start the interval, and this is where we end. Well, we never really

want to end. All numbers that are

greater than 0 are valid, so this goes up to infinity. So we say up to infinity

and we put a parentheses. Infinity will always

have parentheses. Bracket means, like,

kind of a hard stop. Infinity never really

stops, so you can kind of always just go a little bit

more and a little bit closer. In this — in the next example,

g of x equal to 1 over x plus 1, we found that the domain

were all real numbers except negative 1. So, again, let’s start

with a number line. Here’s 0 in the middle,

negative 1, positive 1. Okay. Just going to make

a simple number line. And everything was

valid except negative 1, so 0 all the way up this way. So this was all valid. Negative 0.9, negative 0.999, all of these numbers

here are valid. When I evaluate g of

x at negative 0.9, I don’t get 0 in

the denominator. I get positive 1/10. I can divide 1 by positive 1/10. If I plug in negative

1.001, I’m not going to get 0 in the denominator. I’m going to get a small

number, so these are all valid. I just don’t want to include

negative 1, so not including it, I have a open parentheses. Okay? I can go all the

way this direction. I can go start at negative

1 and go more negative, but I just cannot include

— there’s a hole here. There’s a gap right

there at negative 1. So interval notation,

we go left to right. So I’m going to start

at negative infinity, so open parentheses

negative infinity. And I’m — all of these numbers

are valid up to negative 1, and I use an open —

I use a parenthesis because it cannot be

equal to negative 1. That would be a bracket, like

we saw in the above problem. And then I have more. I have another region,

so there’s two regions. It’s broken up right

here at negative 1, so I can union this together

with negative 1 all the way up to positive infinity. So I put negative 1 up

to positive infinity. Again, if this is really new, if

interval notation’s really new, our textbook has more on this in

section 1.6, so you can go back to that previous section

and look at more examples. Or you can look at the section

that we’re in, domain and range, and see some more examples. I highly recommend doing that. There’s some great examples in

our textbook about functions, domain, writing in

interval notation. Okay. For our final example,

h of x equals 3 of x plus 2, we found that the domain

is all real numbers. So if we were to

make a number line — And we — everything is valid. Everything’s valid. The whole number line is shaded. That would be the

interval negative infinity to positive infinity. So instead of writing out all

real numbers, this is kind of short number for that,

that everything from works from negative infinity

up to positive infinity. Okay. Let’s take a look

at two more examples about finding the domain and then writing it

in interval notation. So, again, domain. If we’re looking at domain, we assume that all

real numbers work, and then we start

to restrict things. And here I have a

denominator, and so I know that a denominator

cannot be equal to 0. So I take my denominator. I say this cannot be 0. When is this 0? So I solve this equation, and

— but we just have a not equal to sign, so how do I solve that? Well, x squared minus 1 factors

into x plus 1 times x minus 1. That cannot be 0. So I have this factored form. This means that neither

of these factors can be 0, so x plus 1 cannot be 0. Or x minus 1 cannot be 0, so that means x cannot

be negative 1, or x cannot be positive 1. So that means we have

two restrictions. There are two values that make

g of x have a denominator of 0. And you can check this,

so I’ll put some check out to the side here. I’ll erase this in a moment

so we have some more space. So g of x. If I say

g of negative 1, that would be negative 1 over

negative 1 squared minus 1, which is negative 1 over negative 1 squared

is positive 1 minus 1. There you see we

have a denominator of 0, which is undefined. Likewise, if I were to

consider x equal to positive 1, if I were to evaluate

g of positive 1, that would be positive 1 over

positive 1 squared minus 1, which would be still 1

minus 1 in the denominator. And that would be 0, which

would still be undefined. So either of those values

make our domain undefined, so we want to write this

in interval notation. Again, I would start with a

number line, and here we have 0, positive 1, negative 1. And we said all real

numbers are valid. Everything’s valid except

negative 1 and positive 1. So all of these are valid

until you get to positive 1. Positive 1’s not valid,

and then we can go on. So you see that we have

three different regions. So how do we write that

in interval notation? So I start from the left side. We start at — let’s see

if I can put a — hmm. I was going to try to cover

up this you try, but — if I start from the left side,

we start at negative infinity. We are valid up to negative

1, but not including it, so we use a parentheses. Union for our second interval. Negative 1 up to positive 1,

union positive 1 to infinity. So this is our domain

in interval notation for that first problem. All right. Had to clear out

some space there, make it a little bit more

space for us to write. In the second example, h of

x is equal to the square root of 2 minus x and then plus 3

outside of the square root. Again, if we’re trying

to find the domain, we assume everything works. And then we start figuring

out, okay, in this situation, this isn’t going to work. Well, we have a square root, and we know that square

roots are only valid in the real number system if you’re evaluating the

square root of things that are greater

than or equal to 0. So it’s not that x needs to

be greater or equal to 0. It’s whatever is underneath

the square root has to be greater than

or equal to 0. That needs to be greater

than or equal to 0. So I’m going to solve

this equation and figure out what values of x make

this 2 minus x bit greater than or equal to 0. So subtract 0 from both

sides, and now I need to divide both sides

by negative 1. Dividing both sides by negative

1 flips that inequality symbol, and now we have x is valid when

it’s less than or equal to 2. So let’s just check

that out to the side. Does that make sense? It’s always good —

does it make sense? So let’s say h is — well,

what would be less than 2 that we would want to check? Oh, 0’s easy to check

often, so h of 0. Square root of 2 minus 0

plus 3 outside of that. So we need to do what’s

inside the square root first. Two minus 0, square

root of 2 plus 3. Yeah, that works. I need to evaluate that,

I could with a calculator, but I can take the

square root of 2. Let’s try 3. Okay? We’re going to

make sure something — we’re going to look and see what

doesn’t work, and it says it has to be less than or equal to 2. So does 3 not work? So the square root of 2 minus

3 plus 3, that would be equal to the square root

of negative 1 plus 3. And here is where

we have a problem in the real number system. Square root of negative 1 is

not a real number, so that means that 3 is not a valid input. So now that makes me a little

more confident that x less than or equal to 2 is

our valid range. So let’s draw a number line, kind of help us write

in interval notation. Here’s 0, 1, 2. And I want all of the

numbers that are less than 2, so this direction. But we can be equal to 2, so that means there’s

a bracket here. So in interval notation, I

start from left to right. Here we have on the left

side negative infinity, so parentheses negative

infinity, up to 2 bracket because it can be equal to 2. What happens when it’s 2? We have 2 minus 2, which is 0. Square root of 0 is valid. It’s just 0 plus

3, so that works. There’s our interval

for number two. All right. Let’s talk about what

should you try before moving on to the next video. So the directions for these

problems are to find the domain. Find and write the

domain — whoop. Not writhe, write,

although maybe some of you might writhe

doing some of this. Find and write the domain

in interval notation. So question one. F of x is equal to

1 over x minus 4. Two, g of x equal to x squared. And the third function to

think about is h of x equal to the absolute value of

x. Try those before moving on to the next video and

bring those answers to class.