and what we're doing today is finding the domain

of a function.

I'm going to look at the very basic function f

of x equals x plus 1.

This is an equation of a line.

What we want to do when we're finding the domain

of a function is find the value of x that we can put

into our function so that the output will be real numbers.

So as I look at the equation on this line,

I have a line that looks like this.

I'm going to cross at positive one, and I'm going

to have a one, one slope.

[ Pause ]

And as I look at the graph of the function,

I realize that no matter what value I put in for x

that every output that I have for every y value

that I have will be ok values.

They'll all be real values.

So for this function, my domain is going to be equal to x

such that x is a real number.

This is called set builder notation.

If you don't like set builder notation, you can write it

with integral notation.

It runs from negative infinity to infinity.

So a real basic function like this, we don't have to worry

about what values of x that will be undefined

or won't be a real number, then I can say that it's going

to all be real numbers.

If my function, say, is f of x is equal to the square root

of x, then if I were to graph this function

[ Pause ]

it would look like something like that.

So as we can see from the graph that I don't get

to have any negative x values over here, and, in fact,

if I were to put a negative x value in here, a negative one

or negative two or something like that,

you'll find that you have a complex number,

and for right now, we want to stay in the real numbers.

So we know that the domain of this one is x,

such that x is greater than or equal to zero

because you can take the square root of zero.

Square root of zero is right there,

and x belongs to the real numbers.

Again, this is set builder notation,

or if you'd like to do an integral notation,

you have the domain is equal to zero to infinity, but you do get

to include zero, so it's a squared off bracket all the way

up to infinity.

This excludes any values that are less than zero.

Let's look at a more complex rational function.

I have x over x minus one.

Now, if I put in zero, I'm going to be fine.

If I put in two, I'm going to be fine.

If I put in negative two, I'm going to be fine.

If I put in one, however,

[ Pause ]

I'm going to be undefined.

So I don't really care what's going on on the top.

I can put in any number I wanted on the top, but what I do want

to prevent is I want to find prevent the denominator

from being zero.

So what I want to do is I want

to find the restrictions only on the denominator.

So since the denominator x minus one not equal to zero,

and then I solve for x. So I have x minus one can't be zero.

Adding one to both sides that means x cannot be one.

This is my restriction.

I build my domain for my restrictions.

[ Pause ]

So what I've got is my domain is equal to x

such that x cannot equal one, and x is a real number.

So that's the domain for that rational function.

Let's look at another function.

We have x minus two over x squared minus 4x minus 5.

Now right now I've got to decide what values

in the denominator would make this to be zero

because that's what x cannot be.

Those are my restrictions.

At this point, I can't always clearly see what it is,

so I have to factor this.

So I know that this factors into x minus five multiplied

by x minus one, x plus one.

Now, looking at this, I know that this factor

of the denominator cannot be zero,

and I know that this factor of the denominator cannot be zero.

These will give me my restrictions.

X minus five cannot be zero, and x plus 1 cannot be zero.

So x cannot be 5, and x cannot be negative one.

These are my restrictions.

[ Pause ]

and from my restrictions, I build my domain.

My domain is equal to x such that x is not equal

to five nor negative one, but it is a real number.

Again, I don't care about what's happening in the numerator,

because a numerator I don't care if it's zero or not.

I only want to make sure

that the denominator is not equal to zero.

What I don't want is the denominator equal to zero.

I just have to prevent zero from being in the denominator