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[MUSIC] So, before we talk about the domain of the square root function, we

just want to remind ourselves what the square root function even is.

So here, I've made a graph of the square root function.

And along the x-axis, I plot the numbers one to sixteen and in the y-axis I've got

the numbers one through four. And then in this green curve here, I've

plotted the the, the square root function.

What is the square root, right? Well, here's an example.

Here, I've got the square root of four. And I'm saying the square root of four is

two. What that means, is if I take the number

two and I square it, I get back four. I don't know, if I move over to, say, the

square root of nine, I get three. And that's because three squared is nine,

alright, or if I move over a little bit further,

the square root of sixteen is four. And that's because four squared is

sixteen. I know there's some crazier values, too.

If I move over here to the square root of two,

well, the square root of two is this sort of crazy number 1.414213 blah, blah,

blah. And maybe it's a little bit surprising,

that if I take that number and square it, I get back two.

So, what's going on here, alright? The square root function takes a number

and spits out a new number, that new number when you multiply it by itself and

you square it, you get back your original number.

Now, here's the question, what sorts of numbers can I take the square root of?

That's asking the question, what's the domain of the square root function?

Now, that we've seen the graph, let's try to write down in words a definition of

the square root function. So, in light of what we just seen, you

might think that the definition of the function f(x) equals square root of x is

a number which squares to x. There's a problem with this though.

Take a look at say, f(9). What would f(9) be?

Well, if you're thinking the square root of x is a number which squares to x, then

you might think that f(9) would be -3, alright?

Because -3^2 is 9. But then, you might also think that half

of nine should be three, right? Because 3^2 is also 9.

This is bad, alright? A function is supposed to be unambiguous.

It's supposed to have one output for each input.

If you take this as the definition of the square root function, just any number

which squares to x, you've introduced some ambiguity, alright?

What's the square root of nine? Is it -3 or is it +3?

Both of those numbers square to nine. So, this is, this is bad, alright?

The solution is to change the definition. Instead of having the the square root

function be just a number which squares to x, you're going to take it to be the

nonnegative number which squares to x. This is better, alright?

In our example here, if I only am allowed to choose the nonnegative number, which

squares to x, then f(9) equals -3, well, -3 is not nonnegative, -3 is negative.

So, that means that this isn't the case, right?

All I'm left with is f(9) = 3, right? Three is the nonnegative number which

squares to nine. Alright.

So, this will be our definition for for, for the square root function.

The square root of x is the nonnegative number which squares to x.

There's one particular place where this plays out and it's extraordinarily

important. So, let's take a look at that now.

We've got our definition. The squared of x is the nonnegative

number which squares to x. Now, there's one popular misconception

that comes up because of this definition. So, in light of the definition of the

square root, right, the square root of a number being the

nonnegative number which squares the number to the radical, you might be

tricked into thinking that the square root of x squared is x.

That's not true and let's see why. Let's do a specific example where say, x

is -4. So, if I replace the x's here by -4, the

left hand side is the square root of -4 squared,

right? Square root of x squared but with x

replaced with -4. Now, - 4 * -4 is 16.

This is the square root of 16 and the square root of sixteen, the definition of

the square root is the nonnegative number which squares to 16.

There's two numbers that square to 16, +4 and -4.

But the square root is by convention, the nonnegative one, so this is equal to 4.

Duh, look at what happened.

-4, square root of -4^2 + 4, that's the x over here.

This is not true, right. You should not be tricked into thinking

that that's the case. Instead, something else is true,

right? What is true is this.

The square root of x squared is the absolute value of x.

And that works in this specific case, right?

When x is -4, the square root of -4 squared, the square root of 16 is 4.

And 4 really is the absolute value of -4. Alright. So, this is a mistake that comes

up quite a bit. People are often tricked into thinking

that the square root of x squared is just x,

alright? They're just trying to cancel the square roots in the squaring.

That's not possible. Instead, what is true is the square root

of x^2 is the absolute value of x. So, we've got a definition of the square

root function and we've seen that the square root of x^2 is not just x,

it's the absolute value of x. Now,

that doesn't actually address the original question, right?

The original question is, what's the domain of this square root function?

What sorts of numbers can I take root of? For instance, can I take the square root

of a negative number? Let's see why not.

Very concretely. Does it make sense, say, to talk about

the square root of -16? Well, if it did that would be some

number. So, I'll call that number k for crazy,

alright? And what do I know about that number k?

Well, k^22 would have to be -16. Remember, the definition of the square

root function? It's a number that I square to get back

the original number. So, if there were a square root of -16,

when I square it, I get back -16. And imagining here that k is some real

number. And that means there's three

possibilities. Either k is positive, k is zero, or k is

negative. If k is positive, then k squared would

also be positive because a positive number times a positive number is still

positive. But that can't be, because k squared is

supposed to be -16. So, this first possibility doesn't

happen. Now, if k were zero, then k squared would

be zero, but k squared is supposed to be -16.

So, k isn't zero. Is k negative?

Well then, what's k squared? That would be a negative number times a

negative number, and that would still be positive.

And that can't be because k squared is supposed to be -16.

So, this possibility also doesn't happen. So, all of our possibilities have been

eliminated, alright? There can't be a real number k,

which is the square root of -16. Because if k were positive, k squared

would be positive but k squared has to be negative.

k can't be zero because then k squared isn't negative and k can't be negative

because then k squared is positive but k squared is supposed to be negative,

alright? The upshot is that it just doesn't make any sense to talk about the

square root of a negative number. In contrast, it does make sense to talk

about the square root of zero, which is just zero,

zero squared is zero. And it also makes sense to talk about the

square root of positive numbers. So, to summarize the situation, we can

say that the domain of the square root function is all the numbers between zero

and infinity, including zero. So, I'm using the square bracket.

But, of course, not including infinity because infinity is not a number.

Sometimes, you're asked to calculate the domain of a function that's more

complicated than, than just the square root of x.

Let's see an example of that. So, let's try this.

Let's try to find the domain of this function g, which is the square root of

2x + 4. And remember, the domain consists of all

the inputs for which the rule makes sense.

So, I just have to think which x values makes sense for this rule?

Well, in order to take the square root of 2x + 4, I'm going to need that 2x + four

is not negative because I can't take the square root of a negative number so I

need to guarantee that 2x + 4 is not negative, meaning greater than or equal

to zero. Now, I can subtract four from both sides

and I get that 2x is at least -4. Then, I can divide both sides by two.

Two is positive, so it doesn't change the inequality.

x is bigger than or equal to -2. So, as long as x is at least -2, then 2x

+ 4 is at least zero, which means it makes sense to take the square root.

So, I can summarize the situation, the domain of g consists of all numbers

greater than or equal to -2. This is our notation for that.

I used a square bracket to include the -2 and the round bracket here on the

infinity, because infinity is not number, it's not part of the domain.

So, that example was a little bit harder. Let's do an even harder example where

I've got multiple square roots, all right, the square root of something plus

the square root of something. And let's figure out the domain of this

function that has two separate square roots.

This is the function T(x) equals the square root of 1 - x plus the square root

of 1 + x. Now, in order for this rule to make

sense, I have to be able to take this square root and also take this square

root. In other words, in order to do this first

square root, I'm going to need that 1 - x is bigger than or equal to zero,

alright? I need the thing under the square root to

be nonnegative in order to do a square root.

In order to take this square root, I need 1 + x to be bigger than or equal to zero.

And both of these things have to be true in order to take both of these square

roots and then add them together. So, I'll put an and between them.

Now, I go to x to both sides and this inequality and I get one is bigger than

or equal to x. And I can subtract one from both sides of

this and I'll get x is bigger or equal to -1.

And again, both of these things have to happen, right?

I need x less than one and x bigger than or equal to -1 in order to evaluate this

function. Let me write this in in a more reasonable

way, right?

Instead of writing one bigger than or equal to x, I can write what I just said,

x less than one. And here, I'll write, this is x bigger

than or equal to -1. Now, I could write these inequalities as

something about an interval. I could say that x is in the interval -1

to 1, alright?

To say that x is less than one and bigger than or equal to -1, exactly means that

your inside this interval. And I'm using square brackets here,

because I've got greater than or equal to, less than or equal to.

And then, I can summarize the situation by writing the domain of T is this

interval, alright?

And this is describing the values of x for which this rule makes sense at the

domain of the function T. Let's do one more example.

some square root problem where I've also got an x squared term.

Let's calculate the domain of this function C.

C of x is the square root of 1 - x^2. So, the domain consists of all the inputs

for which the rule makes sense. So, I'm looking for which values of x

make the thing under the square root nonnegative.

There's lots of different ways to think about which values of x make this true.

one way is to factor 1 - x^2. So, I could factor 1 - x^2 as 1 + x * 1 -

x, alright? That is equal to 1 - x^2.

I'm looking for when that's nonnegative. This is a little bit easier to think

about because now, I just got to figure out when these two terms have the same

sign, alright?

When they're both positive or they're both negative, then their product is

bigger than or equal to zero. So, to think about that,

I'll draw a number line. And I'll first think about when 1 + x is

positive and negative. So, something special happens at -1,

alright? When x is minus one, 1 + x is zero.

When x is less than -1, 1 + x is negative.

And when x is bigger than -1, 1 plus x is positive.