[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. 
Alright. Now, compare this with 1 - x, 
alright? 1 - x, something exciting 
happens at one, alright When X is less 
then one, 1 - x is positive. 
And when x is bigger than one, 1 - x is 
negative. 
Now, I'm not trying really to understand 
1 + x or 1 - x, I'm trying to understand 
their product. 
So, when I multiply those two together, I 
get 1 - x^2 and I want to know, you know, 
when is that positive or or negative. 
Let me mark down the special points -1 
and 1. 
And now, 1 - x^2 is the product of these 
so I can think about various values of x. 
So, when x is less than -1, then 1 + x is 
negative and 1 - x is positive, and a 
negative number times a positive number 
is negative. 
When x is between -1 and 1, then 1 + x is 
positive and 1 - x is also positive in 
that region, so their product is 
positive. 
And when x is bigger than one, 1 + x is 
positive and 1 - x is negative, so their 
product is negative. 
Now, this gets me most of the way there, 
alright? 
Because what I'm trying to understand is 
when this product is nonnegative and I 
can see that it's positive in this 
region, I could also think about what 
happens when I plug in -1 and 1. 
When I plug in -1, I get 1 - 1 which is 
zero. 
And when I plug in one, I get 1 - 1 which 
is zero. 
So, the function is, in fact, is zero in 
between here at -1 and 1. 
So, I'm just trying to figure out which 
values of x make 1 - x^2 nonnegative. 
Well, -1, 1, and anything in between. 
So, one way to summarize the situation is 
to say that the domain of my function C 
consists of all real numbers between -1 
and 1, including -1 and 1. 
So, I'm using the square brackets. 
As long as x is inside here, 
then 1 - x^2 is nonnegative. 
That means it makes sense to take the 
square root and that's the domain of C. 
[MUSIC]