[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]