Domain of a function | Functions and their graphs | Algebra II | Khan Academy
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Domain of a function | Functions and their graphs | Algebra II | Khan Academy


Welcome to my presentation
on domain of a function. So what’s is the domain? The domain of a function,
you’ll often hear it combined with domain and range. But the domain of a function is
just what values can I put into a function and get
a valid output. So let’s start with
something examples. Let’s say I had f of x is equal
to, let’s say, x squared. So let me ask you a question. What values of x can I put
in here so I get a valid answer for x squared? Well, I can really put anything
in here, any real number. So here I’ll say that the
domain is the set of x’s such that x is a member
of the real numbers. So this is just a fancy way of
saying that OK, this r with this kind of double backbone
here, that just means real numbers, and I think you’re
familiar with real numbers now. That’s pretty much every number
outside of the complex numbers. And if you don’t know
what complex numbers are, that’s fine. You probably won’t need
to know it right now. The real numbers are every
number that most people are familiar with, including
irrational numbers, including transcendental numbers,
including fractions — every number is a real number. So the domain here is x —
x just has to be a member of the real numbers. And this little backwards
looking e or something, this just means x is a member
of the real numbers. So let’s do another one
in a slight variation. So let’s say I had f of x is
equal to 1 over x squared. So is this same thing now? Can I still put any x
value in here and get a reasonable answer? Well what’s f of 0? f of zero is equal to 1 over 0. And what’s 1 over 0? I don’t know what it is,
so this is undefined. No one ever took the trouble to
define what 1 over 0 should be. And they probably didn’t do, so
some people probably thought about what should be, but they
probably couldn’t find out with a good definition for 1 over
0 that’s consistent with the rest of mathematics. So 1 over 0 stays undefined. So f of 0 is undefined. So we can’t put 0 in and get
a valid answer for f of 0. So here we say the domain is
equal to — do little brackets, that shows kind of the
set of what x’s apply. That’s those little curly
brackets, I didn’t draw it that well. x is a member of the real
numbers still, such that x does not equal 0. So here I just made a slight
variation on what I had before. Before we said when f of x is
equal to x squared that x is just any real number. Now we’re saying that x is any
real number except for 0. This is just a fancy way of
saying it, and then these curly brackets just mean a set. Let’s do a couple more ones. Let’s say f of x is equal to
the square root of x minus 3. So up here we said, well this
function isn’t defined when we get a 0 in the denominator. But what’s interesting
about this function? Can we take a square root
of a negative number? Well until we learn about
imaginary and complex numbers we can’t. So here we say well, any x is
valid here except for the x’s that make this expression under
the radical sign negative. So we have to say that x minus
3 has to be greater than or equal to 0, right, because you
could have the square to 0, that’s fine, it’s just 0. So x minus 3 has to be greater
than or equal to 0, so x has to be greater than or equal to 3. So here our domain is x is a
member of the real numbers, such that x is greater
than or equal to 3. Let’s do a slightly
more difficult one. What if I said f of x is equal
to the square root of the absolute value of x minus 3. So now it’s getting a little
bit more complicated. Well just like this time
around, this expression of the radical still has to be
greater than or equal to 0. So you can just say that the
absolute value of x minus 3 is greater than or equal to 0. So we have the absolute value
of x has to be greater than or equal to 3. And if order for the absolute
value of something to be greater than or equal to
something, then that means that x has to be less than or equal
to negative 3, or x has to be greater than or equal to 3. It makes sense because x
can’t be negative 2, right? Because negative 2 has an
absolute value less than 3. So x has to be less
than negative 3. It has to be further in the
negative direction than negative 3, or it has to be
further in the positive direction than positive 3. So, once again, x has to be
less than negative 3 or x has to be greater than 3,
so we have our domain. So we have it as x is
a member of the reals — I always forget. Is that the line? I forget, it’s either
a colon or a line. I’m rusty, it’s been
years since I’ve done this kind of stuff. But anyway, I think
you get the point. It could be any real number
here, as long as x is less than negative 3, less than or
equal to negative 3, or x is greater than or equal to 3. Let me ask a question now. What if instead of this it was
— that was the denominator, this is all a separate
problem up here. So now we have 1 over the
square root of the absolute value of x minus 3. So now how does this
change the situation? So not only does this
expression in the denominator, not only does this have to be
greater than or equal to 0, can it be 0 anymore? Well no, because then you would
get the square root of 0, which is 0 and you would get a
0 in the denominator. So it’s kind of like
this problem plus this problem combined. So now when you have 1 over the
square root of the absolute value of x minus 3, now it’s no
longer greater than or equal to 0, it’s just a greater
than 0, right? it’s just greater than 0. Because we can’t have a 0
here in the denominator. So if it’s greater than 0, then
we just say greater than 3. And essentially just get rid of
the equal signs right here. Let me erase it properly. It’s a slightly different
color, but maybe you won’t notice. So there you go. Actually, we should do another
example since we have time. Let me erase this. OK. Now let’s say that f of x is
equal to 2, if x is even, and 1 over x minus 2 times
x minus 1, if x is odd. So what’s the domain here? What is a valid x I
can put in here. So immediately we
have two clauses. If x is even we use this
clause, so f of 4 — well, that’s just equal to 2 because
we used this clause here. But this clause applies
when x is odd. Just like we did in the last
example, what are the situations where this
kind of breaks down? Well, when the
denominator is 0. Well the denominator is 0
when x is equal to 2, or x is equal to 1, right? But this clause only
applies when x is odd. So x is equal to 2 won’t
apply to this clause. So only x is equal to 1
would apply to this clause. So the domain is x is a member
of the reals, such that x does not equal 1. I think that’s all the
time I have for now. Have fun practicing
these domain problems.

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