Natalie Denny MTH 95 “Writing Domain in Interval Notation”

>>Natalie Denny: Welcome back. Now we’re going to look at writing domain in
interval notation. Let’s take a look at f of x is
equal to the square root of x. So in the last video, we
talked about the domain of this function are all numbers that are greater
than or equal to 0. So interval notation
looks like this. We want things that are
greater than or equal to 0. So, actually, I want to
start with this bracket. I’ll explain why in a moment. That means I’m going
to start at 0. It could be equal to 0,
and I can go all the way up to any number that’s greater
than 0 forever, so infinity. So that’s what interval
notation looks like. If this is a review for you,
then you might be able to go through this video
kind of quickly. If not, then you might need
to watch this a couple times or pause every now and then. So if this is new for you,
I usually start to think — until I get used to writing
in interval notation, I think about a number line. And I’m just going to put 0 here
at the center and, oh, you know, 5, 10, okay, negative
5, negative 10. And I want to think
greater than or equal to 0.” So that certainly mean that
0’s valid and 5 is valid, and here in the middle at 2 1/2. Any of these numbers, all
of these numbers are valid. So because it could be equal to
0, I’m going to use a bracket. Okay? So special notation. We need specifically a bracket, and then I would shade all the
way to the right because all of these things are valid. So this bracket is
— that we put on the number line is how we
start that interval notation. This literally reads start at 0. It can be equal to 0. That’s what that bracket means. And then comma. So we separate this interval. This is where we
start the interval, and this is where we end. Well, we never really
want to end. All numbers that are
greater than 0 are valid, so this goes up to infinity. So we say up to infinity
and we put a parentheses. Infinity will always
have parentheses. Bracket means, like,
kind of a hard stop. Infinity never really
stops, so you can kind of always just go a little bit
more and a little bit closer. In this — in the next example,
g of x equal to 1 over x plus 1, we found that the domain
were all real numbers except negative 1. So, again, let’s start
with a number line. Here’s 0 in the middle,
negative 1, positive 1. Okay. Just going to make
a simple number line. And everything was
valid except negative 1, so 0 all the way up this way. So this was all valid. Negative 0.9, negative 0.999, all of these numbers
here are valid. When I evaluate g of
x at negative 0.9, I don’t get 0 in
the denominator. I get positive 1/10. I can divide 1 by positive 1/10. If I plug in negative
1.001, I’m not going to get 0 in the denominator. I’m going to get a small
number, so these are all valid. I just don’t want to include
negative 1, so not including it, I have a open parentheses. Okay? I can go all the
way this direction. I can go start at negative
1 and go more negative, but I just cannot include
— there’s a hole here. There’s a gap right
there at negative 1. So interval notation,
we go left to right. So I’m going to start
at negative infinity, so open parentheses
negative infinity. And I’m — all of these numbers
are valid up to negative 1, and I use an open —
I use a parenthesis because it cannot be
equal to negative 1. That would be a bracket, like
we saw in the above problem. And then I have more. I have another region,
so there’s two regions. It’s broken up right
here at negative 1, so I can union this together
with negative 1 all the way up to positive infinity. So I put negative 1 up
to positive infinity. Again, if this is really new, if
interval notation’s really new, our textbook has more on this in
section 1.6, so you can go back to that previous section
and look at more examples. Or you can look at the section
that we’re in, domain and range, and see some more examples. I highly recommend doing that. There’s some great examples in
our textbook about functions, domain, writing in
interval notation. Okay. For our final example,
h of x equals 3 of x plus 2, we found that the domain
is all real numbers. So if we were to
make a number line — And we — everything is valid. Everything’s valid. The whole number line is shaded. That would be the
interval negative infinity to positive infinity. So instead of writing out all
real numbers, this is kind of short number for that,
that everything from works from negative infinity
up to positive infinity. Okay. Let’s take a look
at two more examples about finding the domain and then writing it
in interval notation. So, again, domain. If we’re looking at domain, we assume that all
real numbers work, and then we start
to restrict things. And here I have a
denominator, and so I know that a denominator
cannot be equal to 0. So I take my denominator. I say this cannot be 0. When is this 0? So I solve this equation, and
— but we just have a not equal to sign, so how do I solve that? Well, x squared minus 1 factors
into x plus 1 times x minus 1. That cannot be 0. So I have this factored form. This means that neither
of these factors can be 0, so x plus 1 cannot be 0. Or x minus 1 cannot be 0, so that means x cannot
be negative 1, or x cannot be positive 1. So that means we have
two restrictions. There are two values that make
g of x have a denominator of 0. And you can check this,
so I’ll put some check out to the side here. I’ll erase this in a moment
so we have some more space. So g of x. If I say
g of negative 1, that would be negative 1 over
negative 1 squared minus 1, which is negative 1 over negative 1 squared
is positive 1 minus 1. There you see we
have a denominator of 0, which is undefined. Likewise, if I were to
consider x equal to positive 1, if I were to evaluate
g of positive 1, that would be positive 1 over
positive 1 squared minus 1, which would be still 1
minus 1 in the denominator. And that would be 0, which
would still be undefined. So either of those values
make our domain undefined, so we want to write this
in interval notation. Again, I would start with a
number line, and here we have 0, positive 1, negative 1. And we said all real
numbers are valid. Everything’s valid except
negative 1 and positive 1. So all of these are valid
until you get to positive 1. Positive 1’s not valid,
and then we can go on. So you see that we have
three different regions. So how do we write that
in interval notation? So I start from the left side. We start at — let’s see
if I can put a — hmm. I was going to try to cover
up this you try, but — if I start from the left side,
we start at negative infinity. We are valid up to negative
1, but not including it, so we use a parentheses. Union for our second interval. Negative 1 up to positive 1,
union positive 1 to infinity. So this is our domain
in interval notation for that first problem. All right. Had to clear out
some space there, make it a little bit more
space for us to write. In the second example, h of
x is equal to the square root of 2 minus x and then plus 3
outside of the square root. Again, if we’re trying
to find the domain, we assume everything works. And then we start figuring
out, okay, in this situation, this isn’t going to work. Well, we have a square root, and we know that square
roots are only valid in the real number system if you’re evaluating the
square root of things that are greater
than or equal to 0. So it’s not that x needs to
be greater or equal to 0. It’s whatever is underneath
the square root has to be greater than
or equal to 0. That needs to be greater
than or equal to 0. So I’m going to solve
this equation and figure out what values of x make
this 2 minus x bit greater than or equal to 0. So subtract 0 from both
sides, and now I need to divide both sides
by negative 1. Dividing both sides by negative
1 flips that inequality symbol, and now we have x is valid when
it’s less than or equal to 2. So let’s just check
that out to the side. Does that make sense? It’s always good —
does it make sense? So let’s say h is — well,
what would be less than 2 that we would want to check? Oh, 0’s easy to check
often, so h of 0. Square root of 2 minus 0
plus 3 outside of that. So we need to do what’s
inside the square root first. Two minus 0, square
root of 2 plus 3. Yeah, that works. I need to evaluate that,
I could with a calculator, but I can take the
square root of 2. Let’s try 3. Okay? We’re going to
make sure something — we’re going to look and see what
doesn’t work, and it says it has to be less than or equal to 2. So does 3 not work? So the square root of 2 minus
3 plus 3, that would be equal to the square root
of negative 1 plus 3. And here is where
we have a problem in the real number system. Square root of negative 1 is
not a real number, so that means that 3 is not a valid input. So now that makes me a little
more confident that x less than or equal to 2 is
our valid range. So let’s draw a number line, kind of help us write
in interval notation. Here’s 0, 1, 2. And I want all of the
numbers that are less than 2, so this direction. But we can be equal to 2, so that means there’s
a bracket here. So in interval notation, I
start from left to right. Here we have on the left
side negative infinity, so parentheses negative
infinity, up to 2 bracket because it can be equal to 2. What happens when it’s 2? We have 2 minus 2, which is 0. Square root of 0 is valid. It’s just 0 plus
3, so that works. There’s our interval
for number two. All right. Let’s talk about what
should you try before moving on to the next video. So the directions for these
problems are to find the domain. Find and write the
domain — whoop. Not writhe, write,
although maybe some of you might writhe
doing some of this. Find and write the domain
in interval notation. So question one. F of x is equal to
1 over x minus 4. Two, g of x equal to x squared. And the third function to
think about is h of x equal to the absolute value of
x. Try those before moving on to the next video and
bring those answers to class.