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How to Find the Domain of Any Function (NancyPi)


Hi guys, I’m Nancy. And I’m going to show
you how to find the domain of any function. The domain is kind of an annoying question
that you’ll get about a function, but it can be very simple. And all it means is all possible
x values that you can have in your function. So all allowable x values that you can have
in your function. To do this you should already know how to solve a quadratic by factoring.
And how to solve an inequality. Okay so these are the five types of functions you might
have to find the domain for. I’m going to go through all of them. If you want to skip
ahead you can. Just look below in the description for the time to skip to. The first case is
really easy. If you have a polynomial – just an x expression like x squared plus 3x plus
1, no square roots, no fractions, your domain answer is just all real numbers. And you can
write that phrase “all real numbers” or you can write interval notation: parenthesis negative
infinity comma positive infinity, end parenthesis. Now if you don’t have just a polynomial and
you have something more complicated like a fraction or a square root or possibly both,
it’s going to be one of these four cases. it’s either just a fraction, it’s a square
on its own, it could be fraction with a square root in the bottom, or a fraction with a square
root in the top (the numerator). And I’m going to explain how to find the domain for all
of those cases. Okay, say you have just a fraction. To find the domain all you have
to do is take the bottom, whatever that may be, and set it not equal to zero. So in this
example we have x squared plus 5x plus 6 “not equal to zero. Notice that it doesn’t matter
what the top is, just take the bottom and set it not equal to zero. And you can solve
this just as if it were a normal equality. And that involves factoring. If you need help
with factoring or solving by factoring you can look at one of my other videos. So in
this equation we get two restrictions two solutions: x cannot equal negative 3 and x
cannot equal negative 2. So when we want to write the domain, the final answer for domain,
you can write it as all real numbers but x cannot equal negative 3 and cannot equal negative
2. Or you can use interval notation and write it as everything from negative infinity, all
the way up to negative 3, not including negative 3, combined with everything from negative
3 to negative 2, not inclusive of either number, combined with everything from negative 2 all
the way up to infinity. So either of these notations is fine – they mean the same thing.
Let’s look at another example. What if you’re given something like this? Where the denominator
is x squared plus 3. Same thing: take the bottom, set it not equal to zero because we
cannot have the denominator equal zero. It’s not defined when it is. Then when you solve
this equation, you end up with x squared cannot equal negative 3. But x squared can never
equal negative 3, so we just can’t have that ever. There’s no x value that would ever make
that happen, so you don’t need to worry about that restriction. All you need to write in
that case would just be your normal all real numbers domain. So you can say all real numbers.
You could say capital R symbol for all real numbers. Or you could say, with interval notation,
everything from negative infinity to positive infinity, with parentheses on either side.
So that would be your three ways of writing the domain for this fraction. Now let me show
you what to do if you have just a square root in your function. I’m going to focus on a
square root, but this is true for any even root – fourth root, sixth root, etc. It’s
not true for odd roots like third root. If you have a square root, or an even root, you
take the expression underneath the radical sign, which is the radicand (everything that’s
underneath the root symbol), so in this case x plus 1, and you need it to be greater than
or equal to zero, because we can’t have a negative number underneath a root. You can
only have a positive number or zero. So what’s underneath must be greater than or equal to
zero. Now you’re going to solve that normally, as you would solve any inequality, and you
get the restriction x is greater than or equal to negative 1. So your domain would be all
real numbers, x greater than or equal to negative 1. Or you can write it in interval notation
as everything from negative 1, all the way up to infinity. Notice that I used a square
bracket for negative 1 because you’re including negative 1, and everything all the way up
to infinity, parenthesis. So either of these ways is fine, as notation for the domain.
Okay let me show you quickly also an example with a quadratic expression under the root.
So here we have x squared plus 5x plus 6 underneath the square root symbol. You still do just
take the expression underneath the root symbol and require that it be greater than or equal
to zero. Okay you need to solve this inequality. It’s a quadratic inequality, which is more
complicated than the last example. If you don’t know how to solve that, you can look
it up somewhere else. Once you have solved this quadratic inequality, you will get that
x can be less than or equal to negative 3, or x can be greater than or equal to negative
2. So your domain you can write either as all real numbers but x less than or equal
to negative 3, x greater than or equal to negative 2. If you were going to write this
in interval notation, it would look like everything from negative infinity all the way up to negative
3, including negative 3, and everything from negative 2 all the way to positive infinity.
So either of those ways are fine to write your domain answer. Okay, if you have a square
root in the bottom of your fraction, all you need to do is take the expression under the
root, x plus 1, and set it greater than zero. Now usually with a square root, we would set
it greater than or equal to zero, but we can’t have that the denominator equal zero, so we
leave that part out. So if you see the square root in the bottom, all you need to do is
take the expression under the root and set it greater than zero, and solve. So in this
case, you subtract one from each side, and you would get that x is greater than negative
1, so your domain would be
all real numbers, x greater than negative 1, or if you wanted to use interval
notation, you would write parenthesis negative 1, everything from negative 1, not including
negative 1, all the way up to infinity. So either of these ways is fine to write your
domain answer. Okay, if your square root is in the top of your fraction, you’ll have two
restrictions. The first one will be that what’s under the square root has to be greater than
or equal to zero. So x plus 1 greater than or equal to zero. And when you solve that
by subtracting 1 from each side, minus 1, minus 1, you get x greater than or equal to
negative 1. The other restriction will come from the bottom. That the bottom can’t be
zero. So we have x squared minus 4 cannot equal zero. Now to solve that as before, you
would have to factor, so you have x plus 2, times x minus 2, can’t be zero. So I’ve re-written
that as a product of two factors, and when you solve each of them separately, set each
of them not equal to zero, you get x cannot be negative 2, and x cannot be 2. Okay so
we have this restriction from the root, and this restriction from the bottom. What you
have to do in this case is take the intersection of the two. So if you think about this, x
has to be greater than or equal to negative 1, which already takes care of negative 2
– if x is greater than or equal to negative 1, we can’t have negative 2 anyway, so you
can ignore that one, throw it away. All you need to write for your domain: x greater than
or equal to negative 1, except it cannot be 2, positive 2, and in interval notation that
would be that you can have everything from negative 1, including negative 1, up to 2
but not including 2, and then everything from 2, not including 2, up to infinity. So either
of these notations is fine, and that’s your answer for the domain. So I hope that was
super fun, finding the domain of your function. It’s okay if you didn’t have fun. You don’t
have to like math. But you can like my video, so if you did, please click like below.

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