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so how do you find the domain of a
function so consider the function 2x
minus 7 what is the domain of this
function what is the list of all
possible X values that can exist in this
function whenever you have a linear
function like the one that's listed the
domain is all real numbers so in
interval notation X could be anything
it could range from any value from
negative infinity to positive infinity
likewise if you have a quadratic
function like x squared plus 3x minus 5
the domain is still our raw numbers or
if you have a polynomial function such
as 2x cubed minus 5x squared plus 7x
minus 3 the domain is the same it's all
real numbers so if there are no
fractions or square roots if you just
have a simple polynomial function this
is going to be the domain now what about
if we have a rational function let's say
if we have a fraction like 5 divided by
X minus 2 how can we find the range I
mean other range but the domain of this
function in this function X can be
anything except a value that's going to
produce a zero in the denominator so for
instance X minus 2 cannot equal zero so
therefore X can't be positive 2 because
if you plug in 2 to minus 2 to 0 and
whenever you have a 0 and the
denominator is undefined you can have a
vertical asymptote so for rational
functions set the denominator not equal
to zero and then you could find the
value of x so how do you represent this
using interval notation so if we draw a
number line X could be anything except 2
so at 2 we're going to have an open
circle it can be greater than 2 or it
can be less than 2 all the way to left
you have negative
Finity all the way to the right positive
infinity so for the left side axe could
be anything from negative infinity to 2
but not including you or it could be
anything from 2 to infinity and so
that's how you can write the domain use
the interval notation for this example
let's try another example let's say if
we have 3x minus 8 divided by x squared
minus 9x plus 20 so we have another
rational function seen by the fraction
that we have so what we need to do just
like before by the way you could try
this problem if you want to we need to
set this not equal to zero so x squared
minus 9x plus 20 cannot equal zero so
how can we find the x values that will
produce a zero in the denominator
well we need to do is we need to factor
this trinomial so what you want to do is
you want to find two numbers that
multiply to 20 but add to the middle
coefficient negative 9 so we know that 4
times 5 is 20
but they add up to 9 so we have to use
negative 4 and negative 5 which stills
multiplies to a positive 20 but add up
to negative 9 so therefore X minus 4
times X minus 5 cannot equal 0 so we can
say that X minus 4 cannot be 0 and X
minus 5 it cannot be 0 and the first one
let's add 4 to both sides so X can be 4
and for the second one X can be 5 now
how do we represent this in interval
notation what I like to do is plot
everything on a number line so if X
can't equal 4 I'm gonna put an open
circle and it can't equal 5 either but
it can be anything else
so now let's write the domain so from
this section its from negative infinity
to 4 but it does include 4 and then
Union we have the second section which
goes from four to five and then Union
the last section which is five to
infinity so X could be anything except
four and five now what about this
example two X minus three divided by x
squared plus four go ahead and find the
domain so let's begin by setting X
square plus four not equal to zero so if
we subtract both sides by four
we'll get this x squared cannot equal
negative four now this will never happen
whenever you square a number you're
gonna get a positive number not a
negative number for example three times
three is nine and negative three times
negative 3 is positive 9 so x squared
will never equal negative 4 so therefore
regardless of what x value you choose
the denominator will never be zero if
you plug into your denominator will be
two squared plus four which is eight and
if you plug in negative two is still
then be eight if you plug in zeros come
before it will never equals zero in the
denominator so therefore for this
particular rational function it's all
real numbers the domain is from negative
infinity to positive infinity now what
if you encounter a square root problem
so for example what is the domain of the
square root of x minus four how can we
find the answer now for square roots or
any radical where the index numbers even
you cannot have a negative number on the
inside if it's odd it could be anything
it's our own numbers but for even
radicals or radicals of even index
numbers you have to set the inside and
greater than or equal to zero it can't
be negative
so for this one only needs to do is add
four to both sides so X is equal to or
greater than four to represent that with
a number line we're gonna have a closed
circle this time so it could be equal to
or greater than so we're going to shade
to the right so to the right we have
positive infinity so the domain is going
to be from 4 to infinity since it
includes 4 Nitze use a bracket in this
case now what about a problem that looks
like this
the square root of x squared plus 3x
minus 28 how can we find the domain of
this function so just like before we're
going to set the inside of the square
root function equal to or greater than 0
now we need to factor so let's find two
numbers that multiply to negative 28 but
that add to 3 so we have 7 + 4 now I'll
need to add up to positive 3 so we're
gonna use positive 7 and negative 4 7
plus negative 4 is positive 3 and 7
times negative 4 is negative 28 so it's
a factor it's gonna be X minus 4 times X
plus 7 so X can equal 4 and X can equal
negative 7 now what I'm gonna do is make
a number line with these two values
now negative seven and four are included
so let's put a closed circle now for
this type of problem we need to be
careful we need to find out which of
these three regions will work so we'll
need to check the signs we need to see
which one is positive and which one's
negative so four let's check this region
first if we pick a number that's greater
than four like five and if we plug it
into this expression will it be positive
or negative well if we plug in five 5
minus 4 is a positive number and 5 plus
7 is a positive number when you multiply
two positive numbers together you're
going to get a positive result now if we
pick a number between negative 7 & 4
let's say 0 and plug it in 0 minus 4 is
negative 0 plus 7 is positive a negative
number times a positive number is a
negative number so if we choose any
number in this region it's going to give
us a negative rezone now if we choose a
number that's less than negative 7 like
a negative 8 negative 8 minus 4 is
negative negative 8 plus 7 is negative
when you multiply two negative numbers
you're going to get a positive result
now we can't have any negative numbers
inside the square root symbol so
therefore we're not going to have any
solution in that region so therefore we
could only shade the positive regions so
now we can have the answer so X can be
less than negative 7 that's to the left
less than or equal to negative 7 or X
can be equal to or greater than 4 now to
represent this using interval notation
it's gonna be from negative infinity to
negative 7 and then Union we're gonna
start back up at 4 to infinity and we
need to use brackets at 7 I mean
negative 7 & 4
because it include those two points we
have a close circle there so that's how
you could find the domain of this type
of function now sometimes you may have a
fraction with a square root so what do
you do if the square root is in the
denominator of the fraction now
if the square root was not in the
denominator we would set the inside
equal to and greater than zero
but we can have a zero in the bottom of
a fraction so this time we can only set
the inside just greater than zero so X
has to be greater than negative three so
the domain is simply going to be from
negative 3 to infinity but not including
negative 3 now let's consider another
example so we're gonna have a fraction
again but with a square root in the
numerator what do you think the domain
for this function is gonna be now if you
have a square root in the numerator you
need to set the inside equal to or
greater than zero so X is equal to and
greater than four now we know that in
the denominator we can't have a zero so
we're going to set it equal or not equal
to zero
now we could factor it so this is going
to be X plus 5 times X minus 5 using the
difference of squares method so X cannot
equal negative 5 and it can't equal 5 so
now let's make a number line so we have
negative 5 4 & 5 so we're gonna have an
open circle at negative 5 & 5 and then X
is equal to or greater than 4 so we're
gonna have a closed circle at 4 and
shade to the right so there's nothing
really to write here because X is not
going to equal to anything less than 4
it equals everything greater than four
included four which is not five so how
do we represent that in interval
notation so this is the first part so
we're going to start with four using
brackets and stop at five using
parentheses since it does not include 5
and then Union for the second part it's
going to go from 5 to infinity so that's
how you can represent the answer using
interval notation now what you do if you
have a fraction that contains a square
root in the numerator and also in a
denominator try this so let's focus on
the numerator we know that X plus 3 is
equal to or greater than zero which
means X is greater than or equal to
negative 3 so if we plot that on our
number line this is what we're gonna
have
so it's 4 negative 3 to infinity now
let's focus on the square root in the
bottom so we know that x squared minus
16 has to be only greater than 0 but not
equal to it because if it's on the
bottom it can't be 0 so if you have a
square root on the top you set it equal
to and greater than 0 if it's on the
bottom simply just greater than 0 so
we'll need to do first is factor this
expression it's gonna be X plus 4 and X
minus 4 so X can't be negative 4 and X
can't be 4 but it can be equal to values
in between so we're gonna make a second
number line now the reason why I can't
equal it is because we don't have the
underline symbol it's only greater than
0 but not equal to 0 so let's start with
an open circle at negative 4 & 4 now
whenever you have like two circles on a
number line due to a square root
function I like to do a scientist
find out which regions it's going to be
negative in this example it's gonna be
positive above negative 3 but negative
below negative 3 now let's plug in some
numbers so if we plug in a 5 to check
the region on the right 5 plus 4 the
using this expression that's going to be
positive and 5 minus 4 is positive so
two positive numbers multiply to each
other will give us a positive result if
we plug in 0 0 plus 4 is positive 0
minus 4 is negative so positive times a
negative number is a negative number
and if we plug in negative 5 to check
that region negative 5 plus 4 is
negative and I get a 5 minus 4 is still
negative 2 negative numbers will
multiply and give you a positive result
so now what should we do at this point
now we know that we can't have any
negative numbers inside a square root
symbol so it's not going to be anything
between negative 4 & 4 so for the square
root on the bottom X can be greater than
4 and it could be less than negative 4
but nothing in between so now what we
need to do is to find the intersection
of these two number lines we got to find
out where is true for both functions so
I'm going to create a hybrid number line
so I'm gonna put negative 4 negative 3 4
and infinity and negative infinity as
well so looking at the first one it's
not gonna work if we have anything
that's less than negative 3 so therefore
we should have nothing on the left side
so this is gonna be irrelevant because
it's true for the second part but it
doesn't work for the first one now we're
not gonna have anything between negative
3 & 4 because this is an empty region
between negative 2 & 4 even though it
works for this one it doesn't work for
the second one so therefore the answer
has to be from 4 to infinity this region
is true for both number lines this
region here applies to this number line
and also this one as well because
somewhere between negative 3 and
infinity there's a 4 now it has to be an
open circle not a closed circle so 4 to
infinity overlaps for this function on
top the square on top and also the
square on the bottom so that's going to
be the answer the domain is going to be
4 to infinity so if you have two square
root functions in the fraction you need
to make two number lines separately and
find the region of intersection where
it's true for both number lines and so
in this example best move
it's infinity and so that's how you
doing so now you know how to find the
domain of a function such as linear
functions polynomial functions rational
functions and also square root functions
