## How To Find The Domain of a Function - Radicals, Fractions & Square Roots - Interval Notation

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

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

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

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

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