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in this video I want to give you a basic

introduction into domain and range and

how to write the expression using

interval notation given the graph of a

function so what is the domain and range

of this particular function now the

domain tells you the x values of the

function the range tells you the y

values of the function so let's focus on

the domain first what is the lowest x

value of the function the lowest x value

as seen here is negative for the highest

x value is stream and this function

contains every x value in between

negative 4 & 3 so in interval notation

we could say that the domain is from

negative 4 to 3 and we're gonna use

brackets because it includes negative 4

& 3 so that's how you can write the

domain for this particular function now

what about the range what's the range of

this particular function well let's

focus on the Y values the lowest Y value

that we see this is negative 5 now the

highest Y value is 4 and there's no

breaks in the graph this graph is

continuous from negative 5 to 4 along

the y axis so therefore the range is

going to be negative 5 to 4 and so

that's a simple way to determine the

domain and range of a function using a

graph now go ahead and try that example

find the domain and range of the

function so let's start with the X

values the lowest x value is at negative

6 and the highest we could see is

positive 6 now notice that we have an

open circle at negative 6 so negative 6

is not included so I'm going to use the

parentheses for that so it's gonna be

negative 6 to 6

but here we have a closed circle so

that's gonna be associated with a

bracket and so that's the domain for

this particular function now what about

the range so let's focus on the Y values

the lowest Y value occurs at negative 4

and the highest Y value occurs at 5 but

negative 4 is not included so it's gonna

be negative 4 to 5 but 5 is included so

that's the range for this particular

function go ahead and try this example

what's the domain and range of that

function so let's start with the X

values the lowest x value is 1 now

notice we have an arrow so this goes all

the way to infinity so therefore the

domain is gonna be from 1 to infinity

now how about the range the lowest Y

value is 2 and because of this arrow

it's gonna go up and to the right

indefinitely

so therefore the highest y-value

technically is infinity because this

doesn't end so the range is going to be

from 2 let me write that better 2 to

infinity and choose included because we

have a closed circle now here's the next

example we have a downward parabola and

how can we determine the domain and

range of it so let's start with the

x-values so once again we have an arrow

that means it's going to go down as it

slowly goes to the left so it's going to

keep going to the left forever and we

have an arrow on the right side so it

keeps going to the right side forever

so therefore the lowest x value because

it goes all the way to the left is

negative infinity and because it goes

all the way to the right the highest of

x value is positive infinity so the

domain is going to be negative infinity

to infinity always use a parenthesis

symbol next to an infinity symbol now

let's focus on a range the Y values

the lowest y-value we can clearly see

that it's a negative infinity because it

keeps going down forever but the highest

y-value is stream it never goes beyond

three so the range starting from the

lowest value to the highest value is

negative infinity to 3 including 3 and

so that's how you can determine the

domain and range of a problem now let's

try some harder examples if he wants you

pause the video and work on this one so

let's start with the domain let's focus

on the X values so the lowest x value is

negative 6 the highest x value is 5 and

notice that there's a jump in the graph

at negative 1 however X can be negative

1 so X could be anything between

negative 6 and 5 except negative 6

because we have an open circle at

negative 6 but it can be negative 1

because we do have a closed circle at

negative 1 so the domain for this one

it's going to be negative 6 to 5 because

it could be any x value between negative

6 and 5 just not negative 6 now what

about the range what about the Y values

so the lowest Y value that we see is

negative 4 and the highest is positive 4

now notice that there's nothing between

negative 1 I mean negative 2 and 1 so Y

can't be anything between there it could

be negative 2 though because we do have

a closed circle at negative 2 but it

can't be negative one negative point 5

point 8 it can't even be 1 because we

have an open circle at 1 so how can we

describe the range using interval

notation in this example

what we need to do is we need to use the

Union symbol that will connect the range

of this expression with the range of

this expression omen and everything in

between since there's nothing there so

we're gonna go in this direction from

the low value to the high value so the

lowest value the lowest Y value that is

is negative 4 and we need to use

parentheses because we have an open

circle so it goes from negative 4 to

negative 2 now we have a closed circle

at negative 2 so we're going to use

brackets and then Union so this is for

the first graph so let's connect it to

the second part of the graph we're gonna

start back up at 1 and end that for now

1 is not included we have an open circle

but we have a closed circle at 4 so 4 is

included and so this represents the

range of this particular function so

hopefully this makes sense to you and

the best way to learn this is to do a

few examples so this is gonna be the

last example for this video go ahead and

determine the domain and range let's see

if you understand how to do it now so

let's start with the X values the first

x value of interest is at negative 8 the

next x value that I want to take note of

which ends this portion of the graph

that's that negative 4 and then the

second part of the graph starts back up

at negative 2 and it ends at 5 now the

third part of the graph it starts back

up again at 7 and then we have an arrow

so it goes to infinity so basically

these numbers that we see here we just

have to use that to write the domain so

the lowest x value it starts at negative

8 and it includes negative 8 and then it

stops at negative 4 but it does include

negative 4 so let's use parentheses

Union now for the second part of the

graph it starts at negative 2

and it includes it and it stops at five

and it does include five and then Union

it starts back up at seven and then ends

at infinity so that's the domain for

this particular graph now let's focus on

the y-values the range so let's focus on

this one the lowest value that I see

here is negative six now this is not the

highest value of this function so I'm

not going to worry about it the highest

value is here which is that one

so this graph includes everything from

negative six to one notice that this

open circle is not relevant it's not

equal to negative four at this point but

it is equal to negative four at this

point if you draw a horizontal line

notice that there's two possible

locations at which you can equal

negative four here and here

it doesn't equal negative four here but

it does equal negative four here so Y

can be negative four so we have

everything from negative six to one now

let's focus on this one I don't need to

worry about this point because it's

already included in this graph the

highest Y value here is two so notice

that Y could be anything from negative

six to two it could be one here and here

so I don't need to worry about this one

now notice that there's nothing between

two and five there's no graph in this

region so that's where I need a union

now the highest y-value is going to be

infinity because of the arrow so these

are the points of interest

so the range is going to be negative 6

to 2 it does include 2 and then Union 5

it does include 5 to infinity now if

this section confuses you here's what

you can do now let's focus on these two

parts separately let's say if we want to

write the range of each one separately

the first one just this portion is going

to be negative 6 to 1 and it includes 1

now the second part is going to be this

starts at negative 3 by the way and it

stops at 2 now if we want to find the

union between these two expressions or

these two sets represented in interval

notation what would it be the union of

those two is going to be negative 6 - 2

let's see if we drew a number line

here's negative six here's one and then

for the second one this is gonna be

negative three and this is gonna be two

so here's the first one negative six two

one and here's the second one negative

three to two so if we combine those two

into one number line if we found a union

between them it's going to look

something like this it's gonna start at

negative six in a different color and

then it's gonna stop at two which gives

us this expression so what you could do

is find the range for each portion of

the graph separately and then just find

a union for any where they overlap and

that would still give you the range or

if you could see it graphically I would

just do it the first way but that's it

for this video thanks again for watching