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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
