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Square Root Functions & Inequalities ( Domain & Range)
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Apr 1, 2025
In this section (6.3) we'll learn how to graph and analyze square root equations and also learn how to graph square root inequalities. Chapters: 00:00 Introduction 01:41 How do you graph a square root function( Domain, Range, Table of values) 03:50 Domain and Range, Graph
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0:00
we're starting the new section so it's
0:02
6.3 and we're going to talk about square
0:04
root functions so we are back to
0:06
functions again we left drawing function
0:10
graphing function now we back to
0:12
graphing I mean yeah want to give give
0:16
my mark but I guess yeah all right so
0:18
we're going to talk about graphing
0:20
radical function square root functions
0:23
okay they are easy to graph and again
0:26
why do we do this because sometimes
0:29
different models can be represented
0:30
using the square root function I've
0:32
already seen some faces that are looking
0:34
dejected as if they're orphans but we're
0:37
not going to listen to that we're going
0:38
to continue to like we're going to
0:40
continue to work right we're not going
0:43
to pay attention to
0:45
thatan I used to
0:47
[Music]
0:48
be a function
0:54
right if a function contains the square
0:56
root of a variable we call that function
1:00
a square root function anytime you have
1:04
that radical here it's called a square
1:07
root function right so example square
1:09
root of x is called a square root
1:12
function square root of x -1 is also
1:16
called a square root function because
1:19
why does it qualify as a square root
1:20
function because you have a variable
1:23
under a radical does that make sense so
1:27
if you see any function that has a
1:28
radical on top of it right a square root
1:31
radical that means that is a square root
1:33
function first thing you need to
1:35
understand is the basics right we need
1:37
to know what a square root function is
1:39
now now that you know assume that you
1:41
know what a square root function is how
1:43
do you graph square root function that's
1:45
what we're going to learn right so to
1:46
graph square root functions we need to
1:49
find a couple of things we need to find
1:51
the domain we need to find the range and
1:53
we also need to find make a table of
1:56
values and then the rest is just
1:57
plugging stuff in and that is pretty
2:00
simple right so as you can look here
2:03
right I gave you an example of a square
2:05
root
2:06
function all right this is a square
2:09
function eyes are not looking at me and
2:11
I don't want to well I didn't understand
2:13
this I want your eyes to pay attention
2:15
here because we moving all right so this
2:17
is a square root function and as you can
2:19
see what do you notice here about this
2:23
function it's curved all right that's a
2:25
good that's a good observation got
2:27
points it got points right starts at
2:29
zero it start at zero that's the main
2:31
thing that I wanted to mention here
2:32
start at zero is going to what Infinity
2:36
on the x- axis right if you look at the
2:38
x- axis this is the x- axis right the x-
2:41
axis deals with what the x-axis deals
2:44
with the domain and the Y AIS deals with
2:47
a range right so if I ask for domain I'm
2:50
looking at the x-axis where does this
2:52
function begin it begins right here at
2:54
zero and how far does it go infity
2:57
Infinity it's going up to Infinity right
2:59
now now if I want to look at the range
3:01
where does this function begin on the y-
3:03
AIS and where how far does it go up
3:06
infinity infinity right the range is
3:07
also 0 to positive Infinity so you see
3:10
what I did here right now we're going to
3:12
learn how to find this thing Al
3:14
algebraically without the function how
3:15
do you find the domain and the range of
3:17
a square root function if you're just
3:20
given uh the the function and you don't
3:23
have a graph so how do you find the
3:25
domain of the range so I'm glad you
3:26
asked you going to learn how to do that
3:28
right so can I ra is
3:30
this do we still need this I need that
3:35
okay I need
3:37
that all
3:39
[Music]
3:40
right I got a strong I need
3:44
that all right so we're going to learn
3:46
how to find the domain of erical
3:49
function right now the first thing we
3:51
need to understand is this when I say
3:53
find the domain of a function what am I
3:55
asking for if you understand something
3:58
it's easier to solve it right
4:00
when I say find the domain of a function
4:01
what I'm asking is this I want you to
4:03
find the values for which the function
4:06
exists does that make sense I want to
4:10
find the values for which the function
4:13
exists right that's the domain the
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domain deals with values that make the
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function uh existent if I can use that
4:22
type of language right so when you when
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you hear the word domain this is what
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you're asking for for what values this
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this function makes sense does that make
4:32
sense yes all right so now I'm assuming
4:36
I can erase the middle
4:39
right thank you for giving me
4:41
permission right
4:43
so yeah so now here's the thing so let's
4:47
consider this function f ofx is equal
4:50
to we have f ofx is equal to ro x - 5
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and I want to find the domain and the
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range of this function so what I do here
5:00
right I want to find the domain so I
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want to find the values for which this
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function will make sense right that's
5:06
what I'm looking for when I want to find
5:07
the domain so for radical function do
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you remember we talked about how can you
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have the square root of a negative
5:13
number no no right so to find the domain
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of this
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function right here's what we want to do
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we want we want x - 5 to be what to be
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greater than or equal to zero this is
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how you find the domain of a radical
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function you want what inside or what's
5:30
under the radical to be what positive so
5:34
any given radical function this is how
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you going to look for the domain does
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that make sense right you want what's
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under the radical this value here to be
5:45
always what positive that's what you
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want for any radical function right so
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if I ask you to find the domain of a
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radical function is always going to be
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setting this value to be greater than or
5:58
equal to zero and then solving for x
6:01
right I don't stop here I need to expand
6:03
I need to finish this right if x - 5 is
6:06
greater than or equal to zero what does
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that mean that means that what x is
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greater than or equal to 5 x is greater
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than or equal to five so therefore the
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domain of this function will be what
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five to what positive Infinity it's very
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simple Stu right so like right now we're
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just looking to find the domain of each
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of these functions we're just going to
6:26
find the domain and then after that
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we're going to learn how to find the
6:29
range
6:30
right so let's try this one here G of
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X so G of X isun 2x Co how do I find the
6:38
domain of this
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function if I want to find the domain D
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so what do I want for this function to
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make
6:47
sense before that what's the setup I
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want what I want this to be
6:54
what I want this to be more than or
6:56
equal to zero and then divide by two
7:00
like you said right and Z so that mean X
7:03
is greater than equal to Z so the domain
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will be
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what0 0 to positive Infinity right so
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the domain will be
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zero to positive Infinity are we on the
7:16
same page here so far yes right all
7:19
right uh next uh Jack what how do I find
7:24
the domain of this function h of X
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is -2x + 1
7:30
- 5 do I care about here does it matter
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no does the matter outside though it's
7:37
outside so does it matter no it's
7:39
irrelevant right so to find the domain
7:41
what do I want want -2x + 1 to be
7:44
greater than or equal to zero right I
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want this to be greater than or equal to
7:48
zero and then now from here I need to
7:50
solve for x right yes tell me would you
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put the on the side you
8:00
it doesn't matter just the Z I mean we
8:03
don't it's not it has no relevance
8:06
because it's not under the radical it
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does not affect what's in here right so
8:10
the only expression that you are
8:12
interested in is what's under the
8:14
radical okay so for this function this
8:18
there's nothing to make this function on
8:21
F what if the was in front of the square
8:24
root it doesn't it still doesn't matter
8:26
okay the only thing that matters is
8:27
what's under the square root right
8:30
everything else is irrelevant because as
8:33
long as this value is positive this
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function makes sense so anything else
8:37
has no uh weight on it okay so now we we
8:41
want -2x + 1 to be greater than or equal
8:44
to Zer so now I'm going to solve this
8:46
for X so what do I do subtract one
8:48
subtract one so I get
8:51
-2X and then what divide by -2 so that's
8:54
X is less than or equal to or POS 1 1
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you see X has to be less than or equal
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because when you divide by negative the
9:03
sign switches to no so what we what
9:08
infity negative Infinity is 12 right so
9:11
negative Infinity is 12 will be the
9:14
domain okay so because it's less ter so
9:17
we want anything that's 12 or less so
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the domain of this function will be
9:21
negative Infinity to one2 now that we
9:24
did the domain the range is very easy
9:27
right you're going to have fun finding
9:29
the
9:30
so how do you find the range of this
9:32
function we're going to start with the
9:34
first one right so the first function
9:36
that we have was F
9:38
ofx is equal to x - 5 and we found the
9:42
domain to be zero no not zero 5 to
9:47
positive Infinity right so to find the
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range here's what you do right you see
9:52
the domain what does the function start
9:54
right here right so to find the r you're
9:56
going to plug this in here the five plug
9:59
it into the x value right to find the
10:02
range so when I plug in five in here
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what do I
10:06
get Z right so if I plug in five here 5
10:09
- 5 is zero
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right do you agree with it yeah now what
10:15
if I plug in Infinity what
10:19
happens of infinity is going to be what
10:22
Infinity right because Infinity is like
10:25
a billion more than a billion if you
10:26
plug in a billion here you're still
10:28
going to get something in a billion
10:30
right so your range will be what the
10:32
lowest value to the highest so it be 0
10:34
to Infinity so the range R is going to
10:38
be 0 to Infinity so what you do is is
10:41
you plug in each of the values in here
10:43
to find what that's going to give you
10:46
right so now let's try for this function
10:49
so let's find the range of this function
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we already have the domain so what do
10:52
you think the range is going to
10:54
be I'm going to plug this in here right
10:58
first so what's s of0 z zero right and
11:02
then what's square root of infinity
11:04
infinity so the range is going to be
11:06
what 0 to Infinity great 0 to Infinity
11:09
so that's pretty simple stuff okay and
11:12
now let's try this guy here right so I'm
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going to start
11:16
with2 then I'm going to plug in 1/2 in
11:18
here so let's do that so if I plug
11:22
in2
11:24
right so I get 1 + one which is what
11:28
zero yeah have a question
11:31
yes so on the one that says x - 5 this
11:35
one yeah you got X and so you put the
11:37
five in the X right there but then when
11:40
it says 12 you put the 12 2 X on yeah on
11:44
that side look look at the the point you
11:47
put in where the Y is why why did you
11:49
switch it like Y is one in the X area
11:52
one's in the Y is not in the Y is still
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12 it's in here I put it in here no but
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point point it's negative infinity and
11:58
12
11:59
y it should be 12 a negative Infinity
12:03
can you go no we never go from a
12:05
positive to a negative it's from
12:07
negative Infinity is that number okay
12:08
why is a negative Infinity all right
12:10
because X is what less than or equal to
12:13
12 okay so that means you're going where
12:16
you're starting where if x is less than
12:18
oh CU you switched the sign yeah I
12:20
flipped it right but I got X is less
12:23
than or equal to 12 so how would you
12:25
represent that is it 12 to negative
12:27
infinity or negative Infinity to 12
12:29
because if you're on the number line
12:31
it's on the left side right so you're
12:32
starting at negative infinity and you
12:34
are going up to 12 and you're stopping
12:36
here okay make sense yeah all right so
12:38
back to this one now I'm trying to find
12:40
the range of this function right so I
12:43
plugged in2 in here all right so 1 + 1
12:47
is what 0 right and I have 0 - 5 do not
12:51
forget this 5 5 in in the r you cannot
12:54
ommit this right it's still in here so
12:57
what's 0 - 5
13:00
right so we have so that's our first
13:02
thing now the next thing we going to
13:04
plug in negative Infinity right if I
13:06
plug in negative Infinity in here what
13:09
happens it becomes positive Infinity
13:12
what's squ of positive
13:13
Infinity posi Infinity so therefore my
13:15
range is what to Infinity so really this
13:20
this exercise is just an exercise of
13:22
observation and looking at what you're
13:24
doing and making sense of it right
13:27
anything that we do we have to use our
13:28
head
13:29
we cannot just assume well I'm just
13:31
going to plug in and infinity no
13:34
everything that we do what one thing I'm
13:36
learning a lot of student is sometimes
13:39
we don't read the instructions and we
13:40
just want to be like robots but we are
13:42
not AI we are actual humans and we can
13:45
think is always to be not necessarily
13:48
not
13:49
necessarily every case is different you
13:51
just have to analyze it right cuz I
13:53
don't want to say yes because every case
13:56
is going to be different you you may
13:57
have a case where that's not the case
13:58
right so it's it's going to be different
14:01
now that we have you know how to find
14:02
the domain of the range right we're
14:04
going to learn how to graph this
14:05
functions okay so now let's learn how to
14:07
graph them so graphing this functions so
14:12
I want a space somewhere I don't know
14:14
where to put
14:15
it put in the middle and then I w't have
14:18
to erase it which I I need but that's
14:20
fine let me wait can't you just eras the
14:22
definition H yeah oh yeah see that's
14:27
think you know you don't use your hand
14:29
you do all right maybe you're Ai No I
14:33
know who's AI I know I'm not telling
14:41
you gra
14:44
right do we have work today what are we
14:47
doing so now we're going to learn how to
14:49
graph them right so let's start with the
14:52
first function right f
14:55
ofx isun x - 5 right
14:59
I found the domain what was the domain
15:01
of this
15:02
function five to positive Infinity right
15:05
yeah and what was the
15:07
range um Z 0 to Infinity right that's
15:12
this function now we have the domain and
15:14
the range and we want to graphic so the
15:16
graphic it is really simple as a as a
15:18
process right so I'm just going to go
15:21
ahead and now my my advice is this for
15:27
uh for um for a radical function yeah
15:30
what is
15:31
[Music]
15:32
theity what is that the Dom
15:35
range I don't have what do you see it's
15:40
it's many points well for a radical
15:44
function you need at least three points
15:46
right for rles you need three points but
15:48
they easy to find the first one is
15:50
always going to be five five right and
15:53
zero because when X is 5 Y is zero right
15:56
the next one choose another Point that's
15:58
higher than five
15:59
six right when X is 6 what's y 6 - 5 is
16:03
1 right 1 is 1 now I'm going to use
16:06
another value that's going to help me
16:07
here right what if I
16:10
use nine or seven yeah no but square of
16:14
two is not like a perfect square I don't
16:16
want to deal with it right I use n okay
16:18
right when X is 9 9 - 5 is 4 of 4 is 2
16:22
right so now I have it now I can graph
16:25
it I'm not saying that you can't use any
16:27
other number but like a decimal decimal
16:30
which I don't like working with decimal
16:31
so I'd rather just use something easy
16:33
right so I can now I can graph the
16:36
function right when X is 5 1 2 3 4 5 Y
16:41
is zero right here right X is 6 Y is 1
16:45
and X is 9 6 7 8 9 Y is 2 right here
16:51
right so it's going to be a curve going
16:53
this in this direction you have to very
16:55
careful don't not some people give me
16:57
like this weird looking light it has be
16:59
a curve right rticles are always going
17:00
to be curves they're going to be curves
17:02
like this so this is how you graph this
17:04
you only really need like three or four
17:06
and you should be fine you just have to
17:08
get the
17:11
yeah yeah mind fig you figure it out
17:15
sure uh yes 9 - 2 is 9 - 5 is two 9 - 5
17:21
is four
17:22
is and S of four is two okay all
17:27
right all now let's
17:33
gra all right let's grab the second one
17:35
right so let's grab the second one here
17:38
so I have GX is 2x okay I'm going to
17:42
make a table for it real
17:44
quick again remember you need to start
17:47
with the first value when X is zero Y is
17:49
what what was
17:51
it zero right yeah because the domain is
17:56
0 to Infinity the range is 0 to Infinity
17:57
when X is0 Y is Z and then let's find
18:00
another value right when X is 2 I use
18:02
two because 4 is a perfect square right
18:06
so when X is
18:08
two this is two right which 2 * 2 is 4 4
18:12
is two right and then I'm also going to
18:14
use
18:16
a let me see for n three of six OT of
18:21
six that's
18:23
f it would be 16 right so if I put here
18:30
it be of 16 which is four right so we
18:33
have that so same
18:36
thing 0 0 1 two one
18:41
two and then 1 2 3 4 5 6 7 8 and then
18:46
four 3 four right here right
18:50
again we're going in this direction
18:53
right now so far we have these functions
18:55
now let's try the next one right the
18:58
next one is going down it's going to go
19:00
down because he has a different
19:01
direction so we're going to see what
19:03
we're going to get here
19:05
okay
19:10
so I put it on computer all right so
19:13
let's try this one here right so for
19:16
this one I have my table of Game X and Y
19:20
right so here uh my first value is uh
19:24
when X is one 12 what do I got um you
19:29
got
19:30
negative five right and I'm going to
19:33
choose something else half so one I can
19:37
choose zero okay right because I I have
19:40
to find Val that are less than 12 oh
19:43
more no because look at my domain my
19:46
domain is what negative Infinity to 12
19:49
so I have to choose values that are
19:51
either 1 12 or less but I'm always going
19:53
to start with 1/2 because that's my
19:55
limiting value on the x- axis right so
19:58
I'm going to choose Z here because 0 is
20:00
less than 1 12 and when X is zero if you
20:03
plug it in here right this is of 1 is 1
20:06
1 - 5 is 4 right so 1 - 5
20:10
is4 so I got that right I'm going to
20:13
choose another one I have a question yes
20:16
why is it that you put four there but
20:17
then for the other tables you do the
20:19
square root so you put what do you mean
20:22
okay no over there right next to where
20:26
it says graphing right here yeah yeah
20:29
for two MH okay for n you put two okay 9
20:34
- 5 is four four four is two so then why
20:37
don't you put the squ all now here let's
20:39
let's wait we're not pluging zero right
20:41
uh so what is this my expression is
20:44
what * 0 - 5 right -2 * 0 is zun of 1 is
20:51
1 1 - 5 is okay than you four right so
20:54
and then we need another one we need a
20:56
third one right so I can put it in
20:59
let's say
21:00
what 1
21:03
22 -2 * -2 is 4 4 + one is five sare of
21:08
five is what 2 point something what is
21:10
it 2.4 2.4 I'm uh let's see six six and
21:17
seven that be good five so find that
21:21
let's let's try that right how negative
21:24
numbers can I multiply until yeah I can
21:25
only put negative numbers cuz I'm going
21:27
in this direction right
21:30
833 or8 will be great no no no4 will be
21:34
great okay it go4 if I put4 2 * -2 * -4
21:40
is 8 8 + 1 is 9un of 9 is 3 3 - 5 is -2
21:46
right so now we have
21:48
this now you can graph it right when X
21:51
is 12 12 is here 5 1 2 3 4 5 right here
21:57
right and then
22:00
Z4 1 2 3 4 and then four uh4 and2 so4 1
22:08
2 3 4 -2 right here right so you see
22:12
what happens now so the function is
22:14
going to go bingo in this Direction all
22:17
right so it's always going to be a
22:20
curve all right it's always going to be
22:23
a curve you're always going to have a
22:25
curve in this case Okay so there always
22:27
going to be a curve so what I want to do
22:29
I want to I want to stop here and then
22:31
maybe give you some some work