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Functions, Equations, Words, Tables & Graphs
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Feb 19, 2025
In our last chapter, we learned how to represent relations as tables and graphs. In this section, we'll use multiple representations to represent functions. We'll also learn how to translate among different verbal, tabular, graphical, and algebraic representations of functions. Chapters: 00:00 Introduction 00:32 What's a Relation? 00:48 Functions: definition 01:45 Visual Explanation of a function ( Domain & Range) 03:00 Example1 & 2 ( functions) 05:47 Non function ( characteristics) 06:44 Real life example of functions and non functions 10:47 What is a function rule? 11:42 Examples 23:38 Equations
View Video Transcript
0:00
so we're going to talk about functions
0:02
equations and graph okay this is fun
0:05
this is not hard at all so don't get
0:08
like intimidated by the amount of words
0:10
that I put on board so this is really
0:12
fun actually okay the first thing we
0:15
want to talk about is last I think on
0:17
Monday we talked about relations right
0:19
and we had a really good conversation I
0:21
felt really encouraged on on Monday when
0:24
we talk about relations who want to
0:26
remind me what a relation is we talk
0:28
about ordered pair right what's on order
0:30
fair you were asking a lot of questions
0:32
about order Fair do you remember what
0:33
they are now do you understand it order
0:35
is like two things that go together
0:37
they're go together right and Order okay
0:40
so now we going to go a little bit
0:41
further we going to talk about function
0:43
what is have you heard the word function
0:45
before function yes how would you define
0:48
a
0:51
function you can read by definition what
0:53
does it say there a function is what a
0:57
function is a relation which each
1:01
Dom what does that mean each remember a
1:04
function is a relation right a relation
1:07
in which each member of the domain is
1:11
fair exactly with one what member in the
1:15
range so how would you explain that to
1:20
somebody each member in the domain is
1:24
paired with exactly one
1:27
member in the range yeah yeah
1:30
[Music]
1:37
uh kind of it could be like that but
1:40
it's this I put an example on the board
1:43
right now here this is a diagram right
1:46
and then you have on the left side you
1:48
have domain and on the right side you
1:50
have what range the domain is also
1:52
called called an like the set of input
1:55
right you input and this is the output
1:58
what's the difference with on input and
2:00
then output input is what you put in and
2:03
output is what you get right you put
2:06
something in you get something back
2:08
right now here if you look at my domain
2:10
I have three names right I have Emily I
2:14
have Gabe and I have L right and then I
2:18
assign them specific things I say that
2:21
Emily owns a dog now I don't know if she
2:23
does own a dog and I don't know I just
2:25
assigned it right this is my function
2:27
right I said Emily owns a dog Gabe owns
2:31
a cat and at least own a fish right yes
2:35
can I have a snake or lizard I don't
2:39
like lizard I don't like reptiles I
2:40
don't like snakes anybody here like
2:43
snakes I'm terrified of snakes I don't
2:46
like snakes my sister wants a snake I
2:48
don't like snakes anyway right that
2:52
yeah yeah yeah that's that's your bunny
2:55
M
2:57
yeah so the domain the domain is the
3:01
input right so I'm saying that I'm
3:03
assigning to Emily a dog and I'm
3:05
assigning to gab a cat right you you
3:08
have a cat well I'm giving you a cat
3:11
right so I'm giving gab a cat and I'm
3:14
giving Le a fish right so if you look
3:16
here if you look at the the definition I
3:19
said what a function is a relation in
3:22
which right each member of the domain so
3:26
how many members do we have in the
3:27
domain here three three right each
3:31
member in the domain so Emily Gabe and
3:34
Le has is spaded exactly one member in
3:38
the range so if you see Emily is sp with
3:39
what a dog G is spare with a cat and Le
3:43
is sp with a fish so this is a function
3:47
so that will be that will qualify as a
3:49
function this will be yes a
3:52
function that's a function yes yes for
3:55
function right yes now if you look here
3:59
and I have another example right now I
4:01
have Emily gave and Le and I gave each
4:04
one of them a dog right so each one of
4:07
them has a dog is this still a function
4:10
yes yes or no who says
4:13
yes no why not I say each member yeah
4:19
what you give each of us a dog I give
4:22
each of you guys a
4:24
dog so is it a
4:28
function yes it is
4:30
right cuz I said what the definition
4:33
let's go back to the definition each
4:35
member of the domain right which is here
4:39
is spared exactly with one member in the
4:43
range right so you fair with a dog
4:46
that's fine gab spare with a dog that's
4:49
fine and Le is spare with a dog so each
4:51
one of them is spare with one yes but if
4:54
we were all sharing one dog you all
4:57
sharing one dog that's fine we wouldn't
5:00
be a function so if you were sharing one
5:03
dog right so you have a dog he has a dog
5:08
and you have a dog that is a function
5:10
still cuz each member is not with two
5:13
different things one you see what I'm
5:17
saying each member is spread with one
5:19
not two yes but if I had like a dog and
5:22
a fish then that will not be a function
5:24
that's right you actually ahead of me
5:26
cuz I'm getting there so this is still a
5:27
function now does anybody understand
5:29
distinction here each member of the
5:31
domain has to have one no more than one
5:35
right if it has two one person's not
5:37
going to have it exactly a function so
5:39
here for example this would not be a
5:40
function because Emily has what a dog
5:43
and Emily also has a fish so that will
5:47
not be a function okay that here will
5:49
not qualify as a function so you be like
5:51
no not a
5:54
function because she's part of the
5:56
domain or the input and she has a dog
5:59
and a fish
6:00
right but if I were to erase
6:02
this then that would be a function
6:05
because em has now one do but if let's
6:08
say if I were to give Gabe a fish then
6:11
that would still not be a function
6:12
because now Gabe has a fish and a dog
6:16
does that make sense yeah definition of
6:17
a function right later on we're going to
6:19
talk about the vertical line test we're
6:21
not doing that yet but here now we just
6:24
understanding what a function is so that
6:26
means that each member in the domain has
6:29
to have one member in the range you
6:32
can't have two if you're part of the
6:34
domain okay does that make sense yeah
6:37
right like again if you had to go to the
6:39
Bible right
6:42
so let's say this let's try to create
6:46
something as a function biblically when
6:48
we use marriage and see if that make
6:50
sense to you marriage all right let's
6:52
talk about marriage as a Rel as a
6:54
function and then tell me this
6:57
is so
7:01
so let's put
7:03
Abigail right let's put um who else
7:07
let's put
7:10
Ebenezer and let's put
7:12
Dexter right and let's put
7:16
here uh actually now I want to put a man
7:19
here let let me put Caleb here right and
7:23
then
7:25
here
7:26
huh all right I give you some different
7:29
Apple Mary instead of cuz I don't and I
7:32
put
7:34
Abigail and then I put in
7:38
Vicki
7:41
right let's this so this will be random
7:45
so this is domain right and this is what
7:48
R
7:54
now right so let's say Caleb Caleb
8:00
Mary right just Rand marri Vicky right
8:06
DX marries AB right let's assume
8:10
that is this a function can it be me yes
8:14
yes all right now let's say it is that's
8:18
the Caleb chooses
8:24
[Applause]
8:26
this you're player is it a
8:31
this is a function no right because
8:33
Caleb has how many wives Caleb has two
8:37
he has
8:39
Mar you say what you say what not
8:48
chance tell not chance
8:52
bust we not going to do that because
8:55
it's one man what one wife one man one
8:58
wife right you know doing that right so
9:01
you see you can use this so does that
9:04
make sense so that means according to
9:06
the Bible God's idea of a function is
9:09
one
9:10
man one wife that's it right
9:17
who well exactly now was Solomon right
9:20
was God pleased with Solomon no if you
9:22
go to the scripture what is God tell
9:24
him it's not biblical was not biblical
9:28
he try to God told him because of he
9:31
married so many wives they led him into
9:34
sin right because they altered his mind
9:38
and he started like have other gods he
9:40
was supposed to
9:41
be who King David I don't know how many
9:45
wies I'm sure you have multiple wives
9:46
too but God did not approve of that
9:48
right I actually don't have anything had
9:50
but do you remember what the Jesus told
9:52
the people he said originally that's not
9:54
what God proclaimed God proclaimed what
9:57
a man shall leave his father and brother
9:59
and cling on to what God his wife he
10:02
didn't say his wives his wife f f e
10:06
right so we're not going to do that so
10:08
in Christian marriage our relationships
10:10
our functions because it's one man one
10:14
woman we not going to pull a Caleb right
10:17
we're not going to have Caleb Mar Mary
10:19
and Vicki at the same time it's not
10:20
happening if he comes to you say Vicki I
10:22
like you also like you tell
10:25
him doing that right we not doing that
10:29
it's one man one woman that's what we
10:32
going to stick with right so therefore
10:35
this is how do another one do another
10:37
one not any
10:40
more so we're not doing that right all
10:43
right so now now the next thing we're
10:45
going to talk about is function rule
10:47
right what is a function rule a function
10:50
rule by definition is an operation
10:53
performed on The Domain value to get the
10:56
range value so basically you do
10:59
something to this to get that that's
11:02
what this means and that's a function
11:03
rule now I'm give you an example and we
11:05
going to use this example to explain it
11:08
right so in the game of wasma rule right
11:12
Kinley made the following rule she chose
11:14
to make the following rule double the
11:16
number and add three that's her rule
11:19
okay now the question is make a function
11:22
table for four different input values
11:25
and write an algebraic expression for
11:28
the rule then State the domain and range
11:31
okay so here's what we're going to do
11:33
I'm going to explain this so let's make
11:36
a table can I ra this yeah all
11:42
right so we're going to make a table so
11:46
let's see this this is called a function
11:48
table right a function table because we
11:51
making a table and we're going to have
11:53
three things in this table we're going
11:54
to have this is the input right input
12:01
we're going to have the
12:04
output right and then we're going to
12:06
have our
12:08
rule right input is usually the X values
12:11
and output is the Y values so now we're
12:14
going to talk about the rules here okay
12:16
so our input Is Random so pick four
12:20
random values because they want us to
12:22
pick to put four random input values so
12:24
pick one give me one excuse say one give
12:28
me a random number five five all right
12:32
um alen give me one number 55 22 22 what
12:37
else 55 86 86 86 all right 86 that's
12:42
fine now listen to her rule so the rule
12:44
that she made is what double a number
12:49
and add what three so how would you
12:52
write that double a number and add three
12:54
so how would you write that 5 * 2/ I
12:57
mean plus three right three let's just
12:59
write the rule in general using X values
13:02
so the rule will look like what double a
13:04
number and add three so we be what two
13:08
what X
13:10
plus three does that make sense yeah
13:13
double a number and then add three now
13:16
let's do this let's apply that rule to
13:18
this are you all paying attention yeah I
13:20
P so how would you do that here what is
13:24
our number here five so be 2 * what
13:29
5 plus and then we going to find we
13:32
going to find the answer and put it here
13:33
right so what would that be 2 * 5 is 13
13:37
all right next is 2 * I actually don't
13:40
know what that is 22 + 3 right so what
13:45
would I be here 47 47
13:48
47 next one 2 *
13:51
55
13:53
113+ three 113 that's 113 all right and
13:58
then 2 * 86 I don't know I don't know
14:01
that one you can use your calculator why
14:04
I my piece of paper what's 2 *
14:10
86 let's see what' you
14:14
get
14:16
172 plus three that's 175 right so
14:20
basically what we did was we just
14:22
created a function table does that make
14:25
sense yeah right so she made up her own
14:28
rule is a number and then R three I
14:31
assign that to you right now create your
14:33
own Rule now I want you all to come up
14:35
with a rule
14:37
no yeah triple number and add five
14:41
Triple A number and add five so what
14:42
would be the rule here Triple A number
14:45
so give it to me in terms of like how do
14:47
you do four 3x + 5 number qu qu number
14:52
is that four uh which one quadruple is
14:56
four right so she said triple number and
14:59
add five that's her rule right and now
15:01
you can find the numbers the thing is
15:04
this the input is random you pick any
15:07
arbitrary values it doesn't have to be
15:09
specific I would like I like my life to
15:11
be easy so I'm going to choose what one
15:13
two three and four I don't want to
15:15
choose big numbers you know so I'll use
15:17
1 2 3 and four if you want to use 5 22
15:19
55 56 that's up to you but you create
15:22
your own Rule and then now you put some
15:24
input and then the output is going to be
15:26
dependent on what the input
15:29
the input is always your own choice but
15:32
the output is
15:34
always uh a result of your input and
15:38
your rule does that make sense okay so
15:41
now you can pick some random values
15:42
let's let's go over the easy ones all
15:44
right 1 2 3 and
15:46
four so she chose 3x + 5 so that'll be 3
15:51
* 1 + 5 and then we find what it
15:56
is h8 8 all right and then 3 * 2 + 5
16:03
that is 11 right yep 3 * 3 + 5 that
16:08
would
16:09
be 14 so as you can see the Gap is 3
16:12
three every time right and 3 * 4 + 5
16:16
that be obviously 17 17 right so now you
16:20
can come up with your own rule somebody
16:22
come up with a different rule that is
16:23
not like quad come up different rule
16:26
Emily
16:33
all right come on the rule let me let me
16:34
hear qu triples plus five Quin triple
16:37
Quin triple is that five five x that's
16:39
six or six no it's not six triple right
16:42
I don't know that's four no quadruple is
16:46
four queen queen that's five right 5 x +
16:50
5 all right all right start with the
16:53
Quin dou and find something else beside
16:56
Quin tuu and I want something else yeah
17:00
um add a number
17:03
to and then divide two that's right add
17:07
the number to eight right so and divide
17:11
it by two right the whole thing
17:13
parentheses are
17:14
that like this sure yeah all right add
17:18
the number to eight and then once you
17:19
get the result divided by two now for
17:22
this case I want to be careful cuz what
17:25
I want to do is this because I like
17:27
things easy I'm going to try to find
17:29
numbers that are what even yeah so
17:31
therefore my input has to be an even
17:33
number right cuz if I choose odd numbers
17:36
I'm going to run into trouble because I
17:38
don't want to deal with our numbers now
17:39
it's not like it's impossible you can
17:41
still do it but I like life easy right
17:43
so I'm going to choose two four six and
17:48
eight because I know that no matter what
17:49
I'm always going to get an even number
17:51
and even numbers are always divisible by
17:54
two two so and then you can go ahead and
17:56
find your rule right okay so this this
17:58
is what is called a function table so
18:01
you have a function a function is a
18:04
relation in which each member of the
18:05
domain is exactly what one member of the
18:09
range right so what I want to do now is
18:11
I'm going to give you some examples and
18:13
you tell me whether this is a function
18:15
or not can I erase this yeah yeah all
18:19
right so let's let's see if this is a
18:21
function here so you guys tell me so if
18:24
I have wait were we doing it on paper
18:27
you can do it on your desk you can do it
18:29
on paper you can do it
18:31
anywhere so here what we have is um
19:24
guys he said friendship okay friendship
19:27
yeah I saw
19:29
all right where am I all right let's see
19:32
I not there don't they don't care about
19:36
is this a function no why not because n
19:40
is because one is with n and 10 all
19:43
right G is this a
19:46
function
19:49
huh is it a
19:51
function is
19:53
it were you paying attention no pay
19:56
attention right CLE it's not a fun
19:58
because Danielle say what one is coupled
20:01
with what n and 10 now this here is a
20:04
friendship right is this a function yes
20:08
why because they all have one they all
20:10
have one right so one is not connected
20:13
to what two so Nora can be friend with
20:15
Vicki Kenzie can be friend with Vicki
20:17
and then Danielle can be friend with
20:18
Vicki too right that's fine
20:26
yes all right let's try to create a
20:30
little bit of a stir let's stir the part
20:32
here
20:34
righty stir the part
20:37
that's put there that's uh who me uh
20:49
KY em okay you
20:52
can't
20:54
next okay put together right now no
21:00
what H huh huh huh huh huh huh what
21:08
[Music]
21:09
is
21:11
what is this a function hey I yes yes
21:17
no tell me why is it is a function
21:20
everyone has
21:21
two together no way it is a function cuz
21:25
everyone has two no no it's not has what
21:29
Kenzie has Vicky and Emily right
21:33
Danielle has what she said Emily KY and
21:37
Dexter so therefore that is not a
21:39
function right he now let's create one
21:45
more so now we
21:51
have uh yes is it okay if I just did the
21:55
first two yeah I didn't write them down
22:02
Caleb you be sending the link I will be
22:06
sending that to your classroom where we
22:09
don't
22:10
have I'm going to create it that's easy
22:12
it takes like two seconds y use it in
22:15
computer he that's no that's the science
22:17
one El
22:19
has the B excuse me
22:24
yeah
22:26
don't I put what you're going to be on
22:29
there no excuse me
22:32
mhm oh
22:34
no no please
22:46
no oh my
22:50
God is this the
22:52
function
22:55
why why is it not a function
22:58
tell me why is not a fun because I'm not
23:00
with the person that I want to be hey at
23:03
least at least this time Caleb only has
23:05
one
23:16
[Music]
23:21
right that that's now we going to talk
23:24
about equations okay
23:28
I mean mram you look to his
23:31
dirty why do you why don't you just do
23:33
marriage with like people that are like
23:34
seniors or
23:37
something I know a lot of the seniors be
23:39
like these you could have put Emily and
23:42
Emily I'm single no no that work em all
23:46
single maybe like one or
23:54
two I don't
23:56
know it was
24:02
whates oh are you surprised that is he
24:05
slapping his friends probably
24:08
not see this's
24:18
[Music]
24:20
different all right so now we're going
24:22
to talk about equations right and by
24:25
definition what is an equation that's
24:27
the simple definition is a maal sentence
24:32
question an equation is a mathematical
24:35
sentence that states that two quantities
24:38
are equal right yep give an example of
24:41
an equation now that you know the
24:44
definition equals what 5 + 5 equals 10
24:49
10 that's an equation right 5 + 5 = 10
24:54
that's an equation right it's is
24:57
mathematical standard that St two
24:58
quantities are equal
25:01
yeah now another equation is this if you
25:04
guys can tell what it is I see if you
25:06
remember 2 L + 2 W what is this p = 2 L
25:10
+ 2
25:12
W
25:16
yeah what dep of what kind of
25:21
figure square or rectangle rectangle
25:24
rectangle right this is the formula for
25:26
what to find the rectangle
25:28
of the perimeter of a rectangle right so
25:31
now the two quantities here this is
25:33
perimeter the perimeter equals what
25:35
twice the length plus twice the width
25:38
this is called an equation okay an
25:40
equation all right now when we get up to
25:43
like later chapters we're going to talk
25:44
about actual equations where you have to
25:46
solve for x and and things of the S now
25:49
here right
25:51
d = RT this is also an equation what is
25:56
this anybody knows d = r *
26:02
yeah no D yeah
26:07
time no that D is d i
26:12
s distance equals what what's
26:17
R range radiation
26:20
no R rate we don't know this
26:26
time so basically this
26:29
equals the
26:31
rate what else can we use for
26:33
rate instead of rate what can you say we
26:36
start with s when you drive what's your
26:40
speed speed right speed so distance
26:44
equal to the speed times the time okay
26:47
distance is equal to the speed multip by
26:50
time right so if I say
26:54
d equals to 60t what am I technically
26:57
saying 160 m I'm driving what 60 m per
27:01
hour per hour right so if I tell you I
27:05
want you to find the distance I travel
27:07
if I if I drove 2 hours what would you
27:10
do 60
27:13
60 times two right if you know the time
27:18
and you know the speed you can find the
27:20
distance that you travel if I'm
27:22
traveling 60 M hour how many miles I
27:24
travel in uh 2 hours yeah you just
27:29
replace it here 12 120 mes right 120 M
27:33
that's like almost going to Ocean City
27:36
and a little more so 60 * 2 that's 120
27:39
mph 120 M hour right so this is a this
27:43
is called an equation okay an equation
27:46
is used and you use it to solve problems
27:49
okay it states that two points are equal
27:51
if you know something you can solve for
27:52
the other right remember we talked about
27:54
the horse um the seahorse he travels how
27:58
many feet per hour you guys remember 5T
28:03
that's so sad it's so slow yeah so sad I
28:06
don't know huh it flaps it thing for few
28:09
seconds this is if you see this 5T that
28:12
will be the distance a seahorse can
28:17
travel in 1 hour right he so 2 hours he
28:21
does 10 fet that's pathetic right for
28:25
but even it's pathetic for for seahorse
28:26
he thinks that he did a lot of work work
28:28
oo I just worked so hard how many 10 ft
28:31
in 2 hours wow you know so it's relative
28:35
because some some animals are like you
28:37
know turos turos are really slow too
28:40
right who knows what the fastest animal
28:42
is uh the humans a cheah cheetah fastest
28:47
what yeah a cheetah is the fast on but
28:50
there's birds
28:56
that cheetah goes I don't know what
28:58
their speed is but cheers are really
29:00
fast but they also get tired quick right
29:02
they can only Sprint and if they can't
29:04
catch the gazelle now they have to wait
29:07
now
29:08
lions are lazy who wants the lion and
29:12
lioness who wants who doesn't hunting
29:15
the lion the lion is hunting and the
29:17
lion sit there and when the food C comes
29:19
in you know he get make choices you know
29:23
but he didn't do anything I just watch
29:25
the
29:26
show no that's not true that is not true
29:31
that is true that is that that is not
29:34
true I don't believe that anyway tell
29:37
this so I don't believe that
#Mathematics
#Teaching & Classroom Resources