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Dear Almighty God,
We thank you for giving us another life,
We thank you for another Beautiful Morning.
As we go on through our lessons today,
May you make us instruments to do good things.
Please enlighten our minds,
Give us the strength to participate in our subject
today.
Thank you for this opportunity to learn and serve
others,
and help us to always remember the Truth of Your
Glory. Amen!
LEARNING OBJECTIVES:
At the end of the lesson, the learners should be
able to:
 illustrate relations and functions;
 verifies if a given relation is a function; and
 display cooperation and oneness in
performing activities.
Methods of
Illustrating
Relations
A relation is the correspondence between
two quantities. It is composed of an independent
variable and dependent variable.
Independent variable - is a variable whose value
freely changes and does not depend on any
quantity.
Dependent variable - is a variable whose value
depends on the changes of values of the
independent variable.
In mathematics, a relation is any set of
ordered pairs. Since it is composed of
ordered pairs, the 𝑥 − and 𝑦 −coordinates
can also be observed whenever a relation is
illustrated. The set of all values for the
𝑥 −coordinate is called the domain of the
relation and the set of all values for the
𝑦 −coordinate is called the range of the
relation.
Mapping Arrow Diagram
X
-3
-1
2
5
Y
2
0
5
Tabular
X
5
4
0
-2
-3
Y
1
2
3
4
5
Ordered Pairs
𝐴=
2,3 , 5,4 , 5,1 , 2, −2
Equation
𝑦 = 7𝑥 + 4
EXAMPLE 1
Illustrating the relation P =
{(Kobe Bryant, Lakers), (Michael
Jordan, Bulls), (Magic Johnson,
Lakers), (Reggie Miller, Pacers)}
using the arrow diagram.
Afterward,
determine
the
domain and range of the
relation.
X
Kobe Bryant
Michael Jordan
Magic Johnson
Reggie Miller
Y
Lakers
Bulls
Pacers
EXAMPLE 2
Illustrate the given relation
below using tabular method.
Afterward, determine the
domain and range of the
relation.
X
Y
5
2
1
0
11
12
19
X
5
2
2
1
0
0
Y
12
11
12
19
11
19
EXAMPLE 3
Give four possible
values for the domain
and range of the
relation 𝒚 = 𝟔𝒙 − 𝟑.
Value for 𝒙
If 𝑥 = 0
If 𝑥 = 1
Solution
𝑦 =6 0 −3
=0−3
= −3
𝑦 =6 1 −3
=6−3
=3
Value of 𝒚
−3
3
Value for 𝒙
If 𝑥 = 2
If 𝑥 = −1
Solution
Value of 𝒚
𝑦 =6 2 −3
9
= 12 − 3
=9
𝑦 = 6 −1 − 3
−9
= −6 − 3
= −9
Functions:
A Special Type of
Relations
A relation can be any correspondence between
two quantities. However, there are instances that the
correspondence exhibits a unique characteristic.
X
Y
1
2
3
0
2
4
X
Y
2
0
2
4
X
Y
1
2
3
0
2
4
one-to-one
correspondence
X
Y
2
0
2
4
one-to-many
correspondence
X
Y
1
2
3
4
many-to-one
correspondence
A function is a relation in which no two
ordered pairs have the same 𝑥 −coordinate. If
the first coordinate is repeated, then the
correspondence is not a function anymore but is
still a relation.
You could deduce from this that all functions
are relations but not all relations are functions.
The notation f(x) is often used when the function
is illustrated in equation form.
EXAMPLE 1
The relation N =
{(3,4), (5,6), (2,7),
(6, -1)}
Since there are no two ordered
pairs that have the same
𝑥 − coordinate, this example is a
function with a domain of D = {2, 3,
5, 6} and a range of R = {-1, 4, 6, 7}.
EXAMPLE 2
The relation W =
{(1,4), (2,4), (3,4),
(4,4)}
Even though the 𝑦 −coordinate appeared
multiple time, the rule to be followed to
consider a relation to be a function only
observes the 𝑥 −coordinate. Since there are
two ordered pairs that have the same
𝑥 −coordinate, this example is still a function
with a domain of D = {1, 2, 3, 4} and a range of
R = {4}.
EXAMPLE 3
The relation Z =
2
𝑥, 𝑦 𝑦 = 𝑥 + 4
If you substitute an arbitrary value for 𝑥, say
2
𝑥 = 5, the resulting equation will become 𝑦 =
9. Since there are two values for 𝑦 (3 and -3),
that
when
squared
becomes
9,
a
correspondence can be illustrated as {(5,3), (5, 3)}. By carefully observing the values of the
𝑥 − coordinate, notice that the two ordered
pairs have the same 𝑥 −coordinate. With this,
the correspondence is not a function but a mere
relation.
ACTIVITY
Determine and illustrate the domain and range of the
given relations.
Group 1
Give four possible
values of the
domain and range
of the relation 𝒚 =
𝟔𝒙 + 𝟐 and
illustrate it using
arrow diagram.
Group 2
Give four possible
values of the
domain and range
of the relation 𝒚 =
𝟖𝒙 − 𝟏 and
illustrate it in
tabular form.
Group 3
Give four possible
values of the
domain and range of
the relation 𝒚 =
𝟐𝒙 − 𝟑 and
illustrate it in
ordered pair form.
A
S
S
I
G
N
M
E
N
T
A. Illustrate the following relations and
determine their corresponding domain and
range.
1. Give three possible values for the domain
and range of the relation 𝑦 = 5𝑥 + 1.
Illustrate the domain and range using the
ordered pair form.
2. Give three possible values of the domain
and range of the relation 𝑦 = 2𝑥 + 4 and
illustrate it in tabular form.
B. Determine if the following correspondence are a
function of a mere relation.
1. The relation H = {(12, 15), (11, 31), (18, 8), (15,
12), (3, 12)}
2. The relation Z = {(5, 0), (0, 5), (8, −8), (−8, 8), (0,
0)}
3. The relation A = {(−3, −1), (2, 0), (5, 1), (3, −8), (6,
−1)}