This document discusses functions and relations. It defines a relation as a set of ordered pairs that associates each element of one set with an element of another set. A function is a special type of relation where each element of the first set is mapped to exactly one element of the second set. The document provides examples of relations and functions using sets, tables, graphs and equations. It describes key characteristics of functions, such as the vertical line test, and discusses classifying relations as functions or not functions. The learning objectives are for students to illustrate relations and functions, verify if a relation is a function, and determine dependent and independent variables.

1. FUNCTIONS AND RELATIONS
Prepared by:
Ms. Ma. Irene G. Gonzales

2. Learning Objectives
At the end of the period, students should be able to:
• Illustrate a relation and a function;
• Verify if a given relation is a function; and
• Determine dependent and independent variables.

3. PAIR-ME-UP ACTIVITY
Group A
Ca
K
H
O
Au
Group B
Oxygen
Gold
Calcium
Potassium
Hydrogen

4. PAIR-ME-UP ACTIVITY
Group A
Manila
Kuala Lumpur
Jakarta
Hanoi
Bangkok
Group B
Indonesia
Malaysia
Philippines
Thailand
Vietnam

5. What is a Relation?
• It involves the association of an individual or object with
another individual or object.
Learning Objective: Students should be able to illustrate a relation and a function.

6. What is a Relation?
• It involves the association of an individual or object with
another individual or object.
Learning Objective: Students should be able to illustrate a relation and a function.

7. Mathematically speaking…
• A relation is any set of one or more ordered pairs.
• There is one value of y that corresponds to EVERY value
of x.
– DOMAIN: set of first coordinates
– RANGE: set of second coordinates
Learning Objective: Students should be able to illustrate a relation and a function.

8. Relation as a Set of Ordered Pairs
• Using the elements of A = {1, 2, 3, 4, 5}, construct the
set of ordered pairs F that describes the relation “The
abscissa is one greater than the ordinate.”
• Solution
F = {(2,1), (3,2), (4,3), (5,4)}
Learning Objective: Students should be able to illustrate a relation and a function.

9. Relation described by a Table
Filipino Heroes Date of Birth
Jose Rizal June 19, 1861
Andres Bonifacio November 30, 1863
Apolinario Mabini July 23, 1864
Emilio Aguinaldo March 22, 1869
Gabriela Silang March 19, 1731
Learning Objective: Students should be able to illustrate a relation and a function.

10. Relation Expressed in Arrow Diagrams
Learning Objective: Students should be able to illustrate a relation and a function.
Manila
Kuala Lumpur
Jakarta
Hanoi
Bangkok
Indonesia
Malaysia
Philippines
Thailand
Vietnam
D R

11. One-to-many
Learning Objective: Students should be able to illustrate a relation and a function.
1
4
9
-3
-2
-1
1
2
3
X Y

12. Many-to-one
Learning Objective: Students should be able to illustrate a relation and a function.
-3
-2
-1
1
2
3
2
4
6
X Y

13. Relation as an Equation or Formula
Learning Objective: Students should be able to determine dependent and independent variables.
• For instance, the equation
y = x + 1
• Between the two variables, which one is the
independent variable? Which is the dependent
variable?

14. Relation as an Equation or Formula
Learning Objective: Students should be able to determine dependent and independent variables.
• For instance, the equation
y = 4x + 1
• Between the two variables, which one is the
independent variable? Which is the dependent
variable?

15. Relation Described Graphically
Learning Objective: Students should be able to illustrate a relation and a function.

16. Differentiate a function from a relation…
• A special type of relation illustrated by the following sets
of ordered pairs:
A = {(1,2), (3,4), (5,6), (7,8)}
B = {(-8,7), (-2,3), (-4,5)}
C = {(4,3), (5,8), (7,9), (-1,2)}
The following sets are relations but not functions:
D = {(1,2), (1,3), (1,4), (1,5)}
E = {(-8,1), (-8,2), (4, 3), (4, 5)}
Learning Objective: Students should be able to verify if a given relation is a function.

17. What is a FUNCTION?
• It is a relation in which each element of the domain
corresponds to exactly one element of the range.
• In simple terms, it is a special type of relation wherein
no two ordered pairs have the same abscissa.
• In other words,
“x is faithful to EXACTLY one y”
Learning Objective: Students should be able to verify if a given relation is a function.

18. Real-Life Relationships of Functions
H = {(Ms. Irene, August), (Joseph, March), (Clyle,
November), (Tim, December)}
J = {(Mr. Perilla, St. Benedict), (Mr. Guevara, St. Bertha),
(Ms. Gonzales, St. Barbara), (Mr. Ulaye, St. Bernard)}
Learning Objective: Students should be able to verify if a given relation is a function.

19. Characteristics of a Function
1. Each element in domain X must be matched with
EXACTLY one element in range Y.
2. Some elements in range Y may not be matched
with any element in domain X.
3. Two or more elements in domain X may be
matched with the same element in range Y.
Learning Objective: Students should be able to verify if a given relation is a function.

20. Vertical Line Test
A graph
represents a
FUNCTION if and only
if no vertical line
intersects the graph in
more than one point.
Learning Objective: Students should be able to verify if a given relation is a function.

21. One-to-many
Learning Objective: Students should be able to illustrate a relation and a function.
1
4
9
-3
-2
-1
1
2
3
X Y

22. Many-to-one
Learning Objective: Students should be able to illustrate a relation and a function.
-3
-2
-1
1
2
3
2
4
6
X Y

23. Function or not a function?
Learning Objective: Students should be able to illustrate a relation and a function.
12
5
8
2
4
X Y

24. Function or not a function?
Learning Objective: Students should be able to illustrate a relation and a function.
12
5
8
2
4
X Y

25. Function or not a function?
Learning Objective: Students should be able to illustrate a relation and a function.
12
5
8
2
4
6
X Y

26. Function or not a function?
Learning Objective: Students should be able to illustrate a relation and a function.
12
5
8
2
4
6
X Y

27. Get ready for the Learning Activity 4.3