1. UNDERSTANDING FUNCTIONS
CHAPTER 1
UNDERSTANDING
FUNCTIONS
Lesson 1
Revisiting Functions

2. R E L A T I O N S H I P

3. Relation – a set of ordered pairs
(mother, child)
(husband, wife)
(God, man)
(employer, employee)
(teacher, student)

4. Relation – a set of ordered pairs
{(mother, child)
mother,
child
(husband, wife)
husband,
wife
(God, man)
God,
man
(employer, employee)
employer,
employee
(teacher, student)}
teacher
student
Domain: The set of the first coordinates of the ordered pair. { }
Domain are x-values, inputs, independent variable.
Range: The set of the second coordinates of the ordered pair. {
, , , , }
Range are y-values, outputs, dependent variable.

5. These are all ways of showing a relationship between two variables.
There are many ways to represent relations:
A set of ordered pairs
Mapping
Table of values
Graph
Equation
These are all ways of showing a relationship between two variables.
y = x+1
Equation

6. TYPES OF RELATION MAPPING
{(Philippines, Manila), (Indonesia, Jakarta), (Thailand, Bangkok)}
Domain
Range
Philippines
Manila
Indonesia
Jakarta
Thailand
Bangkok

7. TYPES OF RELATION MAPPING
{(Mr. Mendoza, STEM A), (Mr. Mendoza, STEM B), (Mr. Mendoza, STEM C)}
Domain
Range
STEM A
Mr. Mendoza
STEM B
STEM C

8. TYPES OF RELATION MAPPING
{(Algebra, Mathematics), (Geometry, Mathematics), (Statistics, Mathematics)}
Domain
Range
Algebra
Geometry
Mathematics
Statistics

9. TYPES OF RELATION MAPPING
{(Statistics, STEM), (Statistics, ABM), (Statistics, TVL), (Calculus, STEM), (Gen Math, STEM), (Gen Math, ABM), (Gen Math, TVL)}
Domain
Range
Statistics
STEM
Calculus
ABM
Gen Math
TVL

10. TYPES OF RELATION One to One Many to One Many to Many Many to Many
Domain
Range
Domain
Range
Philippines
Manila
Algebra
Indonesia
Geometry
Jakarta
Mathematics
Thailand
Bangkok
Statistics
One to One
Many to One
Domain
Range
Domain
Domain
Range
Range
STEM A
Statistics
SHS
STEM
STEM
Mr. Mendoza
Calculus
Calculus
STEM B
ABM
ABM
Gen Math
Gen Math
STEM C
Tech Voc
TVL
One to Many
Many to Many
Many to Many

11. Is a RELATION a FUNCTION ?
A function ( f ) is a correspondence between two sets, the domain and the range, such that for each value in the domain, there corresponds exactly one value in the range.
Is a RELATION a FUNCTION ?
One to One
If the set of ordered pairs have different x-coordinates,
It is a FUNCTION!
If the set of ordered pairs have same x-coordinates,
It is NOT a function
Many to One
One to Many
Many to Many

12. Practice Exercises
I. Determine if the following sets represent a function.
1. { ( 5, -1) , (4, -2) , (3, -3) , (2, -4) , (1, -5)}
{5, 4, 3, 2, 1}
Domain:
Range:
{-1, -2, -3, -4, -5}
Type of Relation:
One to One
Function:
YES!
2. { ( a, 1) , (b, 1) , (c, 1) , (d, 1) , (e, 1)}
{a, b, c, d, e}
Domain:
Range:
{1}
Type of Relation:
Many to One
Function:
YES!
3. { (Manila, UST) , (Manila, Quiapo Church) , (Manila, Divisoria) , (Manila, Rizal Park)}
Domain:
{Manila}
Range:
{UST, Quiapo Church, Divisoria, Rizal Park}
Function:
NO!
Type of Relation:
One to Many

13. {Algebra, Geomerty, Calculus, Statistics}x
Jan
Feb
Mar
Apr
May y 31 29 30 4. 6.
Domain:
Domain:
{Jan, Feb, Mar, Apr, May}
Range:
Range:
{31, 29, 30}
Function:
Function:
YES!
Many to One
Type of Relation: 5.
Algebra
{-4, -1, 0, 2, 4}
Math
Geometry
Domain:
Range:
{-3, 3, -1, 2, 1}
Calculus
Function:
YES!
Statistics
Type of Relation:
One to One
{Math}
Domain:
{Algebra, Geomerty, Calculus, Statistics}
Range:
Function:
NO!
Type of Relation:
One to Many

14. VERTICAL LINE TEST
The set of points in the xy-plane is the graph of a function if and only if every vertical line intersects the graph in EXACTLY one point.

15. If a vertical line passes through a graph more than once, the graph is not the graph of a function.
NOT A FUNCTION!

16. FUNCTION!

17. NOT A FUNCTION!

18. FUNCTION!

19. NOT A FUNCTION!

20. NOT A FUNCTION!

22. Relation: M M M-M
Relation: M M M-M

23. Relation: M M M-M
Relation: M M M-M

24. Relation: M M M-M
Relation: M M M-M

25. Relation: M M M-M
Relation: M M M-M

26. Domain and Range R: (-2, 4) D: [-3, 3) D: -3 ≤ x < 3 D: (-3, 3)
Interval Notation
Set Builder Notation
D: [-3, 3)
D: -3 ≤ x < 3
D: (-3, 3)
R: (-2, 4]
R: -2 < y ≤ 4
Interval Notation
Set Builder Notation
Interval Notation
Set Builder Notation
D: [-3, 3]
D: -3 ≤ x ≤ 3
D: (-3, 3]
D: -3 < x ≤ 3
R: [-2, 4]
R: -2 ≤ y ≤ 4
R: [-2, 4)
R: -2 ≤ y < 4