LESSON 1:
Functions as
Models
Lesson Outline
• Review:
relations and functions
the function as a machine
functions as representations of real life situations
• Functions and relations as a set of ordered pairs
• Functions and relations as a table of values
• Functions and relations as equations
• Functions and relations as a graph in the Cartesian plane
• Vertical line test
WHAT IS RELATION?
• it is a set of ordered pairs
{(-1, 3), (-2, 4), (-3, 5), (-4, 6)}
DOMAIN (x) is the set of all first coordinates. It is also known as
“input”.
RANGE (y) is the set of all second coordinates. It is also known
as “output”.
Domain(x): {-4,-3,-2,-1}
Range(y): {3,4,5,6} 3
FUNCTIONS
It is the relation in which each element of the domain
corresponds to exactly one element of the range.
Functions can be represented in different ways:
Ordered pairs,
A table of values,
Equation
Mapping diagram, and
Graph
Functions by
ordered pairs
Which of the following relations are
functions?
f= 1,2 , −2,2 , 3, −5 , (8,7) FUNCTION
g= 1,3 , 1,6 , 3,5 , 3,7 , (2,4) NOT FUNCTION
Function or not? Why?
f= 1, −3 , 7,4 , −1,4 , (7,0)
g= 1,3 , 2,6 , 3,9 , … (𝑛, 3𝑛)
Functions by
table values
NOT FUNCTION FUNCTION
Input Output Input Output
(x) (y) (x) (y)
8 13 -8 21
18 33 36 15
80 -3 19 6
80 3 30 21
Function or not? Why?
Input Output
(x) (y)
1 2
2 2
3 5
4 5
Functions by
equation
How to determine if the given
equation is a function or not?
The given equation has an equal sign.
Check the variables of x and y.
Lastly, the exponent of y should not be more than 1.
Tell whether the given equation is
function or not a function.
1. 6x+9y=5 FUNCTION
2. 5x+3𝒚𝟐 = 𝟕 NOT FUNCTION
3. 7𝒙𝟐 + 𝟒𝒚 = 𝟕 FUNCTION
4. 8𝒙 > 𝒚 + 𝟔
𝟐
NOT FUNCTION
3. 12𝒙 + 𝟔𝒚 = 𝟏𝟕
𝟐 𝟑 NOT FUNCTION
Functions by
mapping
diagram
• One to One Relation
• One to Many Relation
TYPES OF
RELATIONS
• Many to Many Relation
• Many to One Relation
Types of Relations
1 One to one Relation
Each element in the domain has only one element in the
range and each element in the range is linked to only one
element in the domain.
Types of Relations
2 Many to one Relation
There are elements in the domain that have the same
element in the range. Both the elements a and b in the
example are paired with d.
Types of Relations
3 One to many Relation
There are elements in the domain that have more than one
image in the range. The element a in the example has
images c and d.
Types of Relations
3 Many to many Relation
There are elements in the domain that have more than one
image in the range, and there are elements in the range that
are linked to more than one element in the domain.
Using the types of relations, which are
considered to be function?
1 One to one Relation
2 Many to one Relation
Functions by
Graph
VERTICAL LINE TEST
A graph represents a function if and only if the vertical line
intersects the graph at most once. If any vertical line cuts the
graph of a relation in more than one point, the relation is not a
function.
FUNCTION
FUNCTION
FUNCTION
FUNCTION
NOT
FUNCTION
NOT
FUNCTION
THANK YOU
