Functions & Relations
• An ordered pair, denoted (a,b) is a pair of
elements a and b in which a is considered
to be the first element and b the second.
• A relation is a set of ordered pairs.
• The following are examples of relations:
S  {(1,1), (1, 2), (3, 4), (5, 6)}
T  {(3,5), (4,12), (5,12), (7, 6), (6, 6)}
Domain and Range
• The domain of a relation is the set of all
first elements of the ordered pair.
• The range of a relation is the set of all
second elements of the ordered pair.
• Domain of S  {1,3,5}; range of S  {1, 2, 4, 6}
• Domain of T  {3, 4,5, 7, 6} ;
• range of T  {5,12, 6}
Types of relations
•
•
•
•
One to one.
Many to one.
One to many.
Many to many.
1 
5
2

3

4

8
5 
9
The Rule is ‘ADD 4’
6
7
Ahmed 
Paris
Peter 
London
Ali
Dubai
Jaweria

New York
Cyprus
Hamad 
Has Visited
There are MANY arrows from each person and each place is related to MANY
People. It is a MANY to MANY relation.
Person
Bilal
Has A Mass of


Salma 
Kg
62
Peter
Alaa

George 
Aziz
64
66

In this case each person has only one mass, yet several people have the same
Mass. This is a MANY to ONE relationship
• One to one: linear, square root graph, hyperbola.
• Many to many: circle
• Many to one: parabola, truncus, semicircle.
• One to many: sideways parabola.
FUNCTIONS
• Many to One Relationship
• One to One Relationship
For a relation to be a function it has to be one to one
or many to one relation.
Apply THE VERTICAL LINE TEST to test
whether a relation is a function.
If a vertical line cuts once only anywhere on the graph,
the relation is a function.
If it cuts more than once, the relation is not a function.
(b)
(a)
(c)
(d)
(a) and (c)
How do we test for one-toone functions?
We apply THE HORIZONTAL LINE TEST.
If a horizontal line cuts once only anywhere in
the domain, it is a one-to-one function
(assuming that it passed the vertical line test
for a function first).
Which are one-to-one functions?
b)
a)
y
8
y
6
5
4
3
2
1
-4
6
4
2
-2 -1
2
4
x
-10
-5
5
10
-2
y
c)
10
d)
y
8
4
2
6
-4
4
-2
2
-2
2
-4
-2
a and d
-4
2
4
x
4
x
x
Function notation
equation
implied or restricted domain
all real numbers
f :[0, )  R, f ( x)  x

g : R  R, g ( x)  log 2 ( x)
Re-write the following using
function notation:
4
f ( x) 
 1,
x 3
x  (,3)  (3, )
h( x)  x  1,  1  x  3
2
Examples 10 &11 on page 182
Restricting the domain to
create a one-to-one function.
Consider the function
•
•
•
•
f : R  R, f ( x )  2 x  1
2
Draw a graph
Apply the horizontal line test
Is it a one-to-one function?
Restrict the domain of f so that it is one-to-one.
HINT: It can be done in more than one way.
Not one-to-one
Restricted domain, one-to-one now.
Another way of restricting the domain to create a one-to-one function.
Any other suggestions?
• Consider each of the graphs studied so far:
linear, quadratic, hyperbola, truncus, square
root graph, circle, semicircle.
• For each apply the vertical line test to
determine if it is a function.
• For each function apply a horizontal line test to
determine if it is a one-to-one function.
Exercise 5C
Q2, 3, 4, 5
Q13 a, c
Exercise 5D
Q1, 2