MAT016.
Module 4
RELATIONS AND FUNCTIONS
4.1 Basic Concepts and Definitions
Objectives:
1) To define and give examples of relations and functions.
2) To graph relations and functions.
3) To define and determine the symmetries and graph of the
relations.
4) To define inverse relations and inverse functions.
5) To graph inverse relations and functions
Definition 1. Ordered Pair : If two sets A and B are related by means of
a given condition, each pair of values (x, y) where x Î A and y Î B, is
called an ordered pair.
Definition 2. Cartesian Product A C B : Given two sets A and B, the
Cartesian product A C B is the set of ordered pairs ( x, y ) where
x Î A and y Î B.
Definition 3. Relation : A relation from A to B is a subset of the
Cartesian product A C B. The set A is called the domain of the
relation, and set B is the range of the relation.
Examples : The following are examples of relations.
R1 = { (1, 2) , (3, 4) , (5, 6) }
R2 = { (2, 4) , (2,- 4) , (3, 9) , (3, -9) }
R3 = { (x, y) : x+y=3}
R4 = { (x, y) : x2 + y2 = 4 }
R5 = {(x, y) : y = x2 + 4 }
R6 = { (x, y) : x 2 - y2 = 4 }
Definition 4. Function : Given two sets A and B, a function is a relation
from A to B which assigns to each x Î A one and only one y Î B. In
a function, no two ordered pairs have the same first element.
Examples : The following are examples of functions.
F1 = { (1, 2) , (3, 4) , (5, 6) }
F2 = { (x, y) : x + y = 2 }
F3 = { (x, y) : y = x2 + 4 }
Remark : All functions are relations. But not all relations are functions.
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MAT016. Module 4
Graphs :
The geometric representation of a relation is called the graph of the
relation.
Symmetry :
A graph is said to be symmetric with respect to the x-axis if
(a, b) is on the graph and (a, -b) is also on the graph as shown in
Fig. 1.
A graph is said to be symmetric with respect to the y-axis if
(a, b) is on the graph (-a, b) is also on the graph as shown in Fig. 2.
A graph is said to be symmetric with respect to the origin if
(a, b) is on the graph (-a, -b) is also on the graph as shown in Fig. 3.
(a, b)
b
x
a
-b (a, -b)
Figure 1.
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MAT016. Module 4
(-a, b) b (a, b)
x
-a a
Figure 2
(-a,b) (a,b)
(-a,-b) (a,-b)
Figure 3
The concept of a relation is useful in describing many situations in
real life. A Relation between two variables is evident in the following
examples.
1. The price of goods (p) rises when the supply (s) is low.
2. The number of teachers needed (t) depends upon the number of
students (s).
3. The area of a square (A) increase as the length of the sides (x) is
increased.
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MAT016. Module 4
4.2 Describing Relations
Relations between two variables in mathematics can be described by
equations, tables, ordered pairs or graphs.
Examples:
a. Area of a square with side x.
Equation : A = x2
Tables of values
x 0 1 2 3 4
A 0 1 4 9 16
Ordered pairs: (0,0), (1,1), (2,4), (3,9), (4,16)
Graph: A
9 (3, 9)
A=x 2
4 (2, 4)
(1, 1)
1
x
1 2 3 4
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MAT016. Module 4
b. Equation : y = 2x + 3
x 0 1 2 3
Table of values :
A 3 5 7 9
Ordered pairs : (0,3), (1,5), (2,7 ), (3,9)
A y = 2x + 3
(3, 9)
9
4
(0, 3)
1
x
1 2 3 4
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MAT016. Module 4
c. Equation y = ± x - 1
x 1 2 3 4 5
y 0 ±1 ± 2 ± 3 ±2
3
2
(2, 5)
1 (1, 0)
x
5
-1 1 2 3 4
-2 (-2, 5)
Definition 5. Domain : The set of all first components in a relation
is the domain of the relation. In an equation involving (x, y).
The domain is the set of permissible values of x.
Definition 6. Range : The set of all second components in a
relation is the range of the relation. In an equation involving
(x, y), the range is the set of permissible values of y.
Kinds of Relations
1. Single valued – a relation is single valued if for every x, there is
one y, but y can be paired to more than one x.
2. Single valued, one-to-one – a relation is single valued, one-to-
one if for every x, there is one y and vice versa.
3. Multi-valued – a relation is multi-valued if x can be paired to
more than one value of y.
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MAT016. Module 4
* A function is a single valued relation.
* To test whether a graph is a function, pass vertical lines through
the graph. If every vertical line intersects the graph at only one
point, then it is a function.
Examples: The following are examples of functions.
a. Linear function : y = f ( x ) = ax + b
If y = x - 3, then f (1) = -2, f (2 ) = -1, f (3) = 0, f (4 ) = 1
b. Quadratic function : y = f ( x ) = ax 2 + bx + c
If y = x 2 - 2 x - 3, then
f (1) = -4 f (0) = -3
f (2) = -3 f (- 1) = 0
f (3) = 0 f (- 2) = 5
c. The circumference of a circle C (r ) is a function of the radius.
C = 2pr , C (1) = 2p , C (2 ) = 4p , C (3) = 6p , C (4 ) = 8p
4.3 The Inverse of a Function.
The inverse of a function y = f ( x ) can be found by interchanging the
x and y coordinates and then solving for y. The inverse of a function is
not always a function. The graph of the inverse function is symmetric to
the graph of the original functions about the line x = y. If the function is
one to one, then its inverse is also a function.
Example a. f ( x ) = y = 2 x + 1 Inverse function : x = 2 y + 1
x y x y
- 2 y = -x + 1
0 1 1 0
-x 1
1 3 3 1 y= +
-2 -2
2 5 5 2
x 1
3 7 7 3 f -1 ( x ) = y = -
2 2
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MAT016. Module 4
Graph : y
f (x)
5
4
f –1 (x)
3
2
1
x
1 2 3 4 5 6 7
Example b. f ( x ) = y = x 2 Inverse function : x = y 2
x y x y
0 0 0 0 y2 = x
±1 1 1 ±1
f -1
(x ) = y = ± x
±2 4 4 ±2
±3 9 9 ±3
Graph : Graph f (x) and f-1 (x) below:
6
5
4
3
2
1
x
1 2 3 4 5 6
-1
-2
-3
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MAT016. Module 4
Exercises
Graph each function and its inverse on the same set of axes. Tell whether
or not each inverse is also a function.
1. y = x 3
2. y = 2 x - 3
3. y = x 2 - 2
4. y = - x 2 + 2
