Summary The Higher Arithmetic, ISBN: 9781139643528 Engineering maths

Functions and
Binary Operations
Function
Let A and B be two non-empty sets, then a function f from set A to set
B is a rule which associates each element of A to a unique element of B.
f
It is represented as f : A → B or A → B and function is also called the
mapping.

Domain, Codomain and Range of a Function
If f : A → B is a function from A to B, then
(i) the set A is called the domain of f ( x ).
(ii) the set B is called the codomain of f ( x ).
(iii) the subset of B containing only the images of elements of A is
called the range of f ( x ).
A B
f d
a
b e Range
c f
g
Domain h
Codomain

Characteristics of a Function f : A → B
(i) For each element x ∈ A, there is unique element y ∈ B.
(ii) The element y ∈ B is called the image of x under the function f.
Also, y is called the value of function f at x i.e. f ( x ) = y.
(iii) f : A → B is not a function, if there is an element in A which has
more than one image in B. But more than one element of A may
be associated to the same element of B.
(iv) f : A → B is not a function, if an element in A does not have an
image in B.

, Identification of a Function from its Graph
Let us draw a vertical line parallel to Y-axis, such that it intersects the graph of
the given expression. If it intersects the graph at more than one point, then the
expression is a relation else, if it intersects at only one point, then the
expression is a function.
Y Y



X′ X X′ X
O O



Y′ Y′
(i) (ii)
In figure (i), the vertical parallel line intersects the curve at two points, thus the
expression is a relation whereas in figure (ii), the vertical parallel line intersects
the curve at one point. So, the expression is a function.


Types of Functions
1. One-One (or Injective) Function
A mapping f : A → B is a called one-one (or injective) function, if
different elements in A have different images in B, such a mapping is
known as one-one or injective function.
Methods to Test One-One
A B
(i) Analytically If f ( x1 ) = f ( x2 ) ⇒ x1 = x2 f
4
or equivalently x1 ≠ x2 1
6
2
⇒ f ( x1 ) ≠ f ( x2 ), ∀ x1 , x2 ∈ A, 3 7
then the function is one-one.
(ii) Graphically If every line parallel to X-axis cuts the graph of
the function atmost at one point, then the function is one-one.
Y



X' X
O


Y'
(iii) Monotonically If the function is increasing or decreasing in
whole domain, then the function is one-one.

, Number of One-One Functions
Let A and B are finite sets having m and n elements respectively, then
n P , n ≥ m
the number of one-one functions from A to B is  m
 0, n < m
 n( n − 1)( n − 2) ... ( n − ( m − 1)), n ≥ m
=
 0, n<m

2. Many-One Function
A function f : A → B is called many-one function, if two or more
than two different elements in A have the same image in B.
Method to Test Many-One
(i) Analytically If x1 ≠ x2 ⇒ f ( x1 ) = f ( x2 ) for some x1 , x2 ∈ A, then
the function is many-one.
A B
f
4
1
2 5
3 6


(ii) Graphically If any line parallel to X-axis cuts the graph of
the function atleast two points, then the function is many-one.
Y
y = f ( x)

X' X

Y'

(iii) Monotonically If the function is neither strictly increasing
nor strictly decreasing, then the function is many-one.
Number of Many-One Function
Let A and B are finite sets having m and n elements respectively, then
the number of many-one function from A to B is
= Total number of functions − Number of one-one functions
 n m − n Pm , if n ≥ m
=
 n m , if n < m