Types of Functions Mapping

Let A and B be two non-empty sets.Let A and B be two non-empty sets.

A function f from A to B is an association of every element of A to an

unique element in B. We write this as f : A g B.

A is called the domain.

B is called the co-domain.

Examples:

i)

Because the element c A is not associated with an element of B.

ii)

Because the element a A has no unique association.

Hence y = f(x), y is called the image of x and x is called the pre-image of y.

Range of f

Note:

Example:

Range of f = {1,2,3,4}

Types of functions (Mappings)

One-one function (or an injective map)

A function f : A g B is called a 1-1 mapping, if every element of f(A) does not have more than one pre-image.

Examples:

i)

ii)

Many-one function

A function which is not 1-1.

Example:

Onto function (or surjective map)

Let f : A g B be a function, f is said to be an onto function if f(A) = B.

Examples:

i)

ii)

Bijection

Let f : A g B be a function, f is said to be a bijection if f is 1 - 1 and f is onto.

Example:

Constant function

A function f : A g B is said to be a constant function, if

Example:

Identity function

Note:

Identity function is denoted by I_A.

Example:

Equality of two functions

Two functions f and g are said to be equal if

domain of f = domain of g

codomain of f = codomain of g and

Composite function

Let f : A g B and g : B g C be two functions.

Let h: A g C be function such that h(x) = g(f(x))

h is called the composite of f and g, denoted by h = g o f.

i.e., (g o f) (x) = g (f(x))

Note:

(iii) Domain of gof is Domain of f.

Example:

(gof) (x) = g (f(x)) = g (x+1) = (x+1)²

(fog) (x) = f (g(x) = f (x²) = x² + 1

Inverse of an element

f^-1(b) = {a|b = f (a)}.

Example:

f^-1(1) = {a}

f^-1(2) = {b,c}

Inverse function

Example:

Note:

(i) Inverse of a function f is unique.

(ii) f^-1of = fof^-1 = I_A

(iii) (f^-1)^-1= f

(iv) (fog)^-1= g^-1of ^-1