# DIFFERENT TYPES OF FUNCTIONS

## About the topic "Different types of functions"

Here, we are going to see the different types of functions in sets.

The five types of functions are

(i) One to one or Injective function

(ii) Onto or Surjective function

(iii) One to one and onto or Bijective function

(iv) Constant function

(v) Identity function

Let us discuss the above different types of functions in detail.

## One to one or Injective function

Let f : A ----> B be a function.

The function f is called an one-one function, if it takes different elements of A into different elements of B.

That is, we say f is one-one

In other words f is one-one if no element in B is associated with more than one element in A.

A one-one function is also called an Injective function.

The figure given below represents a one-one function.

## Onto or Surjective function

Let f : A ----> B be a function.

The function f is called an onto function, if every element in B has a pre-image in A.

That is, in B all the elements will be involved in mapping.

An onto function is also called a surjective function.

The figure given below represents a onto function.

## One to one and onto or Bijective function

Let f : A ----> B be a function.

The function f is called an one-one and onto or a bijective function if f is both a one-one and an onto function

More clearly,

f maps distinct elements of A into distinct images in B and every element in B is an image of some element in A.

The figure given below represents a one to one and onto or bijective function.

## Constant function

The function f is called constant function if every element of A has the same image in B.

Range of a constant function is a singleton set.

Let A  =  { x, y, u, v, 1 }, B  =  { 3, 5, 7, 8, 10, 15 }.

The function f : A ---> B defined by f (x) = 5 for every x belonging to A is a constant function.

The figure given below represents a constant function.

## Identity function

Let A be a non-empty set. A function f : A ---> A is called an identity function of A if f (a) = a for all a belonging to A.

That is, an identity function maps each element of A into itself.

For example, let A be the set of real numbers (R). The function f : R ----> R be defined by f (x) = x for all x belonging to  R is the identity function on R.

The figure given below represents the graph of the identity function on R.

## terms related to functions

Let f : A ----> B be a function.

Then, we have

Domain : Set A

Co-domain : Set B

Range : Elements of B involved in mapping.

Note :

In onto function, co-domain  =  Range

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