A relation f from a set A to a set B is said to be a **function **if every element of set A has one and only one image in set B

In other words, a function f is a relation from a non-empty set A to a non-empty set B such that the domain of f is A and no two distinct ordered pairs in f have the same first element

If f is a function from A to B and (a, b) ∈ f, then f (a) = b, where b is called the *image *of a under f and a is called the *preimage *of b under f.

The function f from A to B is denoted by f: A → B

**Example **Let N be the set of natural numbers and the relation R be defined on

N such that R = {(x, y) : y = 2x, x, y ∈ N}.

What is the domain, codomain and range of R? Is this relation a function?

**Solution **The domain of R is the set of natural numbers N. The codomain is also N.

The range is the set of even natural numbers.

Since every natural number n has one and only one image, this relation is a

function

**Example **Examine each of the following relations given below and state in each

case, giving reasons whether it is a function or not?

(i) R = {(2,1),(3,1), (4,2)}, (ii) R = {(2,2),(2,4),(3,3), (4,4)}

(iii) R = {(1,2),(2,3),(3,4), (4,5), (5,6), (6,7)}

**Solution **(i) Since 2, 3, 4 are the elements of domain of R having their unique images,

this relation R is a function.

(ii) Since the same first element 2 corresponds to two different images 2

and 4, this relation is not a function.

(iii) Since every element has one and only one image, this relation is a

function

## Types of function and their graphs

**Identity function** Let R be the set of real numbers. Define the real valued function f : **R → R** by y = f(x) = x for each x ∈ **R**. Such a function is called the identity function. Here the domain and range of f are **R**.

**Graph of identity function**

**Constant function** Define the function f: R → R by y = f (x) = c, x ∈ R where c is a constant and each x ∈ R. Here domain of f is R and its range is {c}.

**Polynomial function** A function f : R → R is said to be polynomial function if for each x in R, y = f (x) = a_{0} + a_{1} x + a_{2} x^{2} + …+ a_{n} x^{n} , where n is a non-negative integer and a_{0} , a_{1} , a_{2} ,…,a_{n} ∈R

functions defined by f(x) = x^{3} – x^{3} + 2 is Polynomial function