# What is a function?

420 Points
13 years ago

Dear Rubankumar

Function can be easily defined with the help of the concept of mapping. Let X and Y be any two non-empty sets. "A function from X to Y is a rule or correspondence that assigns to each element of set X, one and only one element of set Y". Let the correspondence be 'f' then mathematically we write f:X → Y

where

y = f(x), x ε X and y ε Y. We say that 'y' is the image of 'x' under 'f' (or x is the pre image of y).

Two things should always be kept in mind:

(i)     A mapping f: X → Y is said to be a function if each element in the set X has its image in set Y. It is possible that a few elements in the set Y are present which are not the images of any element in set X.

(ii)    Every element in set X should have one and only one image. That means it is impossible to have more than one image for a specific element in set X. Functions can't be multi-valued (A mapping that is multi-valued is called a relation from X to Y)

•   Set 'X' is called the domain of the function 'f'.
•   Set 'Y' is called the co-domain of the function 'f'.
•   Set of images of different elements of set X is called the range of the function 'f'. It is obvious that range could be a subset of co-domain as we may have few elements in co-domain which are not the images of any element of the set X (of course these elements of co-domain will not be included in the range). Range is also called domain of variation.

The set of values for which a function is defined is called the domain of the function. The range of the function is the set of all images of domain of f. In above example, the set A is the domain of the function f. B is not range but the co-domain of the function. The range is the subset of the co-domain.

All the best.

AKASH GOYAL

Please feel free to post as many doubts on our discussion forum as you can. We are all IITians and here to help you in your IIT JEE preparation.

Win exciting gifts by answering the questions on Discussion Forum. So help discuss any query on askiitians forum and become an Elite Expert League askiitian.