Definition

Function can be easily defined with the help of the concept of mapping. Let A and B be any two non-empty sets. “A function from A and B is a rule or correspondence that assigns to each element of set A, one and only one element of set B.” Let the correspondence be “f” then mathematically we write f: A B where y = f(x), x A and y B. We say that “y” is the image of “x” under ‘f’ (or x is the pre-image of y).

• A mapping fA → B is said to be a function if each element in the set A has an image in set B. It is possible that a few elements in the set B are present which are not the images of any element in set A.
• Every element in set A should have one and only one image. That means it is impossible to have more than one image for a specific element in set A. Functions cannot be multi-valued. (A mapping that is multi-valued is called a relation from A and B.)

Let us consider some other examples to make the above-mentioned concepts clear.
1. Let f: R+ R where y2 = x. This cannot be considered a function as each x R+ would have two images namely ±. Thus it would be a relation.
2. Let f: [–2, 2] R, where x2 + y2 = 4. Here y = , that means for every x  [–2, 2] we would have two values of y (except when x = ±2). Hence it does not represent a function.
3. Let f: R R where y = x3. Here for each x R we would have a unique value of y in the set R (as cube of any two distinct real numbers are distinct). Hence it would represent a function.