Maths JEE

Definition of Function

The relation f: A → B is a function if all elements in A have their images in B and to each element in A there exists one and only one image in B.

Number of Functions Formula

If set A has m elements and set B has n elements then the total number of functions from A to B is n^m.

Domain and Range

Let f: A → B then set A is known as the domain of f and set B is known as its co-domain; f(A) is the range of f.

One-One Function

A function from X to Y is one–one iff x1 ≠ x2 ⇒ f(x1) ≠ f(x2).

One-One Function Count

If A and B are finite sets with m and n elements then number of one–one functions from A to B is nPm if n ≥ m and 0 if n < m.

Many-One Function

A function f: A → B is many–one if two or more elements of A have the same image in B.

Onto Function

A function f: A → B is onto if every element of B is the f-image of some element of A.

Onto Function Count

If A and B are two sets with m and n elements such that 1 ≤ n ≤ m then number of onto functions from A to B is ∑ (-1)^(n-r) * C(n

Into Function

A function f: A → B is into if there exists an element in B having no pre-image in A.

Bijection Definition

Function f: A → B is a bijection if it is one–one as well as onto.

Bijection Count

If A and B are finite sets and n(A)=n(B) then number of bijections from A onto B is n!.

Equality of Functions

Two real valued functions f and g are equal iff Domain of f = Domain of g and f(x)=g(x) ∀ x in Domain of f.

Constant Function

A function f defined by f(x)=c for each x ∈ R is called a constant function; Domain: R

Identity Function

A function f defined by f(x)=x for each x ∈ R is called an identity function; Domain: R