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Functions or Mappings

Let A and B be two non-empty sets. Then, a rule or a correspondence f which associates to each  , a unique element  is called a function or a mapping from A to B and we write,  f: A
à B. f (x) is called the image of x and x is called the pre-image of f (x).

A is called the domain of f and  is called the range of f.


Various types of functions

I. A function  f : A à B is said to be:

1.  Many–one If two or more than two elements in A have the same image in B.

2.  One–OneIf distinct elements in A have distinct images in B i.e. f is one-one, if


3.  Into If at least one element in B which has no pre-image in A.


One-one mapping is called injective; onto mapping is called surjective and a one-one, onto mapping is called bijective. 

4.  Onto or Surjective functions Let f : A à B. If every element in B has at least one pre–image in A, then f is said to be an onto function.

5.  Identity function – Let A be a non–empty set. Then, the function I defined by I : A à A : I (x) = x for all  is called an identity function on A.  It is a one–to–one onto function with domain A and range A.

6.  Constant functionLet f : A à B, defined in such a way that all the elements in A have the same image in B, then f is said to be a constant function.

7.  Inverse function Let f be a one–one onto function from A to B. Let y be an arbitrary element of B. Then f being onto, there exists an element x A such that f (x) = y.

A function is invertible if and only if f is one–one onto.



.    (i) Even functions
A function f (x) is said to be even if f (-x) = f (x).

       (ii) Odd functionsA function f (x) is said to be odd if f (-x) = -f (x).




III.  Periodic functionsA function f (x) is said to be periodic with period p, if f (x)=f(x+p)


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