Module 1 : Real Numbers, Functions and Sequences
Lecture 1 : Real Numbers, Functions   [ Section 1.2 : Functions ]
 
1.2     
 FUNCTIONS :
  You already have some familiarity with the concept of a function. Function is a kind relation between various objects. For example, the volume  of a cube is a function of its side  ; in physics velocity of a body at any time is a function of its initial velocity and time, and acceleration; and so on. In mathematics, a function is defined as follows:
   
1.2.1  
 Definition :
(i)


 For sets    and   , a  function from    to    , denoted by  ,  is a correspondence which assigns to every element     ,  a unique element  ( )   . The value of the function at an element in is denoted by  (  ), which is an element in . This is indicated by  .
(ii)


 For a function  , the set    is called the domain of    and the subset  of , (set of images of  ) is called the range of  .  If    , then    is said to be real-valued. If also, then the natural domain of   is the set of all   for which .
 
1.2.2  
 Examples :
(i) 


 Let . Then,   has natural domain . Its range is also , because for any given ,
if  then we get .

(ii)

Let . Then,   has natural domain , and its range is given by .
(iii)
  Let .Then,  has natural domain , and its range is given by .
1.2.3
Definition :
(i)

 A real-valued function  is said to be bounded if its range is a bounded subset of  , that is, there is some
          such that  |   ( ) |     for all    .
(ii)

 Let   :    . The graph of  is the set :
        G (  )  =  { ( ,  (  ) )  :    }   x  .
(iii)


Let  :   B   be a function. We say  is one-one (or injective) if
        .
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