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Functions

Definition of Function:
A relation in which no two ordered pairs have same first coordinates is called function.

Domain and Range:  

If f : A→B is a mapping, then A is called the domain of f,  B the codomain of f and and f(A) is called the range of f.



Types of Functions

i) One - One function (or injection):  f:A→B is one - one if f(x1)=f(x2)   then x1=x2.

ii) Onto function (or Surjection): A function f:A→B is said to be an onto function. If f(A), the image of A equals B. That is f is onto if every element of B the codomain is the image of atleast one element of A the domain.
      f:A→B is onto ⇔ for every y∈B there is atleast one x∈A such that
                         f(x) = y
                     ⇔ f(A) = B

iii) One-one onto function (Bijection): A function f:A→B is said to be a bijection if it is both one-one and onto.


iv)Inverse of a function: If f is a function then the set of ordered pairs obtained by interchanging the first and second cordinates of each ordered pairs in f and is denoted by f-1 and read as "f-inverse"

v)Identity Function : A function f:A→B is said to be an identity function on A (denoted by IA) if f(x)=x for all x∈A.
IA is an identity function on A⇔IA(x) = x for all x∈A

vi)Constant Function: A function f:A→B is a constant function if there is an element c∈B such that f(x) = c for all x∈A.
        That is, a constant function is a function whose range is a singleton set.

vii)Equal Function: Two functions f and g defined on the same domain D are said to be equal if f(x)=g(x)

viii)Composite Fuction: Let f:A→B, g:B→C be two functions. Then the composite function of f and g (denoted gof)
                        gof: A→C is defined by
                        (gof)(x) = g(f(x)).

Note:          In the product function gof
                a) The codomain of f is the domain of g.
                b) The domain of gof is the domain of f.
                c) The codomain of gof and g is the same set.
                d) (gof)(x) = g(f(x)).


Sample Questions

From Functions to Algebra

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