Q. 8 A3.7( 3 Votes )

Let A = [–1, 1], Then, discuss whether the following functions from A to itself are one – one, onto or bijective:


Answer :

TIP: One – One Function: – A function is said to be a one – one functions or an injection if different elements of A have different images in B.


So, is One – One function


a≠b


f(a)≠f(b) for all


f(a) = f(b)


a = b for all


Onto Function: – A function is said to be a onto function or surjection if every element of A i.e, if f(A) = B or range of f is the co – domain of f.


So, is Surjection iff for each , there exists such that f(a) = b


Bijection Function: – A function is said to be a bijection function if it is one – one as well as onto function.


Now, here f: A A: A = [–1, 1] given by function is


Check for Injectivity:


Let x, y be elements belongs to A i.e. such that


f(x) = f(y)



2x = 2y


x = y


1 belongs to A then



Not element of A co – domain


Hence, f is not One – One function


Check for Surjectivity:


Let y be element belongs to A i.e be arbitrary, then


f(x) = y



x = 2y


Now,


1 belongs to A


x = 2, which not belong to A co – domain


Hence, f is not onto function


Thus, It is not Bijective function


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