Q. 5 B5.0( 1 Vote )

Classify the following functions as injection, surjection or bijection:

f : Z Z given by f(x) = x2

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, f : Z Z given by f(x) = x2


Check for Injectivity:


Let x1, – x1 be elements belongs to Z i.e such that


So, from definition


x1 ≠ – x1


(x1)2 = ( – x1)2


f(x1)2 = f( – x1)2


Hence f is not One – One function


Check for Surjectivity:


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


f(x) = y


x2 = y



not belongs to Z for non–perfect square value of y.


Therefore no non – perfect square value of y has a pre–image in domain Z.


Hence, f is not Onto function.


Thus, Not Bijective also


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