Q. 5 D5.0( 1 Vote )

Classify the following functions as injection, surjection or bijection:

f : Z Z given by f(x) = x3

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) = x3


Check for Injectivity:


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


f(x) = f(y)


x3 = y3


x3 – y3 = 0


x = y


Hence f is One – One function


Check for Surjectivity:


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


f(x) = y


x3 = y




Since f attain only cubic number like 1,8,27….


Therefore no non – perfect cubic values of y in Z (co – domain) have a pre–image in domain Z.


Hence, f is not onto function


Thus, Not Bijective also


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