Q. 1 A3.8( 17 Votes )

Give an example of a function

Which is one – one but not onto.

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


Now, Let, given by f(x) = x2


Check for Injectivity:


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


So, from definition


f(x) = f(y)


x2 = y2


x2 – y2 = 0


(x – y)(x + y) = 0


As therefore x + y>0


x – y = 0


x = y


Hence f is One – One function


Check for Surjectivity:


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


f(x) = y


x2 = y



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


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


Hence, given by f(x) = x2 is One – One but not onto.


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