Q. 1 B5.0( 4 Votes )

Give an example of a function

Which is not one – one but 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) = x3 – x


Check for Injectivity:


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


So, from definition


f(x) = f(y)


x3 – x = y3 – y


x3 – y3 – (x – y) = 0


(x – y)(x2 + xy + y2 – 1) = 0


As x2 + xy + y2 ≥ 0


therefore x2 + xy + y2 – 1≥ – 1


x – y≠0


x ≠ y for some


Hence f is not One – One function


Check for Surjectivity:


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


f(x) = y


x3 – x = y


x3 – x – y = 0


Now, we know that for 3 degree equation has a real root


So, let be that root




Thus for clearly , there exist such that f(x) = y


Therefore f is onto


Hence, given by f(x) = x3 – x is not One – One but onto


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