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: R R given by


Check for Injectivity:


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


f(x) = f(y)



xy2 + x = yx2 + y


xy2 + x – yx2 – y = 0


xy (y – x) + (x – y) = 0


(x – y)(1 – xy) = 0


Case i :


x – y = 0


x = y


f is One – One function


Case ii :


1 – xy = 0


xy = 1


Thus from case i and case ii f is One – One function


Check for Surjectivity:


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


f(x) = y



x = x2y + y


x – x2y = y


Above value of x belongs to R


Therefore for each element in R (co – domain) there exists an element in domain R.


Hence, f is onto function


Thus, Bijective function


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