Q. 5 F4.0( 4 Votes )

Classify the foll

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 + x


Check for Injectivity:


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


f(x) = f(y)


x2 + x = y2 + y


x2 – y2 + x – y = 0


(x – y)( x + y + 1) = 0


Either (x – y) = 0 or ( x + y + 1) = 0


Case i :


If x – y = 0


x = y


Hence f is One – One function


Case ii :


If x + y + 1 = 0


x + y = – 1


x ≠ y


Hence f is not One – One function


Thus from case i and case ii f is not One – One function


Check for Surjectivity:


As


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


f(x) = 1


x2 + x = 1


x2 + x – 1 = 0



Above value of x does not belong to Z


Therefore no values of x 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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