Q. 224.0( 2 Votes )

Show that f : R <

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, f : A A given by f(x) = x [x]


To Prove: – f(x) = x [x], is neither one – one nor onto


Check for Injectivity:


Let x be element belongs to Z i.e such that


So, from definition


f(x) = x [x]


f(x) = 0 for


Therefore,


Range of f = [0,1] ≠ R


Hence f is not One – One function


Check for Surjectivity:


Since Range of f = [0,1] ≠ R


Hence, f is not Onto function.


Thus, it is neither One – One nor Onto function


Hence Proved


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