Please do C D and E For the given f Z right arrow Z decide w

Please do C, D, and E

For the given f: Z right arrow Z, decide whether f is onto and whether it is one-to-one. Prove that your decisions are correct.

Solution

c) onto as range of f(x) is Z and f(x) = odd intergers when x is even , f(x) = even intergers when x is odd

Thus even integers + odd integers + 0 (when x is -1) = Z

one to one as no element in x will give repeatable values as functions are increasing functions

d) onto as x/2 covers whole range of intergers

not one to one as as only x/2 covers the whole range (when x is even ) , repeated values will come when x is odd

e) not onto as only multiples of 3 and 2 are getting in the range

not one to one as it will be get repeated many times ( as in case 6 (for x=2) and 6 (for x=3))