Question

Show that the function f: R_* → R_* defined byis one-one and onto, where R_* is the set of all non-zero real numbers. Is the result true, if the domain R_* is replaced by N with co-domain being same as R_*?

Answer 1

It is given that f: R_* → R_* is defined by

One-one:

∴f is one-one.

Onto:

It is clear that for y∈ R_*, there existssuch that

∴f is onto.

Thus, the given function (f) is one-one and onto.

Now, consider function g: N → R_*defined by

We have,

∴g is one-one.

Further, it is clear that g is not onto as for 1.2 ∈R_* there does not exit any x in N such that g(x) =.

Hence, function g is one-one but not onto.