Page No 10:
Question 1:
Show that the function f: R* → R* defined by
is 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*?
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 exists
such 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.
