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Question 1

Show that the function f : R* → R* defined by f(x) = 1/x 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

Answer

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 → Rdefined 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.

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