Q. 13.9( 96 Votes )

# 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*?

Answer :

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

check for one-one:

For a function to be one-one, if f(x) = f(y) then x = y.

f(x) = f(y)

⇒ x = y

⇒Therefore, f is one – one.

We can see that y ϵ R, there exists , such that

⇒ f is onto.

**Therefore, function f is one-one and onto.**

Now, Let us consider g: N → R_{*} defined by

Then, we get,

g(x_{1}) = g(x_{2})

⇒ x_{1} = x_{2}

⇒ g is one–one.

It can be observed that g is not onto as for 1.2 ϵ R there does not exist any x in N such that

Therefore, function g is one –one but not onto.

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