# Q.1 Show that the function  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∗?

Given,  is defined by  .

One - One :

f is one-one.

Onto:

We have  , then there exists      ( Here ) such that

.

Hence, the function is one-one and onto.

If the domain R is replaced by N with co-domain being same as R∗   i.e.    defined by

g is one-one.

For    ,

but there does not exists any x in N.

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

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