Question 1189172
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            I will use the specific of the given function,  and will provide more simple solution.



<pre>
We are given the function  f(x) = x - {{{1/x}}}.


In part (a), the domain of this function is  (-oo,0) U (0,oo).


Notice that there is an identity  f(x) = {{{f(-1/x)}}},  valid for all x from the domain  (check it on your own).


It means that each image of f(x) has two pre-images: x and {{{-1/x}}}.

One of these pre-images is positive number; the other pre-image is negative number.


So, the function f(x), defined on the whole domain, is not one-to-one function; 

THEREFORE, it has no an inverse function.




In part (b), the domain is restricted: now it is the set of all positive numbers.

On this set, the function f(x) = x - {{{1/x}}} is monotonically increasing  
(check it on your own, for example, taking the derivative).


Hence, it is one-to-one map.


Therefore, on the restricted domain, this function has an inverse function.
</pre>

Solved.