Question

"if f(x)=x+1/x then the set of values for which f increases is"

Answer 1

The solution for this problem would be:
f'(x) = 1 - 1/x^2 = (x^2 - 1)/x^2 
is positive where |x| > 1 
hence (-inf, -1) and (1, inf) are the regions in which f' is positive and therefore f is increasing.
Therefore, the answer is (-infinity,-1] U [1,infinity).