y=(1/x)-x, xy=1-x2,
x2+xy-1=0, x=(-y±√(y2+4))/2, which means the inverse can be one of two possibilities.
If g(y)=(-y±√(y2+4))/2, then g(x)=(-x±√(x2+4))/2, that is, g(x)=(-x+√(x2+4))/2 or g(x)=(-x-√(x2+4))/2.
If y=f(x) then there are two possible inverses shown by g(x).
An even function obeys the rule f(x)=f(-x) and an odd function obeys the rule f(x)=-f(-x).
f(-x)=(-1/x)+x=-[(1/x)-x]=-f(x), so it's odd.