Finding the set of all possible values of a function similar to Nesbitt's Inequality 
Let $x,$ $y,$ $z$ be positive real numbers. Find the set of all possible values of
$$f(x,y,z) = \frac{x}{x + y} + \frac{y}{y + z} + \frac{z}{z + x}.$$

This seems extremely similar to Nesbitt's inequality, in which I did some reasearch on this problem to find. Nesbitt's states that for positive real $a, b, c,$ then $$\displaystyle\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}\geq\frac{3}{2}.$$ However I note that the function stated in the problem isn't in the same orientation to apply Nesbitt's, and just similar. I'm stumped on making any progression on this problem, as I've tried combining the denominators to form one big fraction as well as substituting variables to try clearing denominators. I would appreciate some help to start this problem.

I think $1 < f(x,y,z) < 2.$ Indeed, because
$$\frac{x}{x+y} \geqslant \frac{x}{x+y+z}.$$
Equality occur when $x = 0$ or $z = 0.$
Therefore
$$f(x,y,z) \geqslant \frac{x+y+z}{x+y+z} = 1.$$
But $x,y,z$ are positive real numbers, so $f(x,y,z) > 1.$
Another
$$\frac{x}{x+y} < \frac{x+z}{x+y+z},$$
equivalent to
$$\frac{yz}{(x+y)(x+y+z)}\geqslant 0.$$
Equality occur when $yz=0.$ So
$$f(x,y,z) < \frac{2(x+y+z)}{x+y+z}=2.$$
