Let's say that we have $$f(x,y,z) = x/k - (y/k) ((z - x/k)/(z - y/k))$$ $$k = constant \in ]0,1[$$
And I need to show in some way that the variable $x$ is more important in some metric that I don't know which one could be good. I thought about analyze the partial derivatives of that functions, but I don't think that is a good way, because one will only see some restricted path through the surface.
Another approach would be do monte carlo simulation in some domain of interest (it exists, actually is a real world problem) and see that the variance of the function value increases more varying x fixing the another variables rather than the same simulation with $x$ fixed.
In some extent one can thinking in this function as a model, and I need to calculate the variable importance, it could be a way as well.
But I'm really puzzled how to approach this problem.
Any metric or some approach is welcomed!
Thank you very much!