Simple equation Range thing...

I need the Range for

f(x)= Sqrt[3 - |3 x| - |3 x - 2|]
those |x| are absolute values.

I try to get the f(x)^-1 but I don't think it's even possible. I get to:

y^2 - 3 = -|3x| - |3x-2|

I honestly don't know how to get it, I know it's [0,1], because of the graph, but I need to do it manually, not using a graph from Mathematica.

Edit: well the actual question is when does this function reach the values (if it does) of 2, 1, 1/2, 0 and -1. I know it won't get to 2, or -1, because of the graph, but that's about it.

Branch up the funtion in different cases by considering when the argument for the absolute value is higher and lower than zero. Then analyze the resulting function.

The three intervals that derive from doing this are:
(-oo,0) ; [0,2/3) ; [2/3;+oo)

The function value on these intervals is, respectively:
sqrt(1+6x) ; sqrt(1) ; sqrt(5-6x)

This furthers restricts our domain for this funcion to:
[-1/6,0) ; [0,2/3) ; [2/3,5/6]

From there it's easy to see that the funcion will be limited between [0,1]

Branch up the funtion in different cases by considering when the argument for the absolute value is higher and lower than zero. Then analyze the resulting function.

The three intervals that derive from doing this are:
(-oo,0) ; [0,2/3) ; [2/3;+oo)

The function value on these intervals is, respectively:
sqrt(1+6x) ; sqrt(1) ; sqrt(5-6x)

This furthers restricts our domain for this funcion to:
[-1/6,0) ; [0,2/3) ; [2/3,5/6]

From there it's easy to see that the funcion will be limited between [0,1]

I already had the domain but I hadn't thought about that, thanks a lot.