Equation of state
For ideal gases, Pv=nRT and therefore Pv-nRT=0
That is a simple equation, and it shows that there is a function f with f(P,v,T)=0, namely f(P,v,T)=Pv-nRT.
There is another function f* with f*(P,v,T)=1:
As you can see, that function does not help - it just gives an additional constant.
Non-ideal gases have more complicated laws, but the idea is the same.
Edit: Oh, forgot to add n.