What is the largest domain of the f(x)=x^2-x-12 which has an inverse function?
The inverse relation to a vertical parabola is a horizontal parabola. So where is the vertex of the inverse relation? That will be the starting point for your domain.
I don't understand the question. That function cannot have an inverse since it is not a 1-1 function.
No, but you can define new functions having that same "formula" but restricted domains that do have inverses. The crucial point is, as topsquark suggests, that $\displaystyle x^2- x- 12= x^2- x+ 1/4- 49/4= (x- 1/2)^2- 49/4$.