x=-3|y|-3.
When y=0, x=-3. That’s the point (-3,0).
When y>0, x=-3y-3, so 3y=-x-3, y=-(x+3)/3. But -(x+3)/3>0, -x-3>0, x<-3. So to keep y positive, x must be less than -3.
When y<0, x=3y-3, so y=(x+3)/3. But x+3<0, x<-3. To keep y negative, x must be less than -3, the same as for y>0.
In finding points on the inverse function, we now know that x≤-3, and apart from (-3,0), each value of x generates two values for y. For example, x=-4: y=(x+3)/3=-⅓; and y=-(x+3)/3=⅓. A graph would look like: