Remark
Much of what you need to know is in the graph which I've put below. The graph will give the answer and you can confirm it if need be.
Asymptote
The larger the value of x the closer this graph gets to - 1. To confirm this statement, let x be a relatively small number like (say) 20. Put that through the equation.
y = -2(0.5)^20 - 1
y = 2*0.000000953 which is very nearly 0 even when multiplied by 2.
y ~ -1 so if you go out far enough, your calculator will return a 0.for the first term of your equation.
y = - 1 is the equation of the asymptote.
Domain
Any value in the real number system is a possible member of the domain.
- ∞ < x < ∞
Range
The range is a little different. It goes from -∞ < y < - 1
