try sketching the piece-wise function first ...
btw, this problem should be in the precalculus forum ... it's not advanced algebra.
That's certainly one way to do it - although you'd then need to check that any x value you found that makes f zero is in that part of the domain.
To get you started:
Does f(x) = 0 have a solution on the interval x<-1? (That's the interval .)
Solve f(x)=0 on the interval x<-1.
So solve -x=0 on the interval x<-1.
That equation has one solution, x=0, but x=0 isn't in that interval (x<-1, the interval ).
Therefore f(x) = 0 has no solutions on the interval x<-1 ( the interval ).