Is h(4) defined? No - that would have to be = f(2(4)) = f(8), but f is only defined on [-1,1].

Is h(2) defined? No - that would have to be = f(2(2)) = f(4), but f is only defined on [-1,1].

Is h(1) defined? No - that would have to be = f(2(1)) = f(2), but f is only defined on [-1,1].

Is h(1/2) defined? Yes - that would have to be = f(2(1/2)) = f(1) (= 1), since f IS defined on [-1,1].

Is h(1/4) defined? Yes - that would have to be = f(2(1/4)) = f(1/2) (= 1/4), since f IS defined on [-1,1].

Any guesses now what the domain of h might be?

Once you know the domain of h, how to find the range of h will hopefully become a little clearer.