Where is f defined? It's a power series, so you need to find its radius of convergence R. The function will be defined in the interval of convergence (–R,R) (and maybe also at the endpoints of that interval).
Is it monotonic? You can differentiate a power series term by term within its radius if convergence. So do that, and see whether the derivative always has the same sign.
If it is strictly monotonic, then as x goes along the interval of convergence the function will take each value exactly once. The extreme values will be taken at the ends of the interval of convergence. Is the greatest value of the function greater than 1.5, and is the least value less than –0.5? If so, then it must take each of those two values exactly once.
Linguistic note: monotonous means boring and dreary; monotonic means always going the same way (in other words, always increasing or always decreasing).