2014 AMC 12A Problem 24

Let f_0(x)=x+|x-100|-|x+100|, and for n\geq 1, let f_n(x)=|f_{n-1}(x)|-1. For how many values of x is f_{100}(x)=0?

\textbf{(A) }299\qquad \textbf{(B) }300\qquad \textbf{(C) }301\qquad \textbf{(D) }302\qquad \textbf{(E) }303\qquad

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.