A new approach to numerically solve the forward-backward functional differential equation

x'(t) = ax(t) bx(t-1) cx(t 1) f(t) (1)

is presented, where a, b, and c are constant parameters and f is a real-valued continuously differentiable function. The step by step version of the Tau method was applied to approximate the solution of equation (1) by a piecewise polynomial function. Examples of a boundary value problem and an initial value problem were posed, solved with the proposed method, and analyzed. The obtained results in the boundary value problem were compared with those produced by other methods found in the literature. For the initial value problem we provided the approximation of the forward solution of (1). We conclude that the excellent numerical results and the simplicity of the Tau method demonstrate the method`s versatility for solving problems defined by equation (1).