Initial value problem for PDE wave equation using D'Alembert's formula

Question

$$\frac{\partial^2}{\partial t^2}u(x,t)=c^2 \frac{\partial^2}{\partial x^2}u(x,t) , \qquad -\infty<x<\infty, \qquad t>0,$$ \begin{equation}u(x,0)=e^{-x^2}\end{equation} \begin{equation}u_t(x,0)=\cos(2x)\end{equation} I used D'Alembert's formula and obtained the following: \begin{equation}u(x,t)=e^{x^2+c^2t^2}[\cosh(2xct)]+\frac{1}{2c}(\cos(2x)\sin(2ct))\end{equation} this may be correct so far, but my prof said something about extending the solution into the $x<0$ region as D'Alembert's formula is only true for $x>0$. Can someone explain this and help me extend the solution? If D'Alembert's formula is true for $x>0$, why would it not be good for $x<0$?