$$\frac{\partial ^2 u}{\partial t^2}~=~c^2\frac{\partial ^2 u}{\partial x^2}$$ BVP: $\begin{cases}u_{tt}-u_{xx}=0,~-t<x<t,~0<t\\ u(-t,t)=b(t),~0 \leq t\\ u(t,t)=a(t),~~~~0 \leq t\end{cases}$(Boundary conditions)
Asking for verification of the general solution
$\Phi(x,t)~=~\mathcal{F}(x-ct)+\mathcal{G}(x+ct)$(Solution to the BVP)
It is just when domain is infinity or also when the domain is final (like [-1,1])?
Hint:
$\begin{cases}u(t,t)=a(t)\\u(-t,t)=b(t)\end{cases}$ :
$\begin{cases}F((c+1)t)+G((c-1)t)=a(t)\\F((c-1)t)+G((c+1)t)=b(t)\end{cases}$