The equation is $$u_{tt} - u_{xx} = 0, x>0, t>0$$ $$u(x,0)=f(x); u_t(x,0)=0$$ $$u_x(0,t)=ku_t(0,t)$$ and the third condition is hard to deal with, as it isn't three standard type of boundary condition (Dirichlet, Neumann and Robin).
It is trivial to solve it in the region $x\geqslant t$, because it is just the case of d'Lambert formula and doesn't involve the boundary condition. Is there some hint to solve it in $x\leqslant t$?
Your idea to extend $f$ to the whole line is correct I think. An extension like $$ f(-y) = cf(y), \qquad (y>0) $$ seems to work, with $c$ depending on $k$ (except for $k=-1$, that I'll have to think more). Problem Unknown time-dependent boundary data for a BVP involving the wave equation and the reference there is somewhat relevant. In particular, differentiability is not required for a weak soluion.