Domain of Dependence of a Scalar first-order Hyperbolic Conservation Law

Domain of Dependence of a Scalar first-order Hyperbolic Conservation Law

For a linear advection equation: $u_t+cu_x =0$ on
$(x,t)\in(-\infty,\infty)\times[0,\infty)$ With Cauchy initial data :
$u(0,t) = u_0(x)$
The dependence of a solution at a fixed space-time point $u(X,T)$ is a
single point $X-cT$ since $u(X,T)=u_0(X-cT)$,and characteristics are
straight lines with constant slope (constant wave velocity).
What about this scalar pde:
$u_t+f(u)_x =0 $ where $f''(u)<0$ on
$(x,t)\in(-\infty,\infty)\times[0,\infty)$
and $u(0,t) = u_0(x)$
My particular case is a concave, smooth quadratic flux function. Here
$f'(u)$ will produce shock and rarefaction waves, so how do we derive the
domain of dependence of the solution at a fixed space-time point $u(X,T)$
for this case?