A first order hyperbolic problem using method of characteristics

Question

Consider the initial value problem for the equation $u_t+au_x=f(t,x)$ with $u(0,x)=1$ and $f(t,x)=1$ if $x\ge 0$ and $0$ otherwise. Assume that $a$ is positive. Show that the solution is given by $u(t,x)=0$ if $x\leq 0$, $x/a$ if $x\ge 0$ and $x-at\leq 0$ and $t$ if $x\ge 0$ and $x-at\ge 0$