f(x+t)=3 csc (3x+3t+pi)-2
but you know that csc has a period 2pi, that is to say csc(z+2pi)=csc(z)
the domain is the set of all real numbers, except the ones for which csc is undefined (which are the ones that annulate a sine) :
solve for x in 3x+pi=k*pi, for any (positive or negative) integer k.
do these help ?