Buenos dias

For the period, let t be the period, t>0.

f(x+t)=f(x)

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)

so 3t=2pi

--> t=2pi/3

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 ?