ça va, merci

$\displaystyle \csc(z)=\frac{1}{\sin(z)}$

So it's not defined if $\displaystyle \sin(z)=0$.

$\displaystyle \sin(z)=0 \Leftrightarrow z=k \pi ~,~ k \in \mathbb{Z}$ (the set of all integers)

So for $\displaystyle \csc(3x+\pi)$ to be defined, you need $\displaystyle 3x+\pi$ to be different from $\displaystyle k \pi$

This means $\displaystyle 3x+\pi \neq k \pi \Rightarrow 3x \neq k \pi$ (since k is any integer)

So it's not defined for $\displaystyle x=\frac{k\pi}{3}$, for any integer k.

The domain is then $\displaystyle D=\mathbb{R}- \left\{\tfrac{k \pi}{3} ~:~ k \in \mathbb{Z}\right\}$

Does it look clear ?

For the range... the cosecant's range is $\displaystyle \mathbb{R}-(-1,1)$ (see here :

Cosecant -- from Wolfram MathWorld )

So for any z in the domain of the cosecant, csc(z)$\displaystyle \geq$1 or csc(z)$\displaystyle \leq$-1. Thus 3csc(z)$\displaystyle \geq$3 or 3csc(z)$\displaystyle \leq$-3. --> 3csc(z)-2$\displaystyle \geq$1 or 3csc(z)$\displaystyle \leq$-5.

The range of $\displaystyle 3 \csc(3x+\pi)-2$ is hence $\displaystyle \mathbb{R}-(-5,1)$

For the graph, I have no particular method... I'm sorry, because we never learnt csc so I can't help you much than your book on this :s