The question is a bit ambiguous: you may mean
[tex]\sqrt{3x+8}[/tex]
or
[tex]\sqrt{3x}+8[/tex]
Anyway, in both cases, the domain of a function involving a square root is found by imposing the content of the root to be greateer than or equal to zero. So, in the first case you have
[tex]3x+8\geq 0 \iff 3x\geq -8 \iff x\geq -\dfrac{8}{3}[/tex]
In the second case, you have
[tex]3x\geq 0 \iff x\geq 0[/tex]
