Answer
1) Domain: $\left[-\frac{5}{3},\infty\right)$
2) Range: $[0,\infty )$
Work Step by Step
Given \begin{equation}
h(x)=\sqrt[4]{3 x+5}= \sqrt[4]{3 \left(x+\frac{5}{3}\right)}. \end{equation}
a) This is an even root function because the index, $n=4$, is even. The radicand must be positive. So, we require $$3\left(x+\frac{5}{3}\right)\geq 0\implies x\geq -\frac{5}{3}.$$ 1) The domain is $$\left[-\frac{5}{3},\infty\right)$$.
2) The range is $[0,\infty )$.
See the graph for proof.
