Answer
$(-1,2]$
Work Step by Step
The component functions must all be defined.
(The domain is the intersection of the domains of components)
$x(t)=\sqrt{4-t^{2}}$ is defined for
$4-t^{2}\geq 0$
$t^{2}\leq 0$
$-2\leq t \leq 2,\quad t\in[-2,2]$
$y(t)= e^{-3t}$ is defined for all real t's.
$z(t)=\ln(t+1)$ is defined for
$t +1\gt 0$
$t\gt-1,\quad t\in(-1,+\infty)$
The domain of $\mathrm{r}$ is the intersection of all three,
$(-1,2]$
