Answer
$(-3,-2)\cup(-2,3)$
Work Step by Step
The component functions must all be defined.
(The domain is the intersection of the domains of components)
$x(t)=\displaystyle \frac{t-2}{t+2}$ is defined for $t \neq-2$.
$y(t)= \sin t$ is defined for all real t's.
$z(t)=\ln(9-t^{2})$ is defined for
$9-t^{2}\gt 0$
$t^{2}\lt 9, \quad t\in(-3,3)$
The domain of $\mathrm{r}$ is the intersection of all three,
$(-3,-2)\cup(-2,3)$
