Answer
The domain: $D = \left\{ {\left( {r,s,t} \right)| - 4 \le rst \le 4} \right\}$.
The range is the interval $\left[ {0,4} \right]$.
Work Step by Step
The function $P\left( {r,s,t} \right) = \sqrt {16 - {r^2}{s^2}{t^2}} $ is defined only when $16 - {r^2}{s^2}{t^2} \ge 0$ or ${r^2}{s^2}{t^2} \le 16$. So,
$\left| {rst} \right| \le 4$
$ - 4 \le rst \le 4$
Hence, the domain is $D = \left\{ {\left( {r,s,t} \right)| - 4 \le rst \le 4} \right\}$.
Since ${r^2}{s^2}{t^2}$ is nonnegative, the range of $P$ is the interval $\left[ {0,4} \right]$.
