Answer
(a) All points in 2-space where y \geq 2
(b) All points in 3-space where $x^2+y^2+z^2 \leq 25$
(c) All points in 3-space
Work Step by Step
(a) $f(x,y) = xe^{-\sqrt{y+2}}$
We find the domain
$y+2 \geq 0$
$y \geq -2$
Domain = $\{x,y)\in R^2|y\geq-2\}$
The domain is the set of points $(x,y)$ so that $y$ is greater or equal to $-2$.
(b) $f(x, y, z) = \sqrt {25 − x^2 − y^2 − z^2}$
We find the domain:
$25 − x^2 − y^2 − z^2 \geq 0$
$x^2 + y^2 + z^2 \leq 0$
Domain = $\{(x,y,z)\in R^3| x^2 + y^2 + z^2 \leq 25\}$
The domain is the set of point $(x,y,z)$ inside or on the sphere of center $(0,0,0)$ and radius $5$.
(c) $f(x, y, z) = e^{xyz}$
We find the domain.
Here $f$ is defined for all real values of $x, y$ and $z$.
Domain = $\{(x,y,z)\in R^3\}$
