Answer
$z=\sqrt{1-x^{2}-y^{2}}$
Work Step by Step
The whole sphere:$\quad x^{2}+y^{2}+z^{2}=1$
"Upper hemisphere" restricts the z-coordinates to $z\geq 0.$
Combined,
$ x^{2}+y^{2}+z^{2}=1,\quad z\geq 0.$
Express z in terms of x and y
$z^{2}=1-x^{2}-y^{2},\quad z\geq 0.$
Take the square root (result is nonnegative, so we drop the condition on z)
$z=\sqrt{1-x^{2}-y^{2}}$
