Answer
$a.\quad 3$
$b.\quad\{(x,y,z)|\quad x,y,z\geq 0,\quad 4-x^{2}-y^{2}-z^{2} \gt 0\}$,
the interior of a sphere (not the sphere itself) with radius 2, in the 1st octant (see image)
Work Step by Step
$a.$
$f(1,1,1)=\sqrt{1}+\sqrt{1}+\sqrt{1}+\ln(4-1^{2}-1^{2}-1^{2})=3+\ln 1=3$
$b.$
Because of the restrictions for square roots, x,y, and z must be non-negative
(points in the 1st octant or on the positive axes).
Because of the restriction for logarithmic functions,
$4-x^{2}-y^{2}-z^{2} \gt 0$
$4 \gt x^{2}+y^{2}+z^{2}$
which is a region bounded by the circle of radius 2, centered at the origin.
(the sphere itself is not included in the domain)
Combining the two restrictions, the domain of f is
$\quad\{(x,y,z)|\quad x,y,z\geq 0,\quad 4-x^{2}-y^{2}-z^{2} \gt 0\}$,
the interior of a sphere with radius 2, in the 1st octant (see image)
