Answer
$[-4,0) \cup (0,1]$
Work Step by Step
Interpret $f(x) $ as $u(x)-v(x).$ The domain of $f$ is the intersection of domains of $u$ and $v$.
$u(x)=\sqrt{x+4}$ is defined for all $x$ such that $x+4\geq 0.$
The domain of $u$ is $[-4,+\infty)$.
$v(x)=\frac{\sqrt{1-x}}{x}$ is defined for $x$ satisfying:
$\left\{\begin{array}{ll}
1-x\geq 0, & and\\
x\neq 0 &
\end{array}\right.$
that is, for
$\left\{\begin{array}{ll}
x\leq 1, & and\\
x\neq 0 &
\end{array}\right.$
that is, $(-\infty,0) \cup (0, 1]$
The domain of f is the intersection of these two: $[-4,0) \cup (0,1]$
