Answer
$(-1, +\infty)$
Work Step by Step
Recall:
The function $y=\ln{x}$ is defined only when $x\gt0$.
Thus, the function $f(x)=\ln{\frac{1}{x+1}}$ is defined only when $\dfrac{1}{x+1}\gt0$.
Note that $\dfrac{1}{x+1}$ is greater than zero then $x+1\gt0$.
Hence, the given function is defined only when:
\begin{align*}
x+1&\gt0\\
x&\gt -1
\end{align*}
Therefore, the domain of the given function is $(-1, +\infty)$.