Find the domain of the function.
\[
f(x)=\log _{3}\left(\frac{3}{x+2}\right)
\]

Step 1 ：The domain of a function is the set of all possible input values (often referred to as 'x' values) which will output real numbers.

Step 2 ：For the function \(f(x) = \log_3\left(\frac{3}{x+2}\right)\), the argument of the logarithm, which is \(\frac{3}{x+2}\), must be greater than zero. This is because the logarithm of a negative number or zero is undefined in the real number system.

Step 3 ：So, we have the inequality: \(\frac{3}{x+2} > 0\)

Step 4 ：To solve this inequality, we first find the critical points by setting the numerator and denominator equal to zero.

Step 5 ：For the numerator, 3 = 0, there are no solutions.

Step 6 ：For the denominator, x + 2 = 0, the solution is x = -2.

Step 7 ：Now, we test the intervals determined by the critical point x = -2.

Step 8 ：For x < -2, the expression \(\frac{3}{x+2}\) is positive because a positive number divided by a negative number is negative, and the logarithm of a negative number is undefined.

Step 9 ：For x > -2, the expression \(\frac{3}{x+2}\) is positive because a positive number divided by a positive number is positive, and the logarithm of a positive number is defined.

Step 10 ：Therefore, the domain of the function \(f(x) = \log_3\left(\frac{3}{x+2}\right)\) is x > -2.

Step 11 ：\(\boxed{x > -2}\)