Find the domain of the function. \[ f(x)=\frac{x-2}{\sqrt{x+1}} \] Write your answer as an interval or union of intervals.

To find the domain of the function \[ f(x) = \frac{x-2}{\sqrt{x+1}}, \] we need to consider the conditions under which the function is defined. 1. The denominator \(\sqrt{x+1}\) must be positive because we cannot divide by zero and the square root must be defined. Therefore, we require: \[ x + 1 > 0. \] This simplifies to: \[ x > -1. \] 2. Since the square root must be positive (not just non-negative), we have: \[ \sqrt{x+1} > 0 \implies x + 1 > 0 \implies x > -1. \] Thus, the only restriction on \(x\) is that it must be greater than \(-1\). Combining these conditions, we find that the domain of \(f(x)\) is: \[ (-1, \infty). \] Therefore, the domain of the function \(f(x)\) is \[ \boxed{(-1, \infty)}. \]