Find the domain of the function. \[ f(x)=\sqrt{\frac{10}{x+2}} \] Write your answer as an interval or union of intervals. \begin{tabular}{rrr|}\hline\( \frac{\square}{\square} \) & \( \infty \) & \( -\infty \) \\ \hline\( (\square, \square) \) & {\( [\square, \square] \)} & \( (\square, \square] \) \\ \hline\( \square, \square) \) & \( \square \) & \( \square \cup \square \) \\ \hline\end{tabular}

To find the domain of the function \( f(x) = \sqrt{\frac{10}{x+2}} \), we need to ensure that the expression inside the square root is non-negative. This means that the denominator, \( x + 2 \), must be positive (since we cannot divide by zero or take the square root of a negative number). Setting up the inequality: \[ x + 2 > 0 \] Solving this gives: \[ x > -2 \] At \( x = -2 \), the function is undefined because it leads to division by zero. Thus, the domain of the function can be expressed as: \[ \text{Domain: } (-2, \infty) \] Now, let’s take a fun detour into the historical roots of square roots and functions! Did you know that the ancient Babylonians were among the first to use square roots, around 2000 BC? They had methods to approximate square roots, and they wrote them on clay tablets, laying the groundwork for mathematical concepts we still use today! Also, a common mistake when dealing with functions like these is to forget about the restrictions posed by denominators! Always check whether the denominator can equate to zero or go negative, as it can lead to undefined points in your function. So remember, when you play with functions, check their limits before your reach!