Find the domain of the function. \( f(x)=\frac{1}{\sqrt{x+12}} \) The domain of \( f(x) \) is \( (-12, \infty) \). (Type your answer in interval notation.)

To determine the domain of the function \( f(x) = \frac{1}{\sqrt{x+12}} \), you first need to recall that the expression under the square root must be positive for the function to be defined. This means we need \( x + 12 > 0 \), leading us to \( x > -12 \). Given that the square root function cannot be zero (as it would make the denominator undefined), we arrive at the conclusion that \( x + 12 \) must be strictly greater than zero. Thus, the domain of \( f(x) \) is \( (-12, \infty) \). Now let's steer away from the strict definitions and dive into some colorful context! Did you know that the concept of negative and positive numbers has been around since ancient times? The ancient Egyptians used a system that included a form of negatives, but it wasn't until the Middle Ages that Europeans began to truly embrace and understand negative numbers as real quantities, not just debts or deficits! If you're ever puzzled about similar functions, remember that checking for values that make a denominator zero or a square root negative can save you from wrong answers! A common mistake is to forget about the strict inequalities when dealing with square roots. Always keep in mind that the expression inside the root needs to be positive, not just non-negative!