The function given is $f(x) = \sqrt{-7 - x}$.

To determine the domain of this function:

1. The square root function is defined for non-negative values (i.e., the expression inside the square root must be greater than or equal to zero).

2. Therefore, we need: $-7 - x \geq 0$ Solving this inequality: $-x \geq 7$ Multiply both sides by $-1$, which reverses the inequality: $x \leq -7$

Domain:

The domain of the function is all values of $x$ such that $x \leq -7$.

Expressing in interval notation:

The domain is $(-\infty, -7]$.

Would you like more details or have any questions?

Here are some related questions:

1. What happens to the domain if the function is $f(x) = \sqrt{x + 7}$?
2. How does the domain change for $f(x) = \sqrt{-7 - 2x}$?
3. What is the domain of $f(x) = \sqrt{x^2 - 4x - 12}$?
4. How can we graph the function $f(x) = \sqrt{-7 - x}$?
5. What is the range of the function $f(x) = \sqrt{-7 - x}$?

Tip: Always remember that square roots of negative numbers are undefined in the real number system.