QUESTION 12 - 1 POINT
Determine the domain and range of the following parabola.
\[
f(x)=-(x+5)^{2}-4
\]

Step 1 ：The domain of a function is the set of all possible input values (x-values) which will produce a valid output. For a parabola, the domain is all real numbers because you can plug any real number into the function and get a real number out.

Step 2 ：The range of a function is the set of all possible output values (y-values). For a downward-opening parabola like this one (since the coefficient of the \(x^{2}\) term is negative), the range is all y-values that are less than or equal to the vertex of the parabola.

Step 3 ：The vertex of the parabola is the point at the top of the parabola (for a downward-opening parabola) and it has coordinates (h, k) where h is the x-value that makes the expression inside the parentheses equal to zero and k is the y-value when you plug h into the function.

Step 4 ：To find the range, we first need to find the vertex of the parabola. The x-coordinate of the vertex is -5 (since that makes the expression inside the parentheses equal to zero) and the y-coordinate is the value of the function when x = -5.

Step 5 ：The y-coordinate of the vertex is -4. So, the range of the function is all y-values that are less than or equal to -4.

Step 6 ：The domain of the function is all real numbers, and the range of the function is \(\boxed{y \leq -4}\).