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The function provided is a piecewise function:

\[
f(x) =
\begin{cases}
-x^2 - 10 & \text{if } 0 < x < 6 \\
-4x^2 - 7 & \text{if } x \geq 6
\end{cases}
\]

### Domain Analysis:
- The first piece, \( -x^2 - 10 \), is defined for \( 0 < x < 6 \), meaning the function is valid for all \( x \) strictly between 0 and 6 (but not including 0 or 6).
- The second piece, \( -4x^2 - 7 \), is defined for \( x \geq 6 \), which means the function is valid for \( x \geq 6 \).

### Domain of the Function:
The domain includes values where each piece is defined. So, the domain is:

\[
x \in (0, \infty)
\]

This means \( f(x) \) is defined for all \( x > 0 \), since the function handles values both less than and greater than 6 through its piecewise definition.

Would you like further clarification or more details?

Here are some related questions to expand on this concept:

1. What are the ranges of the two pieces of the function?
2. How does the function behave at \( x = 6 \), given that the pieces meet there?
3. Could there be any discontinuities in the function based on its piecewise definition?
4. How would you compute \( f(6) \), if required?
5. Can we graph the function to visualize how it behaves across its domain?

**Tip:** When finding the domain of piecewise functions, always carefully analyze the conditions under which each piece is defined.

Find the domain of the following piecewise function: f(x) = { -x^2 - 10 if 0 < x < 6, -4x^2 - 7 if x >= 6 }

Learn how to find the domain of a piecewise function with quadratic components. This problem involves analyzing the function's defined intervals and ensuring continuity. Suitable for high school students in Grades 10-12.

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