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To determine the domain and range of the function \( f(x) = \sqrt{-3 - 8x} \), we need to follow these steps:

### Domain
Since the function involves a square root, the expression inside the square root must be non-negative to avoid complex numbers. Thus, we require:
\[
-3 - 8x \geq 0
\]
Solving for \( x \):
1. Add 3 to both sides:
   \[
   -8x \geq 3
   \]
2. Divide by -8 (note that dividing by a negative reverses the inequality):
   \[
   x \leq -\frac{3}{8}
   \]

So, the domain of \( x \) is \( (-\infty, -\frac{3}{8}] \).

### Range
For the range, we analyze the values that \( f(x) \) can take based on the square root function.

Since \( f(x) = \sqrt{-3 - 8x} \), the square root function always yields non-negative values, starting from 0 (when \( x = -\frac{3}{8} \)) and increasing as \( x \) decreases further.

Thus, the range of \( f(x) \) is \( [0, \infty) \).

### Final Answer
- **Domain**: \( x \in (-\infty, -\frac{3}{8}] \)
- **Range**: \( y \in [0, \infty) \)

Would you like further details on any part of this solution? Here are some related questions to deepen your understanding:

1. How would the domain change if the function were \( f(x) = \sqrt{3 + 8x} \)?
2. What would be the range if the square root were replaced by a cube root?
3. How do we determine the domain and range for functions without square roots?
4. Can you explain why dividing by a negative number reverses the inequality sign?
5. How does the presence of a negative coefficient affect the domain of functions with square roots?

**Tip:** When working with square roots, remember that the expression under the square root must be non-negative to keep the function within real numbers.

Find the domain and range of the function f(x) = √(-3 - 8x).

Learn how to find the domain and range of the function f(x) = √(-3 - 8x). This solution explains the steps to determine the domain by setting the expression inside the square root non-negative and analyzing the range based on the square root properties. Suitable for high school students studying functions.

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