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The problem is asking to explain why the square root of \(-x\), represented as \(\sqrt{-x}\), is not necessarily undefined and how that affects the domain and range of the function \(f(x) = \sqrt{-x}\).

### Explanation:
1. The square root of a **negative number**, such as \(\sqrt{-144}\), is undefined in the real number system because there is no real number that, when squared, gives a negative result. In the real number system, only positive numbers have real square roots.

2. However, for \(\sqrt{-x}\), the expression depends on **x**. When \(x\) is positive, \(-x\) becomes negative. For example, if \(x = 144\), then \(\sqrt{-144}\) is undefined (as already stated).

3. Now, if we examine \(f(x) = \sqrt{-x}\), to ensure that the expression is **defined** for real numbers, we must ensure that the expression inside the square root, \(-x\), is non-negative (since square roots of negative numbers are undefined in the real number system). This means:
   \[
   -x \geq 0 \implies x \leq 0.
   \]
   Therefore, the domain of \(f(x)\) consists of values of \(x\) that are less than or equal to zero (i.e., \(x \leq 0\)).

4. In this case, the square root of \(-x\) is defined for all non-positive values of \(x\) (i.e., \(x \leq 0\)) because when \(x\) is non-positive, \(-x\) becomes non-negative, and we can take the square root of a non-negative number.

### Domain and Range of \(f(x) = \sqrt{-x}\):
- **Domain**: As we concluded, the function is only defined when \(x \leq 0\). Therefore, the domain is \( (-\infty, 0] \).
  
- **Range**: Since \(\sqrt{-x}\) results in a non-negative number (because square roots of non-negative numbers are non-negative), the range of \(f(x)\) is \( [0, \infty) \).

### Conclusion:
The function \(f(x) = \sqrt{-x}\) is defined for \(x \leq 0\), and its range is all non-negative real numbers.

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### Related Questions:
1. What is the difference between the square root of a negative number and a complex number solution?
2. How would the domain change if the function were \( f(x) = \sqrt{x^2 - 4} \)?
3. What is the geometric interpretation of the function \( f(x) = \sqrt{-x} \)?
4. How does the concept of imaginary numbers relate to the square root of negative numbers?
5. How would the range change if the function were \( f(x) = -\sqrt{-x} \)?

### Tip:
Remember, the square root of a negative number is undefined in the real number system but can be expressed in the complex number system as an imaginary number (e.g., \( \sqrt{-1} = i \)).

The square root of a negative number, such as √-144, is undefined. Explain why the square root of -x, √-x, is not necessarily undefined and what this means about the domain and range of f(x) = √-x.

Explore why the square root of -x, represented as √-x, is not always undefined and learn how to determine the domain and range of the function f(x) = √-x. This explanation covers key mathematical concepts including square root properties and inequalities, suitable for high school students in Grades 9-12.

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