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To solve this problem, we need to understand the function given: \( f(x) = -\sqrt{x} \). This function is defined only for \( x \geq 0 \) because the square root of a negative number is not defined in the real number system.

For each of the given points \((x, y)\), we will check if the \( y \)-value matches \( f(x) \).

### Step-by-step evaluation:
1. **\( f(x) = -\sqrt{x} \)** means we find the square root of \( x \) and take its negative.

2. We'll evaluate the function for the provided options:

   - **Option A: \((-9, 3), (-4, 2), (-1, 1)\)**
     - \(\sqrt{-9}\) is not defined in the reals.
     - \(\sqrt{-4}\) is not defined in the reals.
     - \(\sqrt{-1}\) is not defined in the reals.
     - None of these points are valid.

   - **Option B: \((1, 1), (4, 2), (9, 3)\)**
     - For \((1, 1)\): \( f(1) = -\sqrt{1} = -1\), not \( 1 \).
     - For \((4, 2)\): \( f(4) = -\sqrt{4} = -2\), not \( 2 \).
     - For \((9, 3)\): \( f(9) = -\sqrt{9} = -3\), not \( 3 \).
     - None of these points are valid.

   - **Option C: \((-9, -3), (-4, -2), (-1, -1)\)**
     - \(\sqrt{-9}\), \(\sqrt{-4}\), and \(\sqrt{-1}\) are not defined in the reals.
     - None of these points are valid.

   - **Option D: \((1, -1), (4, -2), (9, -3)\)**
     - For \((1, -1)\): \( f(1) = -\sqrt{1} = -1\).
     - For \((4, -2)\): \( f(4) = -\sqrt{4} = -2\).
     - For \((9, -3)\): \( f(9) = -\sqrt{9} = -3\).
     - All three points match.

### Conclusion:
**Option D: \((1, -1), (4, -2), (9, -3)\)** contains points that lie on the graph of \( f(x) = -\sqrt{x} \).

Would you like further details or have any questions?

### Related Questions:
1. How would the graph of \( f(x) = \sqrt{x} \) differ from \( f(x) = -\sqrt{x} \)?
2. What is the domain and range of the function \( f(x) = -\sqrt{x} \)?
3. How do you find the square root of a negative number in complex numbers?
4. What transformations are applied to the function \( f(x) = \sqrt{x} \) to obtain \( f(x) = -\sqrt{x} \)?
5. How would you plot \( f(x) = -\sqrt{x} \) using a graphing calculator?

### Tip:
Remember, for square root functions like \( f(x) = \sqrt{x} \) or \( f(x) = -\sqrt{x} \), the domain is restricted to non-negative values of \( x \).

Use the graphing calculator to graph the function f(x) = -√x. Which list contains three points that lie on the graph of the function?

Explore how to determine which points lie on the graph of the function f(x) = -√x. Learn step-by-step evaluations of each option, highlighting the domain restrictions and transformations. Suitable for high school students in Grades 9-12.

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