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To determine which label represents the function \( f(x) = \ln(-x) \), we need to analyze the properties of this logarithmic function. Here’s a step-by-step breakdown:

1. **Domain of \( f(x) = \ln(-x) \):** This function is only defined for \( x < 0 \), since the argument of the natural logarithm must be positive. So, we are looking for a curve that only exists in the left half of the graph (for negative \( x \) values).

2. **Shape and Behavior of \( \ln(-x) \):** The function \( f(x) = \ln(-x) \) has the general shape of a logarithmic curve. As \( x \) approaches 0 from the left (i.e., \( x \to 0^- \)), \( \ln(-x) \to -\infty \). As \( x \) becomes more negative, the function gradually increases.

3. **Analyzing the Graph:** Looking at the graph, we can observe:
   - **F(x)**: This curve exists only in the left half of the graph and has a shape that matches the expected behavior of \( \ln(-x) \), going to \( -\infty \) as \( x \) approaches 0 from the left and increasing as \( x \) becomes more negative.
   - **Other Curves (G(x), H(x), Q(x))**: These either extend into the right half of the graph or do not match the logarithmic shape we expect from \( \ln(-x) \).

4. **Conclusion**: The function \( f(x) = \ln(-x) \) is represented by the curve labeled **F(x)**.

Would you like more details on why the other curves don't match, or do you have any other questions?

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### Related Questions:
1. What are the properties of the natural logarithm function \( \ln(x) \)?
2. How does the domain of \( \ln(-x) \) differ from \( \ln(x) \)?
3. Why does \( f(x) = \ln(-x) \) tend to \( -\infty \) as \( x \) approaches 0 from the left?
4. How do transformations like reflections affect the domain and range of functions?
5. What are the typical characteristics of exponential and logarithmic graphs?

### Tip:
When dealing with logarithmic functions, always check the argument's sign to determine the function's domain.

Which label represents the function f(x) = ln(-x) in the given graph?

Learn how to identify the graph of the function f(x) = ln(-x) from a set of curves. This problem involves understanding the properties of logarithmic functions and graph analysis, suitable for Grades 11-12.

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