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?

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.