The function you've provided is \( f(x) = -3 \ln(x) \), where \( \ln(x) \) represents the natural logarithm of \( x \).

### Key Points to Analyze:
1. **Domain**: Since \( \ln(x) \) is only defined for \( x > 0 \), the domain of the function is \( (0, \infty) \).
2. **Range**: The logarithmic function \( \ln(x) \) tends to negative infinity as \( x \to 0^+ \) and increases without bound as \( x \to \infty \). Multiplying it by -3 reverses the direction:
   - As \( x \to 0^+ \), \( f(x) \to \infty \).
   - As \( x \to \infty \), \( f(x) \to -\infty \).
   - Thus, the range is \( (-\infty, \infty) \).
3. **Derivative**: To find the rate of change, we can differentiate:
   \[
   f'(x) = \frac{d}{dx} \left( -3 \ln(x) \right) = -\frac{3}{x}
   \]
   This tells us that the function is decreasing for all \( x > 0 \), since \( f'(x) < 0 \) for all \( x > 0 \).

4. **Critical Points and Behavior**: 
   - No critical points as \( f'(x) = -\frac{3}{x} \) is never zero.
   - The function has a vertical asymptote at \( x = 0 \).

5. **Graph Behavior**: The graph of \( f(x) \) shows a decreasing curve with a vertical asymptote at \( x = 0 \), heading towards negative infinity as \( x \) increases.

Would you like more detailed information or explanations on any part of this?

### Follow-up Questions:
1. How would the graph of \( f(x) = -3 \ln(x) \) compare to \( f(x) = \ln(x) \)?
2. What happens to \( f(x) \) if we apply a horizontal shift to the logarithm, such as \( f(x) = -3 \ln(x - 2) \)?
3. How can we find the points where \( f(x) \) intersects the \( x \)-axis, if they exist?
4. What are the applications of the logarithmic function in real-world problems?
5. How does the negative coefficient (-3) affect the behavior of the natural logarithm function?

### Tip:
When dealing with logarithmic functions, always check the domain carefully, as \( \ln(x) \) is only defined for positive \( x \)-values.

Explore the function f(x) = -3 ln(x) with a detailed analysis of its domain, range, derivative, and graph behavior. Understand how multiplying by a negative coefficient affects the natural logarithm, suitable for advanced high school and college-level calculus students.

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