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𝒇𝒂 (𝒙) = 𝒂 ∗ 𝒍𝒏 ( 𝟏 : x^2+ 𝟒) 

The image you uploaded contains a math problem involving the function \( f_a(x) = a \cdot \ln\left(\frac{1}{x^2 + 4}\right) \) along with a graph. The problem seems to ask various questions related to this function and its characteristics.

### Here's a breakdown of the task:

1. **Definition Domain (Definitionsbereich) and Range (Wertebereich):**
   - The domain (Definitionsbereich) of the function \( f_a(x) \) is the set of all real numbers \( x \) for which the function is defined.
   - The range (Wertebereich) is the set of all possible values \( f_a(x) \) can take.

2. **Symmetry:**
   - You need to determine whether the function \( f_a(x) \) is symmetric, and if so, whether it's symmetric with respect to the y-axis or origin.

3. **Extreme Points:**
   - This involves finding the critical points of the function by setting the derivative \( f_a'(x) \) equal to zero and solving for \( x \).

4. **Behavior for \( a \to 0 \):**
   - Examine how the function behaves as the parameter \( a \) approaches 0.

5. **Function Properties:**
   - Analyze how the graph changes with different values of \( a \), and find key points like maximum, minimum, or inflection points.

6. **Graph Intersection with the x-axis:**
   - Determine where the function intersects the x-axis by solving \( f_a(x) = 0 \).

7. **Limit Behavior (Verhalten im Unendlichen):**
   - Study the behavior of the function as \( x \) approaches infinity or negative infinity.

8. **Interpretation of the Graph:**
   - The graph provided shows the function for a specific value of \( a \), and you might be asked to interpret various features based on this visual representation.

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### Steps to Address the Problem:
1. **Domain**: Determine the set of \( x \) values for which the argument of the logarithm is positive.
2. **Symmetry**: Analyze whether \( f_a(-x) = f_a(x) \) (even function) or \( f_a(-x) = -f_a(x) \) (odd function).
3. **Extrema**: Compute the first derivative and find where it equals zero.
4. **Limit as \( a \) approaches 0**: Consider the limiting behavior of \( f_a(x) \) as \( a \) approaches 0.
5. **Intersection and Asymptotic Behavior**: Solve for \( x \) where \( f_a(x) = 0 \) and investigate the limit as \( x \) goes to infinity or negative infinity.

Would you like to go through these steps in detail? Or would you prefer focusing on a specific part of the problem?

Explore the function f(a, x) = a * ln(1 / (x^2 + 4)) in detail, including its domain, symmetry properties, critical points, limit behavior as a approaches 0, and graph interpretation. Suitable for advanced high school or college-level mathematics.

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