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.

---

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?