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The image shows a question written in German: 

"Untersuchen Sie den Graphen der Funktion \( f(x) = \cos(x^3) \) auf Symmetrie."

Translation: 
"Examine the graph of the function \( f(x) = \cos(x^3) \) for symmetry."

### Symmetry Analysis:
To investigate the symmetry of the function \( f(x) = \cos(x^3) \), we typically check for two types of symmetry: 
1. **Even Symmetry (Symmetry about the y-axis)**: A function is even if \( f(-x) = f(x) \).
2. **Odd Symmetry (Symmetry about the origin)**: A function is odd if \( f(-x) = -f(x) \).

#### 1. **Even Symmetry**:
Check if \( f(-x) = f(x) \):
\[
f(-x) = \cos((-x)^3) = \cos(-x^3)
\]
Using the fact that \( \cos(-\theta) = \cos(\theta) \) (cosine is an even function):
\[
f(-x) = \cos(x^3)
\]
Thus, \( f(-x) = f(x) \), meaning the function is **even** and symmetric about the y-axis.

#### Conclusion:
The function \( f(x) = \cos(x^3) \) is an even function, meaning it exhibits symmetry with respect to the y-axis.

Would you like more details on any specific aspects? Here are five related questions to deepen your understanding:

1. What is the difference between even and odd functions in terms of symmetry?
2. How would you graph \( f(x) = \cos(x^3) \) and verify its symmetry visually?
3. How do transformations (like shifts or stretches) affect the symmetry of a function?
4. Can you apply this symmetry concept to other trigonometric functions, such as \( \sin(x^3) \)?
5. What is the role of function composition (like the \( x^3 \) inside \( \cos \)) in determining symmetry?

**Tip:** When checking symmetry, always test both \( f(-x) = f(x) \) for even symmetry and \( f(-x) = -f(x) \) for odd symmetry.

Untersuchen Sie den Graphen der Funktion f(x) = cos(x^3) auf Symmetrie.

Explore the symmetry of the function f(x) = cos(x^3). This detailed solution checks for even and odd symmetry using mathematical techniques and explains why the function is even and symmetric about the y-axis.

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