The function you've provided is \( f(x) = e^{-x^2} \).

This function is a Gaussian function, commonly known as the normal distribution curve when normalized. The function is defined for all real numbers \( x \) and has the following key characteristics:

### Key Characteristics:
1. **Symmetry**: The function is symmetric about the y-axis, meaning \( f(x) = f(-x) \). This is because the exponent \( -x^2 \) remains unchanged when \( x \) is replaced with \( -x \).
   
2. **Maximum Value**: The maximum value of \( f(x) \) occurs at \( x = 0 \), where \( f(0) = e^0 = 1 \).

3. **Asymptotic Behavior**: As \( x \) approaches \( \pm \infty \), \( f(x) \) approaches 0, but it never actually reaches 0. This is due to the nature of the exponential function.

4. **Inflection Points**: The second derivative \( f''(x) \) can be used to find inflection points, which are points where the concavity of the function changes.

### Graph of the Function:
The graph of \( f(x) = e^{-x^2} \) is a bell-shaped curve that peaks at \( x = 0 \) and gradually approaches the x-axis as \( |x| \) increases.

Would you like a detailed exploration of the derivatives, integration, or other aspects of this function?

### Related Questions:
1. What are the first and second derivatives of \( f(x) = e^{-x^2} \)?
2. How can the function \( f(x) = e^{-x^2} \) be integrated?
3. What are the applications of the function \( f(x) = e^{-x^2} \) in probability and statistics?
4. How does the function \( f(x) = e^{-x^2} \) behave as \( x \) approaches infinity?
5. How can the Gaussian function be normalized to create a probability density function?

**Tip**: The function \( f(x) = e^{-x^2} \) is often used in statistics as the basis for the normal distribution, one of the most important distributions in probability theory.

Explore the Gaussian function f(x) = e^{-x^2}, its properties, and applications. Learn about its symmetry, maximum value at x = 0, asymptotic behavior, and graph characteristics. Suitable for advanced high school students or college-level mathematics.

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