The function you've provided is:

$f(x) = x e^{\frac{1}{x}}$

If you'd like to explore its properties, we can start with things like the derivative, limits, or behavior at specific points. Would you like to explore any of these or something else specific regarding this function?

Here are some related questions you might find useful:

1. What is the derivative of $f(x)$?
2. What is the limit of $f(x)$ as $x$ approaches 0?
3. How do we analyze the behavior of $f(x)$ for large values of $x$?
4. How does the function behave as $x \to \infty$ or $x \to -\infty$?
5. Is there any point where $f(x)$ has a local maximum or minimum?

Tip: The product rule can be very useful for differentiating functions that are products of two terms, like $x$ and $e^{\frac{1}{x}}$.