https://file.aiquickdraw.com/mathbot/file/1726403886703obco3qcl.jpg

The function provided in the image is:

\[
f(x, y) = \frac{xy}{|xy|}
\]

### Analysis:

This function returns the **sign** of the product \(xy\):
- If \(xy > 0\), then \(f(x, y) = 1\) (since \(\frac{xy}{|xy|} = 1\)).
- If \(xy < 0\), then \(f(x, y) = -1\) (since \(\frac{xy}{|xy|} = -1\)).
- If \(xy = 0\), the function is **undefined** (because division by zero occurs when \(xy = 0\)).

#### Breakdown:
- \(xy\) is the product of \(x\) and \(y\).
- \(|xy|\) is the absolute value of that product, which ensures a positive result for the denominator.
  
Thus, \(f(x, y)\) is essentially a signum function for the product of \(x\) and \(y\), excluding the case when either \(x = 0\) or \(y = 0\), which makes \(xy = 0\).

Would you like more details on this or to explore specific cases?

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Here are 5 related questions to deepen your understanding:
1. What are the values of \(f(x, y)\) in each of the four quadrants of the Cartesian plane?
2. How would the function change if we define a separate value for \(xy = 0\)?
3. What is the behavior of this function for large positive or negative values of \(x\) and \(y\)?
4. Can you define a modified version of this function where \(xy = 0\) is assigned a specific value?
5. How does this function relate to the concept of the Heaviside step function?

**Tip:** Functions involving absolute values are often used to distinguish between positive and negative values, so analyzing when the expression inside the absolute value is zero is crucial.

This webpage provides a detailed explanation of the function f(x, y) = xy / |xy|, analyzing its behavior based on the sign of the product xy. Explore how the function returns 1 for positive values, -1 for negative values, and is undefined for xy = 0. Suitable for undergraduate students studying calculus and multivariable functions.

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