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?

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.