1. Problem: Determine the domain $D$ of the function $f(x,y) = \frac{x}{y}$ and represent it geometrically. 2. Formula and rules: The domain of a function is the set of all input values for which the function is defined. For $f(x,y) = \frac{x}{y}$, the denominator $y$ cannot be zero because division by zero is undefined. 3. Intermediate work: - The only restriction is $y \neq 0$. - Therefore, the domain is all pairs $(x,y)$ such that $y \neq 0$. 4. Explanation: This means the function is defined for every real number $x$ and every real number $y$ except $y=0$. 5. Geometric representation: The domain is the entire $xy$-plane except the line $y=0$ (the $x$-axis). Final answer: $$D = \{(x,y) \in \mathbb{R}^2 : y \neq 0\}$$