To find the domain of the function $f(x, y) = \sqrt{xy - 3y - 2x + 6}$, we need to ensure that the expression inside the square root is non-negative because the square root of a negative number is undefined in the real number system.

Step 1: Set the expression inside the square root to be greater than or equal to zero:

$xy - 3y - 2x + 6 \geq 0$

Step 2: Analyze the inequality:

The domain consists of all pairs $(x, y)$ that satisfy this inequality. The inequality defines a region in the $xy$-plane where the expression inside the square root is non-negative.

To solve this inequality, let’s consider various approaches:

• Factorize the expression, if possible.
• Analyze specific cases for particular values of $x$ or $y$ to better understand the behavior of the function.

In this case, the complexity of the expression requires a careful approach to determining the region where the inequality holds true.

Conclusion:

The domain of $f(x, y)$ is the set of all pairs $(x, y)$ such that $xy - 3y - 2x + 6 \geq 0$.

Would you like me to further simplify or analyze this inequality?

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Here are 5 related questions to explore further:

1. How would the domain change if the expression inside the square root were a polynomial?
2. What is the range of the function $f(x, y)$ based on the domain found?
3. How do we find the domain of a function with multiple variables, such as $f(x, y)$?
4. What if the function were $f(x, y) = \sqrt{x^2 + y^2}$? How would the domain differ?
5. How does factoring help in finding the domain of an expression like this?

Tip: When dealing with square roots, always remember to check that the radicand (the expression inside the square root) is non-negative for the domain to be valid.