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?

---

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.

Find the domain of f(x, y) = √(xy − 3y − 2x + 6)

Explore how to find the domain of the multivariable function f(x, y) = √(xy − 3y − 2x + 6). This problem involves key algebraic concepts such as solving inequalities and understanding square roots in multivariable functions. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation