91Ó°ÊÓ

Square root functions play a significant role in mathematics, and understanding their domain is crucial. A function involving a square root typically looks like this: \( f(x) = \sqrt{expression} \). Here, the term within the square root must be greater than or equal to zero. This is due to the mathematical rule that you cannot take the square root of a negative number in the set of real numbers.
In the given exercise, the function \( g(x, y) \) includes a square root: \( \sqrt{\frac{x y}{x^{2}+y^{2}}} \). Hence, the expression \( \frac{x y}{x^{2}+y^{2}} \) must be non-negative.
This means we have to solve the inequality \( \frac{x y}{x^{2}+y^{2}} \geq 0 \) to find where the function is defined. Solving such inequalities is an essential skill in understanding where square root functions are valid across their domain.

Inequalities are a foundational concept in calculus and vital when determining the domain of functions, especially those involving square roots. When we encounter an inequality like \( \frac{x y}{x^{2}+y^{2}} \geq 0 \), it means we need to identify all the values of \( x \) and \( y \) for which this condition holds true.
This inequality tells us:

• If \( x \) and \( y \) have the same sign, the product \( xy \) is positive, and hence the entire fraction is non-negative.
• But when \( x \) and \( y \) have opposite signs, the product \( xy \) becomes negative, making the entire fraction less than zero.
• The special case where \( x \) or \( y \) is zero needs careful consideration, as it could lead to an undefined situation, especially if the denominator becomes zero simultaneously.

These insights allow us to sketch the portions of the coordinate plane where the inequality holds, effectively helping to determine the function's domain.

Defining the domain of a function involves identifying all possible input values that produce valid outputs. For functions like \( g(x, y) = \sqrt{\frac{x y}{x^{2}+y^{2}}} \), we impose restrictions for two main reasons: the need for the expression under the square root to be non-negative and to avoid division by zero.
The expression's denominator \( x^{2} + y^{2} \) is zero when both \( x \) and \( y \) are zero. To avoid division by zero, both terms cannot simultaneously be zero.
The condition \( \frac{x y}{x^{2}+y^{2}} \geq 0 \) further restricts the domain, ensuring both x and y share the same sign (both positive or both negative). Such restrictions yield the domain:

• \( (x, y) \in \{ (x, y) \in \mathbb{R}^2 : x > 0, y > 0 \} \)
• \( (x, y) \in \{ (x, y) \in \mathbb{R}^2 : x < 0, y < 0 \} \)

This means the domain consists of all points where both variables are positive or both are negative, effectively capturing all valid inputs for which the function behaves properly without any restrictions.