Question

Describe the region \(R\) in the \(x y\) -plane that corresponds to the domain of the function. $$ f(x, y)=\sqrt{x y} $$

Short answer

The region \(R\) in the \(xy\)-plane, which corresponds to the domain of the given function \(f(x, y) = \sqrt{xy}\), is the Quadrants I and III, including the \(x\) and \(y\) axes. That is, \(R\) is the set of all \((x, y)\) pairs for which either \(x \geq 0\) and \(y \geq 0\) (first quadrant and axes), or \(x \leq 0\) and \(y \leq 0\) (third quadrant and axes).

Step-by-step solution

Identify the Function Type

The function \(f(x, y)=\sqrt{xy}\) is a square root function. For square root functions, the quantity under the square root sign (the radicand) must be greater than or equal to zero, because square root of a negative number would result in a complex number.

Define the Condition for the Domain

The condition that needs to be satisfied for the domain of \(f(x, y)=\sqrt{xy}\) is that the radicand satisfies \(xy \geq 0\). Thus, the product \(xy\) can be either positive or zero.

Determine the Solution Space

By the rule of product of two real numbers, \(xy \geq 0\) if and only if \(x\) and \(y\) are both positive, or both negative. So, the region \(R\) in the \(xy\)-plane are Quadrants I and III, including the axes.