Problem 25 - College Algebra and Calculus: An Applied Approach

Question

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

Step-by-Step Solution

Answer

The domain of the function \(h(x, y)=x \sqrt{y}\) corresponds to the region \(R\) in the \(x y\)-plane where \(x\) can be any real number and \(y\) is a non-negative real number.

Step 1: Understanding Defined Regions

The domain of a multi-variable function is the set of all ordered pairs \((x, y)\) such that the function is defined. For the function \(h(x, y)=x \sqrt{y}\), the domain will be the ordered pairs \((x, y)\) for which the square root of \(y\) is defined.

Step 2: Define Condition for Square Roots

In mathematics, the square root of a number is only defined for non-negative numbers. Hence, \(y\) should be greater than or equal to zero. Also, \(x\) is not under the root, thus its values can be any real number.

Step 3: Define the Domain

Thus, the domain \((x, y)\) of the function \(h(x, y)=x \sqrt{y}\) is defined by the set of all ordered pairs \((x, y)\) where \(x\) is any real number and \(y\) is a non-negative real number.