Question

Describe the domain and range of the function. $$ g(x, y)=x \sqrt{y} $$

Short answer

The domain of the function \(g(x, y) = x \sqrt{y}\) is \(D = \{(x,y) : x \in \mathbb{R}, y \geq 0\}\) and the range is all real numbers \(R = \{ g(x,y) : g(x,y) \in \mathbb{R} \}\).

Step-by-step solution

Determine the domain

The domain of the function is the set of all possible input values for which the function is defined. First, note that the variable x does not impact the definition of the function, because x can be any real number. The square root function sqrt(y) is undefined for negative values of y. Therefore, to satisfy this condition, y has to be greater than or equal to 0. So the domain will be \(D = \{(x,y) : x \in \mathbb{R}, y \geq 0\}\).

Determine the range

The range of a function is the set of all possible output values (y-values). The function \(g(x, y) = x \sqrt{y}\) can generate every possible real number. Here's why: For y>0, as x varies from negative to positive infinity, the function will also vary from negative to positive infinity. For y=0, the function is always 0 regardless of the value of x. Therefore, the range of the function is all real numbers \(R = \{ g(x,y) : g(x,y) \in \mathbb{R} \}\).