Question

Describe the domain and range of the function. $$ g(x, y)=\frac{1}{x y} $$

Short answer

Domain of function: \(R^2\) - {(x, y): x = 0 or y = 0}. Range of function: \(R\) - {0}.

Step-by-step solution

Identify the Domain of the Function

The domain of the function \(g(x, y) = \frac{1}{xy}\) consists of all real numbers (x, y), except those for which the denominator would equal zero. In this case, the function will be undefined for \(x = 0\) or \(y = 0\). Therefore, all points except those on the x-axis and y-axis are in the domain of g, i.e., the domain is \(R^2\) - {(x, y): x = 0 or y = 0}.

Identify the Range of the Function

The range of this function is all the possible output values. The function \(g(x, y) = \frac{1}{xy}\) results in undefined outputs when either x or y equals 0. But for all other real pairs (x, y), the function provides a real and defined output. This means the range of this function is \(R\) - {0}, all real numbers except 0.