what is domain of the two variable function f(x,y)=sqrt(xy) and graph is function in space

Introduction : This is the step by step solution of the problem what is domain of the two variable function f(x,y)=sqrt(xy) and graph is function in space.

Question: What is domain of the two variable function

$$f(x,y)=\sqrt(xy)$$

and graph this function in space.

Solution:

A square root is defined only for non-negative numbers.

that is function function f(x, y) is defined only when xy>=0 and this possible if x>=0,y>=0 or x<=0,y<=0

Combining these we can say that domain of f(x,y) is

$$D={(x,y):x \ge 0,y\ge 0 }∪{(x,y):x\le 0,y\le 0}$$

If we sketch this domain in xy plane then it is look like as follows

Graph of the function $f(x,y) =\sqrt(xy)$ looks like below