Question

Find and Sketch the domain of the function.\(f(x,y) = \sqrt {xy} \)

Short answer

The domain of the function\(f(x,y)\)is \(\{ (x,y)/(y < 0,x \le 0),(x \le 0,x \ge 0),(x \ge 0,y > 0)\} \)

Step-by-step solution

Step 1:- Consideration

Let \(f(x,y) = \sqrt {xy} \)be a equation

Step 2:- Finding different cases:-

For the existence of the function \(f(x,y)\sqrt({}){{xy > 0}}\)such that \(xy \ge 0\)

To exhibit \(xy \ge 0\) the possible cases are

1. \(y < 0\)and \(x \le 0\)
2. \(y = 0\)and\(x \le 0,x \ge 0\)
3. \(y > 0\)and \(x \ge 0\)

These are the possible three cases such that \(xy \ge 0\)

Step 3:- Solving case (i)

If\(y < 0\)and\(x \le 0\)then\(xy \ge 0\)is possible

Step 4:- Graph

The domain of the function \(f(x,y)\)can also be find through the graph

Hence, the domain of the graph is

Step 5:- Solving case (ii)

The function \(f(x,y)\)to exist\(\sqrt {xy} \)must be greater than 0 for that

If\(y = 0\)and\(x \le 0\),\(x \ge 0\)then \(\sqrt {xy} > 0\)

Step 6:- Graph

Domain can also be find through the graph also

Step 7:- The domain

Therefore the domain of case(ii) through the graph is

\(\{ (x,y)/y = 0,x \le 0,x \ge 0\} \)

Step 8:- Solving  case (iii)

If\(y > 0\)and \(x \ge 0\) then such that the function is real

Step 9:- Graph

The domain of the function can also be find through the graph

Step 10:- The domain if \(x \ge 0\) and \(y > 0\)

Therefore, the domain of the function if\(x \ge 0\)and\(y > 0\)is\(\{ (x,y)/x \ge 0,y > > 0\} \)

Hence, the domain of the function

\(f(x,y) = \sqrt {xy} \)is \(\{ (x,y)/(y < 0,x \le 0,y = 0,x \le 0,x \ge 0,y > 0,x \ge 0\} \)