Question

Find and sketch the domain of the function \(f(x,y) = ln(9 - {x^2} - 9{y^2})\)

Short answer

The domain of the function is \(\left\{ {\left( {x,y} \right)/{x^2} + 9{y^2} < 3} \right\}\)

Step-by-step solution

Step 1:- Consideration:

Let \(f(x,y) = \ln (9 - {x^2} - 9{y^2})\) is the function

Step 2:- Finding relation

For the function f(x,y) to be exist \(\ln (9 - {x^2} - 9{y^2})\) must be real

If \(9 - {x^2} - 9{y^2} > 0\) the f exists

Therefore the domain of the function of \(\left\{ {\left( {x,y} \right)/{x^2} + 9{y^2} < 3} \right\}\) I.e. set of all points inside the ellipse \({x^2} + 9{y^2} = {(3)^2}\) but the ellipse is not included

Step 3:- Graph

Hence the domain of f is \(\left\{ {\left( {x,y} \right)/{x^2} + 9{y^2} < 3} \right\}\)