Geometric description of the domain of a function

Question

A real function f of two real variables is given by $$f(x,y)=\sqrt{4 \sqrt{3}y-13x^2-7y^2-6 \sqrt{3}xy-4x}$$ Give a thorough geometric descrition of the domain of f using terms as open, ended, finite, infinite limited, connected and non conncted. Note: The terms have been directly translated and might not be the exact appropriate translation for for mathematics. Original language is Danish where Open: Åben, Ended: Afsluttet, Finite: Begrænset, infinite: Ubegrænset, Connected: Sammenhængene Non connected: Ikke-sammenhængene

How can this question be approached?

Answer 1

If you take a pure translation $$ x = p - \frac{1}{2} \; \; , \; \; y = q + \frac{\sqrt 3}{2} \; \; , $$ you reach $$ 13 p^2 + 6 pq \sqrt 3 + 7 q^2 \leq 4 \; \; . $$ Follow with a pure rotation, $$ p = \frac{u \sqrt 3 - v}{2} \; \; , \; \; q = \frac{u + v\sqrt 3}{2} \; \; , $$ you reach $$ 4 u^2 + v^2 \leq 1 \; \; . $$ Which is an ellipse with its interior. Closed, bounded, and connected.

Depends which order you are writing the rotation, anyway the matrix $$ \left( \begin{array}{rr} \frac{\sqrt 3}{2} & - \frac{1}{2} \\ \frac{1}{2} & \frac{\sqrt 3}{2} \end{array} \right) $$ is orthogonal, top row is sine and cosine of $120^\circ$