Answer
The shaded region in the figure below indicates the domain $D$ of $f\left( {x,y} \right)$:
$D = \left\{ {\left( {x,y} \right)|4{x^2} - y > 0} \right\}$
Work Step by Step
The function $f\left( {x,y} \right) = \ln \left( {4{x^2} - y} \right)$ is defined only when $4{x^2} - y > 0$ or $y < 4{x^2}$. Thus, the domain consists of all points $\left( {x,y} \right)$ lying below the parabola $y = 4{x^2}$. The shaded region in the figure below indicates the domain $D$ of $f\left( {x,y} \right)$:
$D = \left\{ {\left( {x,y} \right)|4{x^2} - y > 0} \right\}$
