Find the domain of each of the following functions. Then graph the domain.
Solution (a) Since the expression under the radical must be nonnegative, the domain of \(f\) consists of all points in the plane for which \[ \begin{eqnarray*} 16-x^{2}-y^{2} &\geq &0 \\[4pt] x^{2}+y^{2} &\leq &16 \end{eqnarray*} \]
The domain is all the points inside and on the circle \(x^{2}+y^{2}=16\). The shaded portion of Figure 3 illustrates the domain.
(b) Since the logarithmic function is defined for only positive numbers, the domain of \(f\) is the set of points \((x,y) \) for which \(y^{2}-4x>0\) or \(y^{2}>4x\). To graph the domain, we start with the parabola \(y^{2}=4x,\) and use a dashed curve to indicate that the parabola is not part of the domain. The parabola \(y^{2}=4x\) divides the plane into two sets of points: those for which \(y^{2}<4x\) and those for which \(y^{2}>4x\). To find which points are in the domain, choose any point not on the parabola \(y^{2}=4x\) and determine whether it satisfies the inequality. For example, the point \((2,0)\) is not in the domain, since \(0^{2}<(4)(2)\). The set of points for which \(y^{2}>4x\), the domain of \(f\), is shaded in Figure 4.