We will denote by D( f ) the maximal domain of definition.
(a) The function can be defined everywhere, where x ≠ 0. Therefore
D( f ) = f{(x, y) : x≠0}=R2\ {x=y};
the domain consists of the whole plane except for the y-axis.
b) The function can be defined everywhere, where x - y ≠ 0. Therefore
D( f ) = f{(x, y) : x ≠ y}=R2\ {x=y};
the domain consists of the whole plane except for those points lying on the line x = y.
(c) Here the denominator is 0, when x2 + y2 = 1. Therefore
D(f) = {(x,y) :x2+y2 ≠ 1}= R2{x2+y2=1};
the domain conists of the whole plane except for the circle around the origin with radius 1.
(d) For the square root to be well-defined we need the inequality y-x2≥0 to be satisfied therefore
D(f)={(x,y):y ≥x2};
the domain consists of the part of the plane, that **** on and above the parabola y = x2.
(e) Here we need x ≥ 0 for the square root to be well-defined and y> sin x to evaluate the logarithm. Note that one inequality is strict, while the other is not. Thus the domain is
D(f ) = {(x, y) : x ≥ 0 and y > sin x} ;
(f) To be able to evaluate f (x, y) we need −x > 0 as well as y − ln(−x)≥ 0 to be satisfied. These inequalities can be rewritten as
D(f ) = {(x, y) : x < 0 and y≥ ln(−x)} .
Note again, that one inequality is strict and the other is not.
