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**Unformatted text preview: **-3-2-1123-2-112Section 13.2C13S02.001:Becausef(x, y) = 4−3x−2yis defined for allxandy, the domain offis the entiretwo-dimensional plane.C13S02.002:Becausex2+ 2y2=0 for allxandy, the domain off(x, y) =px2+ 2y2is the entiretwo-dimensional plane.C13S02.003:If eitherxoryis nonzero, thenx2+y2>0, and sof(x, y) is defined—but not ifx=y= 0.Hence the domain offconsists of all points (x, y) in the plane other than the origin.C13S02.004:Ifx6=ythen the denominator inf(x, y) is nonzero, and thusf(x, y) is defined—but not ifx=y. So the domain offconsists of all those points (x, y) in the plane for whichy6=x.C13S02.005:The real numberzhas a unique cube rootz1/3regardless of the value ofz. Hence the domainoff(x, y) = (y−x2)1/3consists of all points in thexy-plane.C13S02.006:The real numberzhas a unique cube rootz1/3regardless of the value ofz. But√2xis realif and only ifx=0. Therefore the domain off(x, y) = (2x)1/2+ (3y)1/3consists of all those points (x, y)for whichx=0.C13S02.007:Because arcsinzis a real number if and only if−15z51, the domain of the given functionf(x, y) = sin−1(x2+y2) consists of those points (x, y) in thexy-plane for whichx2+y251; that is, theset of all points on and within the unit circle.C13S02.008:Because arctanzis defined for every real numberz, the only obstruction to the computationoff(x) = arctan(y/x) is the possibility thatx= 0. This obstruction is insurmountable, and therefore thedomain offconsists of all those points (x, y) in thexy-plane for whichx6= 0; that is, all points other thanthose on they-axis.C13S02.009:For every real numberz, exp(z) is defined and unique. Therefore the domain of the givenfunctionf(x, y) = exp(−x2−y2) consists of all points (x, y) in the entirexy-plane.C13S02.010:Because lnzis a unique real number if and only ifz >0, the domain off(x, y) = ln(x2−y2−1)consists of those points (x, y) for whichx2−y2−1>0; that is, for whichy2< x2−1. This is the regionbounded by the hyperbola with equationx2−y2= 1, shown shaded in the following figure; the boundinghyperbola itself isnotpart of the domain off.C13S02.0011:Because lnzis a unique real number if and only ifz >0, then domain off(x, y) = ln(y−x)consists of those points (x, y) for whichy > x. This is the regionabovethe graph of the straight line withequationy=x(the line itself isnotpart of the domain off).1C13S02.012:Because√zis a unique real number if and only ifz=0, the domain of the given functionf(x, y) =p4−x2−y2consists of those points (x, y) for whichx2+y254. That is, the domain consistsof all those points (x, y) on and within the circle with center (0,0) and radius 2.C13S02.013:Ifxandyare real numbers, then so arexy, sinxy, and 1 + sinxy. Hence the onlyobstruction to computation off(x, y) =1 + sinxyxyis the possibility of division by zero. So the domain offconsists of those points (x, y) for whichxy6= 0;that is, all points in thexy-plane other than those on the coordinate axes....

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