a function that maps half planes Define
$H^{+}=\{z:y>0\}$
$H^{-}=\{z:y<0\}$
$L^{+}=\{z:x>0\}$
$L^{-}=\{z:x<0\}$
$f(z)=\frac{z}{3z+1}$  maps which portion onto which from above and vice-versa? I will be glad if any one tell me how to handle this type of problem? by inspection?

HINTS


*

*Fractional transformations/Möbius transformations take circles and lines to circles and lines, i.e. they are 'circilinear.' They also preserve connected regions.

*If you find out what happens to the boundaries, you'll know almost everything (except for in which side of the boundary the image resides); in one of those silly word plays, the image of the boundary is the boundary of the image.

*Once you know where the upper half plane, say, goes, you know where the lower half plane goes automatically.

