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?

$f(z)=\frac{z}{3z+1}=\frac{x+iy}{3(x+iy)+1}=\frac{x+iy}{3x+1+i3y}=\frac{(x+iy)(3x+1-i3y)}{(3x+1)^2+9y^2} \implies \Im (f(z))=\frac{y}{(3x+1)^2+9y^2}$. So, If $y > 0$, then $\Im (f(z))>0$.If $y < 0$, then $\Im (f(z))<0.$
