Transformation of region

Question:- Show that the image of the set $$S = {z \in \mathbb{C} | Im(z) \geq 0, |z|\geq 1}$$ under the map $$w = u+\iota v = z+ \frac{1}{z}$$ is the upper half plane of $$v\geq 0$$

My approach Let $$z= re^{\iota \theta}$$. Then $$w=re^{\iota \theta} + \frac{1}{r} e^{-\iota\theta}$$ The mapping given in question is semicircle covering above half of $$z$$ plane, and the real axis.

Therefore i have taken three regions for mapping onto $$w-plane$$

case-1$$z= x, x\geq1$$ Case-2$$z= -x, x\geq 1$$ Case-3 points on boundry of surface of the semi circle.

I understand the first two cases, but don't have any approch to third one. hints are highly appreciated

thankyou

Points on the semicircle are of the form $$e^{i\theta}$$ with $$0\leq \theta \leq \pi$$. For $$z=e^{i\theta}$$ we have $$z+\frac 1 z = e^{i\theta}+e^{-i\theta}=2\cos\, \theta$$ and $$\cos\, \theta$$ takes all values between $$-1$$ and $$+1$$ so the image of the semicircle is the interval $$[-2,2]$$ of the real axis.