Prove $\sqrt{ z − p} − i \sqrt{p}$ sends exterior of $y ^2 = 4px$ to half-plane $x>0$ (Complex analysis)

Question

How to solve Q4 in Exercises 13.6 of the book "Complex Analysis" by Ian Stewart? Here's the problem:

With a suitable choice of the square root, show that $f(z) = \sqrt{ z − p} − i \sqrt{p}$ maps the exterior of the parabola $y ^2 = 4px$ ($p > 0$) onto the right-hand half-plane $x > 0$.