Hello, I am wondering whether we can have different function f(z) for this question.

In the posted solution, Yilin set $ f(z) = \lambda \frac{z-a}{1-\bar{a}z}$ $| \lambda | = 1 $ and get final result as $ f(z) = \frac{5+z}{1+5z}$.

Based on a hint in Textbook 3.3 Example 1, I try to set $ f(z) = \lambda \frac{a-z}{1-\bar{a}z}$ and $| \lambda |=1 $then I did the following computation.

let $\lambda = e^{it}$ , $ a = re^{i \theta}$

$f(0) = 5$ -> $\lambda a = 5$ and $ |\lambda a| = |a| = 5 $ so $ a = 5e^{i\theta}$

$\lambda a = 5e^{it} e^{i\theta} =5$ so $e^{-it} = e^{i\theta}$

$ f(-1) = \lambda \frac{a+1}{1+\bar{a}} = -1$ so $e^{-i\theta} = -1$ ---> $e^{it} =1$ and $ \theta = \pi $

so $\lambda = 1$ and $a = -5$

$f(z) = \frac{-5-z}{1-5z}$

I am not sure if there is a computation mistake for my solution or we could set f(z) in many forms.

Thank you.