numerics – Finding a valid equation for fixed point problem

I currently am working on learning more about fixed point method.
Finding equations that satisfy the constraints of a g function can sometimes
require a bit of engineering.
I have come across one that many would consider simple. Yet, I have been stuck on
it for some time now.

Here it is
$ f(x) = x^2 – x – 2 = 0 $ on $(1.5,3)$.

I have tried many things; however, I have yet to successfully discover
one that maps domain to range for both $g$ and $g^prime$.

Would anyone be able to give me a guiding hand?