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?