Verify (without Banach Contraction Priciple), that the function g(x) = 1 + x - (1/8)x^3 has a unique fixed point

Question

I know how to show that there exist a solution by the intermediate value theorem but I'm not sure how to show that the root is unique?

Answer 1

Hint: A fixed point is one where $g(x)=x$, so you might look for roots of $h(x)=g(x)-x$