Prove that $x^3+x+1$ has a root on $[-1,1]$ using the function $g(x)=-{x^2+x\over x^2+x+1}$ and the Fixed Point Iteration Theorem.
Another question asked the same about $g(x)=-{1\over x^2+1}$ which was immediate. By the Fixed Point Iteration Theorem, $g(x)$ has a fixed point $x$ on the interval, and setting $g(x)=x$, one arrives at the desired root of the above expression. Here I am a little challenged. I arrived at the following equality: $$-x-{x+1\over x^2+x+1}=-{x^2+x\over x^2+x+1}$$. Should I define a new variable to be the LHS? How can I make sure all is preserved? I am almost certain the LHS is monotonic which guarantees injectivity. As for Surjectivity, I am not sure. I could use help. Update: Under $[-1,1]$, the LHS is sent to $[-1{2\over 3},1]$. Assuming monotone, it is also a one-to-one correspondence. It still not so assured in my view.