Control-drag $f\left(x\right)\=\dots$ 
Context Panel: Assign Function

Write $f\left(x\right)\=0$ and press the Enter key.

Context Panel: Solve≻Solve

Write $g\left(x\right)\=\dots$
Context Panel: Assign Function

Calculate $x_1\=g\left(3\right)$ as the first iterate.

Calculate $x_2\=g\left(x_1\right)$ as the second iterate.

The second iterate is the same as the initial iterate, so Newton's method will simply cycle back and forth between these two values. It will never converge to one of the two roots $x\=1 \pm \sqrt{2}$.

The cycling is examined more closely below.

First iterate: Write $g\left(a\right)$ and press the Enter key.

Context Panel: Simplify≻Simplify

$\frac{a\+1}{a-1}$

Second iterate: Apply $g$ to the simplified $g\left(a\right)$.
Press the Enter key.

Context Panel: Simplify≻Simplify
The second iterate is the same as the initial point, so the iteration will cycle between $a$ and $\left(a\+1\right)\/\left(a-1\right)$.