Let's find the inflection points, we'll need to solve the equation for this
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
the second derivative$$\frac{x \operatorname{sign}{\left(x - 1 \right)} - \frac{x \left|{x - 1}\right|}{x - 1} + 2 \left|{x - 1}\right|}{x - 1} = 0$$
Solve this equationSolutions are not found,
maybe, the function has no inflections