KöMaL 2019 B5017

Is there a function $f\colon \mathbb{R}\to \mathbb{R}$ with the following properties: (1) if $x_1\ne x_2$ then $f(x_1)\ne f(x_2)$, (2) there exist appropriate constants $a,b>0$ such that $f(x^2)- \big(f(ax+b)\big)^2\ge \frac14.$ for all $x\in\mathbb{R}$?