Find a rational function $f:\mathbb{R} \to\mathbb{R}$ with range $f(\mathbb{R})=[0,1]$
(Thus $f(x)=\dfrac{P(x)}{Q(x)}$ for all $x\in \mathbb{R}$ for suitable polynomials $P$ and $Q$, where $Q$ has no real root).
Any suggestions would be much appreciated as I have no clue where to begin.
The image of the rational function $x\mapsto\dfrac{x}{1+x^2}$ is a compact interval. Compose it with a suitable affine function to obtain a rational function with image $[0,1]$.