Find a rational function $f: \mathbb R \rightarrow \mathbb R$ with range $f(\mathbb R)=[-1,1]$
(Thus $f(x)=\frac{P(x)}{Q(x)}$ for all $x \in \mathbb R$ for suitable polynomials P and Q, where Q has no real root).
I'm not entirely sure where on this question to start? Any suggestions would be appreciated.