Find a rational function $f: \mathbb R \rightarrow \mathbb R$ with range $f(\mathbb R)=[-1,1]$

Question

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.

Answer 1

How about the following function? $$\dfrac{2x}{x^2+1}$$

Answer 2

One way to significantly restrict the range is by using functions like $x \mapsto x^2$ : its range is half of the real numbers. By composing this with a rational function sending $0,1,\infty$ to $-1,0,1$, you get what you want.

For example, $x \mapsto \frac{x-1}{x+1}$ does this, so after composing we get : $x \mapsto \frac{x^2-1}{x^2+1}$