Function satisfying $f\colon \mathbb{R} \to [-2,2]$ non-trigonometric

Question

If there any non-trigonometric function $$f\colon \mathbb{R} \to [-2,2]$$

The following function $f(x) = 2\sin bx$ will satisfy the above criteria. This came up for discussion with my students.

Answer 1

Try $f(x) = \dfrac{4x}{x^2+1}$:

Answer 2

You can easily make up as many piece-wise defined functions as you'd like to. For example, let $f(x)=2x^3$ if $x \in [-1,1]$, and you can assign any value from $[-2,2]$ to the other values. For example, $f(x)=-2$ if $x<-1$ and $f(x)=2$ if $x>1$.

Answer 3

You could use $$ f(x)=\frac4{1+e^{-x}}-2. $$ This is increasing, with $\lim_{x\to+\infty}f(x)=2$ and $\lim_{x\to-\infty}=-2$.

Answer 4

For a non-trig periodic example, try a sawtooth function like $\; f(x) = 8 \cdot \big| x - \left\lfloor x+ 0.5 \right\rfloor \big| - 2\,$: