Example of a diffeomorphism from all of $\mathbb{R}$ to itself

Question

I can think of diffeomorphisms from an interval to $(a,b)\rightarrow \mathbb{R}$, scaling the tangent function, and from the punctured plane, polar coordinates, or some odd polynomial, but does anyone have a nontrivial example that doesn't break down at the origin?

Answer 1

The most trivial example is the identity map $f(x)=x$. Slightly less trivial are linear polynomials $f(x)=ax+b$ (for $a\neq 0$).

A less trivial but still simple example is $f(x)=x^3+x$. Since $f'(x)=3x^2+1$ is positive everywhere, $f$ is strictly increasing and thus injective. Since $f(x)\to\pm\infty$ as $x\to\pm\infty$, $f$ must be a bijection, and it is thus a diffeomorphism since its derivative never vanishes.