On homeomorphisms and quasi-isometries of the real line

We show that the group of piecewise-linear homeomorphisms of $\mathbb{R}$ having bounded slopes surjects onto the group $QI(\mathbb{R})$ of all quasi-isometries of $\mathbb{R}$. We prove that the following groups can be imbedded in $QI(\mathbb{R})$: the group of compactly supported piecewise-linear homeomorphisms of $\mathbb{R}$, the Richard Thompson group $F$, and the free group of continuous rank.