Group of Orientation-preserving Homeomorphisms of the Reals.

Question

Let $h: \mathbb{R}\rightarrow\mathbb{R}$ ; $\mathbb{R}$ Reals be an orientation-preserving homeomorphism.

I can see $h$ includes linear maps $h=ax+b$ with $a>0$ . Can we say that

every orientation-preserving homeomorphism an order-preserving

automorphism and that every orientation-preserving homeo. is a

monotonic map ?

Thanks.