Still getting used to MathJax, we'll see if this renders correctly!

1. \$$f(x)=x\$$ is _order-preserving_, whereas \$$f(x)=|x|\$$ is not.
2. \$$f(x)=x\$$ is _metric-preserving_, whereas \$$f(x)=2x\$$ is not.
3. \$$f(x)=2x\$$ is _addition-preserving_, whereas \$$f(x)=x^2\$$ is not.

intuitively, I'm thinking of _order-preserving_ functions as ones which don't fold the _domain_ into the _codomain_, in a sense. _metric_ and _addition-preserving_ functions I think of as ones which don't stretch the _domain_ into the _codomain_. I'm sure there are other examples, and I'm not sure the intuition holds up exactly, but those are the ones that come intuitively to mind for me.