Using your Head is Permitted

Let f be an invertible function to and from ℜ⁺xℜ⁺ (where ℜ⁺ is the positive real numbers) that satisfies the following property: Consider ℜ⁺xℜ⁺ as the first quadrant of the plane. Subsets of ℜ⁺xℜ⁺ can be described as geometrical loci. For example, any subset composed of only a single element in ℜ⁺xℜ⁺ can be said to be a "point", the set of all solutions to a linear equation can be said to be a "line", the set of points that are weighted averages of any two points A and B can be said to be a (closed) segment of the straight line passing between A and B, etc.. The property of f is that it maps closed segments of straight lines to closed segments of straight lines.

The question: list all potential candidates for f (parametrically, if you have to). That is, we want to find all bijections, f:ℜ⁺xℜ⁺→ℜ⁺xℜ⁺ that satisfy the desired property.

Note that this month readers are not required to provide proof that the list is exhaustive. However, as a bonus question readers are welcomed to come up with such proofs. The solution will outline a proof. On the other hand, because no proof is required, only the first solution sent by any specific solver will be considered, so as to eliminate the possibility for trial-and-error solving.

To be considered a solver, make sure your list

1. Includes all functions satisfying the property.
2. Does not include any functions not satisfying the property.
3. Does not list the same function twice (possibly by different names).

To be considered a solver of the bonus question, prove the result in a way that allows extending the conclusion to functions f:(ℜ⁺)ⁿ→(ℜ⁺)ⁿ.

List of solvers: