Bijection from the real numbers to the negative real numbers

Question

I'm trying to construct a bijection from $\mathbb{R}$ to $\mathbb{R}^{-}$. The only thing I've been able to find is $f(x)=-e^{x}$ and I was wondering if there is any other functions that work? Would I be right in assuming any function of the form $-a^{x}$ where $a\in \mathbb{R}^{+}$ works?

Answer 1

One branch of a hyperbola: $$ f(x) = x - \sqrt{x^2+1} $$

Answer 2

There are lots and lots of them. Most cannot be described, but even among those that can there are many options. You can compose your exponential with any bijection $\Bbb {R \to R}$ and get a new function. You can take $[0,1_)$ to $[-18,-17), [1,2)$ to $[-783,790)$ and just generally jumble the pieces around.