Is there a way to map $[0, \infty[ $ onto $]- \infty, \infty[$ and vice versa?

Question

As the title says. I'm actually writing a program and a part of it has to do these mappings. I just wish to do something like 2 functions f(x) and g(x), where the first maps $[0, \infty[ $ to $[- \infty, \infty]$ and g(x) does the opposite, preferably in an uncomplicated way. It does not have to be continuous.

Answer 1

Take $$f(x) = \begin{cases} \ln(x) & \text{if } x \in (0,+\infty)\\ 0 & x = 0 \end{cases}.$$

Answer 2

There should be plenty of ways, but here's mine: Given a number $x$ write this value as $n + q$ where $n$ is the integer part. Then define $ f(n+q) = -n -2q$ for $q \in (0,0.5]$ and f(n + q) = n + 2q for $n\in (0.5,1]$. Just make sure that if say you had $f(6)$ this needs to be written as $f(5 + 1)$. It's quite easy from this to work out your other function $g$, and also the inverses of these functions.

Answer 3

I propose $f(x)=\begin{cases}\frac x{x-1}&x\in[0,\infty)\setminus\{1\}\\1& x=1\end{cases}$ for $f:[0,+\infty)\to\mathbb R$

Unfortunately for the other direction, this is not as simple as Martin R's comment pointed out.

We have to proceed in two steps $\mathbb R\overset{g_1}{\mapsto}(0,+\infty)\overset{g_2}{\mapsto}[0,+\infty)$

For instance $g_1(x)=e^x$ and $g_2(x)=\begin{cases}x-1&x\in\mathbb N\\x& \text{otherwise}\end{cases}$

Note: we do not have $f^{-1}=g_2\circ g_1$, but if you need a proper reciprocal, you can proceed the same way with $f_1(x)=\ln(x)$, can you see what $f_2$ should be in this case.

Answer 4

Speaking of digital programming, there is a maximum and minimum in any signed or unsigned data type. Some data types support the representation of infinity, but not all of them. It is practical to think of a linear mapping such as $f:[0,M) \rightarrow (m,M)$ defined by $$f(x) = \frac{M - m}{M}x + m$$ where $m$ and $M$ is the smallest and the largest number the data type can represent. It has good properties to handle; linear, continuous, and bijective. Furthermore, it is efficient in the sense of computational overhead.