Function inverse mapping [0, +inf) to [0, 1)

Question

I have a measure ($x$) which the domain is $[0, +\infty)$ and measure some sort of variability. I want to create a new measure ($y$) that represents regularity and is inverse related to $x$.

It is easy, just make $y = -x$. However I want this new measure to be positive, to make it more interpretable. Linearly, it is impossible, because I would have to map 0 to $+\infty$. However I can create a non-linear measure that follows:

$$ f(x) = y $$ $$ f(0) = 1 $$ $$ x \to +\infty => y \to 0 $$

The logarithmic transformation almost do this, but it does not inverse the relation. What function would do this mapping?

Would be interesting if I could set a "precision", like if $x > 100$, the step in $f(x)$ can be smaller than 0.01.

Answer 1

Try something like $y=f(x)= \dfrac{1}{1+x}$.

This has the properties:

• $f(x)$ decreases as positive $x$ increases
• $f(0)=1$
• $f(1)=\frac12$
• $f(x)\lt 0.01$ for $x \gt 99$ and so also for $x \ge 100$
• $f(x) \to 0$ as $x \to +\infty$

Its inverse is $x=g(y)=\dfrac{1}{y}-1$.