How to make a function lie in the interval [0,1]

Question

Is there a way to convert a function g(x) so that the result lies between [0,1]? Thank you in advance.

Answer 1

If you function is $\ge 0$, use the function $$ x\to \frac x{1+|x|} $$

In general, use $$ x\to \frac 12 \left(1+ \frac x{1+|x|}\right) $$ You hence have a one to one transformation: link to the plot

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Another one is of course $$ x\to \frac 1\pi \arctan x + \frac 12 $$