How to to map one range onto another?

Question

Take an arbitrary real function $f(x)$ with range $f(x) \ge 0$.

Is it possible to find an operator / function $g$ with the following mapping properties:

• $f(x)>1 \rightarrow 0<g(f(x))<1$
• $f(x)=1 \rightarrow g(f(x))=1$
• $0<f(x)<1 \rightarrow 0<g(f(x))<1$
• $f(x)=0 \rightarrow g(f(x))=0$?

I suspect not, but would very much appreciate any input.

Answer 1

Here's a nice smooth rational function that fits the bill: $$g(x) = \frac{2x}{x^2 + 1}$$