Group/Examples/inv x = 1 - x/Lemma 1

Lemma for Group Example: $x^{-1} = 1 - x$

Define $f: \openint 0 1 \to \R$ by:

$\map f x := \map \ln {\dfrac {1 - x} x}$

and $g: \R \to \openint 0 1$:

$\map g z := \dfrac 1 {1 + \exp z}$

Then:

$\map {f \circ g} x = x$

$\ds \map {f \circ g} x$

$=$

$\ds \map f {\map g x}$

$\ds $

$=$

$\ds \map \ln {\frac {1 - \map g x} {\map g x} }$

$\ds $

$=$

$\ds \map \ln {\frac {1 - \frac 1 {1 + \exp x} } {\frac 1 {1 + \exp x} } }$

$\ds $

$=$

$\ds \map \ln {\frac {1 + \exp x - 1} 1}$

multiplying both numerator and denominator by $1 + \exp x$

$\ds $

$=$

$\ds \map \ln {\exp x}$

simplifying

$\ds $

$=$

$\ds x$

$\blacksquare$