" f o f = g ; f a half effect of g "

I here give some very simple cases in order
to approach the meaning of half effect.

We suppose g(x) <> x , g real continuous .

A first illustration : starting with x/(x +1) = x/(x+1) (1)
x > 1 and g(x) such as g(x) /(g(x) +1 ) = 4 * x/(x+1)
when in (1) x => g(x) , RHS becomes 4 * x/(x+1) ;
for half effect sqrt(4) = 2 we've got :
f(x) /(f(x) +1 ) = 2 * x/(x+1)
That is to say g(x) = 4*x/(1 -3*x) ; f(x) = 2 *x/(1 -x) .

Example 1: g(x) = 4*x -3
g(x) -1 = 4*(x -1)
f(x) - 1 = 2(x -1) , f(x) = 2*x - 1.

Example 2: g(x) = x^2+ 6*x + 12 ; x > 3
g(x) - 3 = (x -3)^2 , sqrt(2)^2 = 2
f(x) - 3 = (x -3)^sqrt(2)
and f(x) = (x -3)^sqrt(2) +3 .

Example 3: g(x)^2 = x^2+1
f(x)^2 =x^2 +1/2 , f(x) =sqrt(x^2 +1/2)

Example 4: g(x) =3*x^5 , x > 0
or (x*3^(1/4) )^5 / 3^(1/4)
and f(x) =(x*3^(1/4) )^sqrt(5) / 3^(1/4) =
(1/4) )^(sqrt(5) - 1) * x^sqrt(5)

I do neither search all the solutions nor discuss at length
the intervals where the given functions are defined,

Alain

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