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TOT 1998 Autumn AS6 f (x) = f_1 (f_2( ... f_{n-1}(f_n (x) )... ))

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May 11, 2020
quadraticsfunctioncompositionalgebra

Problem Statement

In a function f(x)=(x2+ax+b)/(x2+cx+d)f (x) = (x^2 + ax + b )/ (x^2 + cx + d) , the quadratics x2+ax+bx^2 + ax + b and x2+cx+dx^2 + cx + d have no common roots. Prove that the next two statements are equivalent: (i) there is a numerical interval without any values of f(x)f(x) , (ii) f(x)f(x) can be represented in the form f(x)=f1(f2(...fn−1(fn(x))...))f (x) = f_1 (f_2( ... f_{n-1} (f_n (x))... )) where each of the functions fjf_j is o f one of the three forms kjx+bj,1/x,x2k_j x + b_j, 1/x, x^2 .
(A Kanel)
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