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functional sequence

Source: flanders '94

September 27, 2005
functionalgebradomainquadratics

Problem Statement

Let (fi)(f_i) be a sequence of functions defined by: f1(x)=x,fn(x)=fn−1(x)−14f_1(x)=x, f_n(x) = \sqrt{f_{n-1}(x)}-\dfrac14. (n∈N,n≥2n\in \mathbb{N}, n\ge2) (a) Prove that fn(x)≤fn−1(x)f_n(x) \le f_{n-1}(x) for all x where both functions are defined. (b) Find for each nn the points of xx inside the domain for which fn(x)=xf_n(x)=x.
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