Consider the function ƒ : [0, ∞) → [0, ∞) defined by X f(x) = 1². 1+x Prove that the following equation holds for all n E N and all x = [0, ∞): ƒn (x) = X 1+nx (Note: fn is the function obtained by composing f with itself n times. For example, f² = ƒoƒ and f5 = ƒoƒoƒoƒoƒ, etc.) O

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Use proof by induction

 

 

Consider the function ƒ : [0, ∞) → [0, ∞) defined by
X
f(x) = 1².
1+x
Prove that the following equation holds for all n E N and all
x = [0, ∞): f (x)
=
x
1+nx
(Note: fn is the function obtained by composing f with itself n
times. For example, ƒ² = ƒoƒ and ƒ5 = ƒoƒoƒoƒoƒ,
O
etc.)
Transcribed Image Text:Consider the function ƒ : [0, ∞) → [0, ∞) defined by X f(x) = 1². 1+x Prove that the following equation holds for all n E N and all x = [0, ∞): f (x) = x 1+nx (Note: fn is the function obtained by composing f with itself n times. For example, ƒ² = ƒoƒ and ƒ5 = ƒoƒoƒoƒoƒ, O etc.)
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