b) Consider the function f(x) = = x - 2k x + k 2k , k = 1, 2, 3, and π, where is the floor function. b.1) Show that f is 2k- periodic function and f (x) = x for all x = (−k, k). b.2) For k 1, consider the function g (x) = f² (x) = (x· = x- 2k[]). b.2.1) Show that 9 is periodic with period 2. [Hint: use b.1 and above a)] b.2.2) Show that g (x) = x², for −1 < x < 1. b3) Assume that f (x) is the restriction of a 2k- periodic function g (x) to the interval (—k, k). Show that the 2p- periodic 9 (x) function can be described by the function g (f (x)) for all x Є R.
b) Consider the function f(x) = = x - 2k x + k 2k , k = 1, 2, 3, and π, where is the floor function. b.1) Show that f is 2k- periodic function and f (x) = x for all x = (−k, k). b.2) For k 1, consider the function g (x) = f² (x) = (x· = x- 2k[]). b.2.1) Show that 9 is periodic with period 2. [Hint: use b.1 and above a)] b.2.2) Show that g (x) = x², for −1 < x < 1. b3) Assume that f (x) is the restriction of a 2k- periodic function g (x) to the interval (—k, k). Show that the 2p- periodic 9 (x) function can be described by the function g (f (x)) for all x Є R.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Please write neatly and explain everything clearly
![b) Consider the function
f(x) =
= x - 2k
x + k
2k
, k = 1, 2, 3, and π,
where is the floor function.
b.1) Show that f is 2k- periodic function and f (x) = x for all x = (−k, k).
b.2) For k 1, consider the function g (x) = f² (x) = (x·
=
x-
2k[]).
b.2.1) Show that 9 is periodic with period 2. [Hint: use b.1 and above a)]
b.2.2) Show that g (x) = x², for −1 < x < 1.
b3) Assume that f (x) is the restriction of a 2k- periodic function g (x) to the interval (—k, k).
Show that the 2p- periodic 9 (x) function can be described by the function g (f (x)) for
all x Є R.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1dc6cf95-b131-44cb-bec5-e5cfc8b95147%2F4b15e5b4-6a7a-4476-9e15-a04b9585ca4e%2Frpxrs8_processed.png&w=3840&q=75)
Transcribed Image Text:b) Consider the function
f(x) =
= x - 2k
x + k
2k
, k = 1, 2, 3, and π,
where is the floor function.
b.1) Show that f is 2k- periodic function and f (x) = x for all x = (−k, k).
b.2) For k 1, consider the function g (x) = f² (x) = (x·
=
x-
2k[]).
b.2.1) Show that 9 is periodic with period 2. [Hint: use b.1 and above a)]
b.2.2) Show that g (x) = x², for −1 < x < 1.
b3) Assume that f (x) is the restriction of a 2k- periodic function g (x) to the interval (—k, k).
Show that the 2p- periodic 9 (x) function can be described by the function g (f (x)) for
all x Є R.
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