3.20 † Let f be any real-valued function on a domain DC R. Define Sf(x) f*(x) = if f(r) > 0, if f(r) < 0. and let f (포) = -f(x) if f(x)< 0, if f(z) > 0 for all r € D. Prove that f(x) = f*(x) – ƒ¯(x) and |f(z)| = f+(x)+ ƒ¯(x) for all r e D. (Hint: Just check the cases based on the sign of f(r).)
3.20 † Let f be any real-valued function on a domain DC R. Define Sf(x) f*(x) = if f(r) > 0, if f(r) < 0. and let f (포) = -f(x) if f(x)< 0, if f(z) > 0 for all r € D. Prove that f(x) = f*(x) – ƒ¯(x) and |f(z)| = f+(x)+ ƒ¯(x) for all r e D. (Hint: Just check the cases based on the sign of f(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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![3.20 † Let f be any real-valued function on a domain DCR. Define
st(x) = {f(z) if f(x) > 0,
if f(r) < 0.
and let
f (x) =
-f(x) if f(x) < 0,
if f(x) > 0
for all r € D. Prove that
f(x) = f+(x) – f (x) and |f(x)| = f*(x)+ f¯(x)
for all æ e D. (Hint: Just check the cases based on the sign of f(x).)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fadd5b56d-97e3-4d3c-9aa9-5b563ad77377%2F51cf991a-2b5a-4173-8eba-49242990e97f%2Fo6zfbhf_processed.png&w=3840&q=75)
Transcribed Image Text:3.20 † Let f be any real-valued function on a domain DCR. Define
st(x) = {f(z) if f(x) > 0,
if f(r) < 0.
and let
f (x) =
-f(x) if f(x) < 0,
if f(x) > 0
for all r € D. Prove that
f(x) = f+(x) – f (x) and |f(x)| = f*(x)+ f¯(x)
for all æ e D. (Hint: Just check the cases based on the sign of f(x).)
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