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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
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).)
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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