Proof That f(x) is 0 if Both Odd and Even

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Homework Help Overview

The discussion revolves around the proof that a function f(x) cannot be both odd and even unless it is the constant function 0. Participants are exploring the definitions and implications of odd and even functions in the context of this proof.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants are questioning the definitions of odd and even functions, with some clarifying the correct forms of these definitions. There is a discussion on how these definitions relate to the proof that f(x) must be 0 if it is both odd and even.

Discussion Status

The discussion is currently focused on establishing correct definitions of odd and even functions. Some participants have pointed out errors in the definitions provided, and there is an ongoing exploration of how these definitions can lead to the conclusion regarding the function being constant 0.

Contextual Notes

There appears to be some confusion regarding the definitions of odd and even functions, which is affecting the progress of the proof. Participants are working through these definitions to clarify the assumptions necessary for the proof.

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Homework Statement


Prove that if f(x) is both odd and even (functions) then f(x) must be the constant function 0. Basically prove that no other function other than 0, can be both odd and even.
 
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How do you define an odd function and an even function?
 
neutrino said:
How do you define an odd function and an even function?


Odd function is when f(-x) = -f(x) and even is if f(-x) = f(-x)
 
STAR3URY said:
even is if f(-x) = f(-x)

That's not quite right.

But in a way, that is what you have to use once you have the definitions of both types of functions. :)
 
As neutrino said, you got the definition wrong. Get the definition right, and then you can make progress.
 
f(-x) = f(x) is even

f(-x) = -f(x) is odd
 
The definitions are correct. Now if a function has to fullfill both of these, it fullfills the product. Use this together with the fact that a square is always... and the fact that something is positive and negative at the same time must be...
 
Basically f(-x) = -f(-x) i.e. f = -f. The rest is algebra.
 

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