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

[STAR3URY]

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.

 

[neutrino]

How do you define an odd function and an even function?

 

[STAR3URY]

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)

 

[neutrino]

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. :)

 

[Hurkyl]

As neutrino said, you got the definition wrong. Get the definition right, and then you can make progress.

 

[STAR3URY]

f(-x) = f(x) is even

f(-x) = -f(x) is odd

 

[coomast]

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...

 

[andytoh]

Basically f(-x) = -f(-x) i.e. f = -f. The rest is algebra.