Is Continuous Function Rewritable as E(x)+ O(x)?

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Discussion Overview

The discussion revolves around whether a continuous function f(x) can be expressed as the sum of an even function E(x) and an odd function O(x), with both E(x) and O(x) also being continuous. Participants explore this concept through examples, particularly focusing on the function e^x and its relationship with hyperbolic functions.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Homework-related

Main Points Raised

  • One participant questions the validity of expressing f(x) as E(x) + O(x), particularly for power functions like e^x.
  • Another participant suggests that f(x) + f(-x) is even, prompting further exploration of generating an odd function.
  • A participant introduces hyperbolic functions, stating that cosh(x) is even and sinh(x) is odd, and shows that cosh(x) + sinh(x) equals e^x.
  • One participant expresses a desire to understand the reasoning behind the statement rather than seeking specific answers, indicating a focus on the underlying principles.
  • Another participant provides formulas for cosh(x) and sinh(x) in relation to e^x, suggesting a generalization of the concept.
  • One participant expresses frustration with their understanding but acknowledges the help received from others.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the original claim regarding the decomposition of continuous functions into even and odd components. There are multiple viewpoints and ongoing exploration of the topic, particularly concerning specific functions like e^x.

Contextual Notes

Participants express uncertainty about the methods to derive odd functions and the implications of the even-odd decomposition for various types of functions. The discussion includes attempts to apply the concepts to specific examples without resolving all mathematical steps.

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is the following statement ture?

A continuous function f(x) can be rewritten as f(x)=E(x)+ O(x)
where:
1) E(x)is even function and O(x) is odd function.
2) Both E(x),O(x) are continuous function too
3) f,E,O are defined in (-oo,oo)

my classmate say it is ture but can not prove it
i think it can't be right since i can't figure out a way to rewrite a power function to E(x)+ O(x). eg e^x

any good sugguestion?
thx
 
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Can you show f(x)+f(-x) even?

Can you generate an odd function like it?

Can you now recover f?
 
Can you show f(x)+f(-x) even?
yeah,this one is easy

Can you generate an odd function like it?
hmmmm...more hints please
 
Here's a much better way of getting you to think of the answer for yourself:

you wanted to do this for e^x?

Do you know what hyperbolic trig functions are?

cosh(x) is even, sinh(x) is odd

cosh(x)+sinh(x) = e^x

if you need to, look up these at, say, wolfram.

If you need more just say, but it's always best to give you the means, especially if it's in terms of stuff you know, and it seems reasonable if you're doing continuity, that you know what cosh and sinh are.
 
well... i am not looking at any specific answer.
i am trying to find out why it is true.

cosh(x)+sinh(x) = e^x
thx for telling me that, i just check it out and it is true


Can you generate an odd function like it?
i am still following ur hint(i have played with it for 2hrs )


if we let g(x)=f(-x)+f(x)
then g(-x)=f(x)+f(-x)=g(x)...(1)
therefore g is even

rearrange (1) gives
-f(x)=f(-x)-g(x)...(2)
and
f(-x)=g(x)-f(x)...(3)

but it doesn't work
what should i do next??
 
So for e^x we have

cosh(x)=(e^(x)+e^(-x))/2

and

sinh(x) = (e^(x)-e^(-x))/2


and remember f(x) = e^x here, and f(-x)=e^(-x)

can you see how that generalizes?
 
:frown: omg i just found out how stupid i am
i want to shoot myself...errr

but really thanks matt
u r the man
 
Hope you think it's better to figure these things out some times than just be told them; I'm trying to stick to Polya's views on teaching.. reminds me to start a thread on that some time
 

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