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

[x_endlessrain_x]

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

 

[matt grime]

Can you show f(x)+f(-x) even?

Can you generate an odd function like it?

Can you now recover f?

 

[x_endlessrain_x]

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

 

[matt grime]

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.

 

[x_endlessrain_x]

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??

 

[matt grime]

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?

 

[x_endlessrain_x]

omg i just found out how stupid i am
i want to shoot myself...errr

but really thanks matt
u r the man

 

[matt grime]

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