Can Any Function Be Decomposed into Odd and Even Subfunctions?

[thenewbosco]

Hello, the question here says:

Show that any given function can be decomposed into the sum of manifestly odd and even subfunctions.

What i have done is just assumed a continuous, differentiable function, with a number a in the domain of the function, then shown that a taylor series for a function alternates between even and odd functions as the powers of x change from even to odd numbers. Is this enough for this question or is there something i haven't seen?

thanks

 

[Hurkyl]

Is this enough for this question or is there something i haven't seen?

No! Surely you see that you have only proven it for functions that have Taylor series expansions at every point?

(Incidentally, even infinitely differentiabe functions can fail to have Taylor series)



This is the sort of problem where you just write down what things mean, and ideas should become evident. What does it mean for the function f(x) to be decomposed into an odd and an even subfunction? What does it mean for a function to be even? What does it mean for a function to be odd?

(Incidentally, you should always ask yourself questions like this anytime you get stuck. In fact, it usually helps to ask these questions before you get stuck)

 

[kyle8921]

Hi there,

Thank you for sharing your approach to solving this question. Your idea of using the Taylor series expansion to show that a function can be decomposed into even and odd subfunctions is a valid approach. However, it would be helpful to provide a more formal proof to support your assumption. Additionally, it would be beneficial to provide an example of a specific function and its decomposition into even and odd subfunctions to further illustrate your point.

Another approach to proving this statement is to use the definition of even and odd functions. An even function is one that satisfies f(-x) = f(x) for all x in the domain, while an odd function satisfies f(-x) = -f(x) for all x in the domain. Using this definition, you can show that any given function can be decomposed into an even and odd subfunction by splitting the function into its even and odd parts, f(x) = g(x) + h(x), where g(x) is the even part and h(x) is the odd part. This can be done by plugging in -x and x into the original function and solving for g(x) and h(x) separately.

In summary, while your approach using the Taylor series expansion is a valid one, it would be helpful to provide a more formal proof and an example to support your argument. Alternatively, you can use the definition of even and odd functions to show the decomposition. I hope this helps!