Are all functions odd or even?

Gold Member
For functions when doing Fourier Transforms, when a function is odd bn = 0 and when a function is even an = 0.

Are all functions odd or even, or are there some cases where they're both / neither?

Hechima

An even function has the property:
f(-x)=f(x)

An odd function has the property:
f(-x)=-f(x)

Suppose you have a function f(x) that is neither odd nor even.

Now, define a function $$g(x)=\frac{f(x)+f(-x)}{2}$$.

So that, $$g(-x)=\frac{f(-x)+f(-(-x))}{2}=\frac{f(x)+f(-x)}{2}$$.

This means that g(x) is an even function, regardless of the properties of f(x)!

Similarly, let $$h(x)=\frac{f(x)-f(-x)}{2}$$.

It's easy to show that h(x) must be odd.

So, functions may be broken up into even and odd components.

Gold Member
Ok thanks.

HallsofIvy

Homework Helper
For example, the function f(x)= x+ 1 has the property that f(2)= 3 while f(-2)= -1. Since neither f(2)= f(-2) nor f(2)= -f(-2) is true, this function is neither even nor odd.
In fact, most functions are neither even nor odd.

In order that a function be both even and odd, we would have to have both f(x)= f(-x) and f(x)= -f(-x) for all x. That is the same as saying f(x)= -f(-x)= -f(x) so that 2f(x)= 0 for all x. The is only true for the "zero" function, f(x)= 0 for all x.

Outlined

the exp function is not odd nor even.

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