Are all functions odd or even?

In summary, functions can be categorized as even, odd, or neither. An even function has the property f(-x) = f(x), while an odd function has the property f(-x) = -f(x). However, a function can be broken down into even and odd components, and most functions fall into the neither category. In order for a function to be both even and odd, it must be the zero function. The exponential function is an example of a function that is neither even nor odd.
  • #1
madmike159
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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?
 
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  • #2
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 [tex]g(x)=\frac{f(x)+f(-x)}{2}[/tex].

So that, [tex]g(-x)=\frac{f(-x)+f(-(-x))}{2}=\frac{f(x)+f(-x)}{2}[/tex].

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

Similarly, let [tex]h(x)=\frac{f(x)-f(-x)}{2}[/tex].

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

So, functions may be broken up into even and odd components.
 
  • #3
Ok thanks.
 
  • #4
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.
 
  • #5
the exp function is not odd nor even.
 

1. Are all functions either odd or even?

No, not all functions are either odd or even. Some functions, such as logarithmic functions, do not have a defined parity (odd or even).

2. How can I determine if a function is odd or even?

To determine if a function is odd or even, you can use the following test: If f(-x) = f(x), then the function is even. If f(-x) = -f(x), then the function is odd.

3. Can a function be both odd and even?

No, a function cannot be both odd and even. It can only be one or the other.

4. Do odd and even functions have different properties?

Yes, odd and even functions have different properties. For example, the integral of an odd function over a symmetric interval is always 0, while the integral of an even function over a symmetric interval is twice the integral over half the interval.

5. What are some examples of odd and even functions?

Examples of odd functions include f(x) = x^3 and f(x) = sin(x). Examples of even functions include f(x) = x^2 and f(x) = cos(x).

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