What are even and odd functions? These two terms get used very frequently in order to simplify problems such as integration or when finding Fourier coefficients, for example.

### Even Function

An even function is such that for all *x* in the domain

**f(x) = f(-x)**

An even function is symmetric with respect to the y-axis.

**Examples:**

- Any polynomial
*p*, where*n*is even for all x^{n}. For example f(x) = x^{2}+2x^{4}– 6. Note that (x+1)^{2}is not even. - f(x) = cos(x)

### Odd Function

An odd function is such that for all *x* in the domain

**f(x) = -f(-x)**

An odd function is symmetric with respect to the origin.

**Examples:**

- x, x
^{3}, x^{5},… etc. Note that unlike even functions, an expression such as x^{3}+ 1 is not odd. - f(x) = sin(x)

### Odd and/or Even?

A function can be neither odd n**or** even, for example x^{3 }− x + 1:

Additionally, the only function that is even **and** odd is f(x) = 0.

### Properties:

- Adding two even (odd) functions will give an even (odd) function.
- Adding an even and an odd function will give a function that is neither odd nor even.
- Both the product of two even functions and the product of two odd functions is even.
- The product of an even function and an odd function is odd.

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I would like to add also the interesting property that every function can be written in a unique way as a sum of an even and of an odd function.

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