# Even and Odd Functions

They are special types of functions

## Even Functions

A function is "even" when:

f(x) = f(−x) for all x

In other words there is symmetry about the y-axis (like a reflection):

This is the curve f(x) = x^{2}+1

They are called "even" functions because the functions x^{2}, x^{4}, x^{6}, x^{8}, etc behave like that, but there are other functions that behave like that too, such as cos(x):

Cosine function: f(x) = cos(x)

It is an even function

But an even exponent does not always make an even function, for example (x+1)^{2} is **not** an even function.

## Odd Functions

A function is "odd" when:

−f(x) = f(−x) for all x

Note the minus in front of f(x): **−f(x)**.

And we get origin symmetry:

This is the curve f(x) = x^{3}−x

They are called "odd" because the functions x, x^{3}, x^{5}, x^{7}, etc behave like that, but there are other functions that behave like that, too, such as** sin(x)**:

Sine function: f(x) = sin(x)

It is an odd function

But an odd exponent does not always make an odd function, for example x^{3}+1 is **not** an odd function.

## Neither Odd nor Even

Don't be misled by the names "odd" and "even" ... they are just **names** ... and a function does **not have to be** even or odd.

In fact most functions are neither odd nor even. For example, just adding 1 to the curve above gets this:

This is the curve f(x) = x^{3}−x**+1**

It is **not an odd function**, and it is **not an even function** either.

It is neither odd nor even

## Even or Odd?

### Example: is f(x) = x/(x^{2}−1) Even or Odd or neither?

Let's see what happens when we substitute **−x**:

**f(−x)**= (−x)/((−x)

^{2}−1)

^{2}−1)

**−f(x)**

So f(−x) = −f(x) , which makes it an **Odd Function**

## Even and Odd

The only function that is even **and** odd is f(x) = 0

## Special Properties

Adding:

- The sum of two even functions is even
- The sum of two odd functions is odd
- The sum of an even and odd function is neither even nor odd (unless one function is zero).

Multiplying:

- The product of two even functions is an even function.
- The product of two odd functions is an even function.
- The product of an even function and an odd function is an odd function.