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) = x²+1

It is called "even" because even exponents like x², x⁴, x⁶, etc behave like that, but there are other functions that do that, too, such as cos(x):

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

Odd Functions

A function is "odd" when:

-f(x) = f(-x) for all x

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

We get origin symmetry:

This is the curve f(x) = x³-x

It is called "odd" because odd exponents like x, x³, x⁵, etc behave like that, but there are other functions that do that, too, such as sin(x).

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 function are neither odd nor even. For example, just adding 2 to the curve above gets this:

This is the curve f(x) = x³-x+2

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

Even or Odd?

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.