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).

And 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):

Sine function: f(x) = 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.