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):

Even Function

This is the curve f(x) = x2+1

They got called "even" functions because the functions x2, x4, x6, x8, etc behave like that, but there are other functions that behave like that too, such as cos(x):

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: -f(x).

And we get origin symmetry:

Odd Function

This is the curve f(x) = x3-x

They got called "odd" because the functions x, x3, x5, x7, etc behave like that, but there are other functions that behave like that, too, such as sin(x):

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 x3+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:

Not Even Nor Odd Function

This is the curve f(x) = x3-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/(x2-1) Even or Odd or neither?

Let's see what happens when we substitute -x:

Put in "-x":   f(-x) = (-x)/((-x)2-1)
Simplify:     = -x/(x2-1)
      = -f(x)

 

So f(-x) = -f(x) and hence it is an Odd Function

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.