MATHS BITE: Even or Odd?

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 xn. For example f(x) = x2+2x4 – 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, x3, x5,… etc. Note that unlike even functions, an expression such as x3 + 1 is not odd.
• f(x) = sin(x)

Odd and/or Even?

A function can be neither odd nor even, for example x− 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.