Given a function f: AB, we say that f is **pair** if and only if f (x) = f (-x) for all x A. That is: the symmetrical values must have the same image. The following diagram shows an example of an even function:

For example, the function f: IRIR defined by f (x) = x^{2} is an even function because f (x) = x^{2}= (- x)^{2}= f (-x). We can notice the parity of this function by looking at its graph:

We note in the graph that there is a symmetry with respect to the vertical axis. Symmetrical elements have the same image. Elements 2 and -2, for example, are symmetrical and have the image 4.

On the other hand, given a function f: AB, we say that f is **odd** if and only if f (-x) = - f (x) for all x A. That is: symmetrical values have symmetrical images. The following diagram shows an example of odd function:

For example, the function f: IRIR defined by f (x) = x^{3} is an odd function because f (-x) = (- x)^{3}= -x^{3}= -f (x). We can notice that the function is odd looking at its graph:

We note in the graph that there is symmetry with respect to origin 0. Symmetrical elements have symmetrical images. Elements 1 and -1, for example, are symmetrical and have images 1 and -1 (which are also symmetrical).

Note: A function that is neither even nor odd is called *no parity function*.

Exercise solved:

Sort the following functions into even, odd, or no parity:

a) f (x) = 2x

f (-x) = 2 (-x) = -2x f (-x) = -f (x), so f is **odd**.

b) f (x) = x^{2}-1

f (-x) = (-x)^{2}-1 = x^{2}-1 f (x) = f (-x), so f is **pair**.

c) f (x) = x^{2}-5x + 6

f (-x) = (-x)^{2}-5 (-x) +6 = x^{2}+ 5x + 6

As f (x)f (-x), so f is not even.

We also have to -f (x)f (-x), so f is not odd.

Because it is neither even nor odd, we conclude that f is a function **No parity.**