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Even and odd function


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) = x2 is an even function because f (x) = x2= (- 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) = x3 is an odd function because f (-x) = (- x)3= -x3= -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) = x2-1
f (-x) = (-x)2-1 = x2-1 f (x) = f (-x), so f is pair.

c) f (x) = x2-5x + 6
f (-x) = (-x)2-5 (-x) +6 = x2+ 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.

Next: Ascending and Descending Functions