## Wednesday, July 11, 2012

### Even - Odd Functions

Every function can be classified as an even function, an odd function, or neither. Even functions have the characteristic that f (x) = f (-x). They are symmetrical with respect to the y-axis. A line segment joining the points f (x) and f (-x) will be perfectly horizontal , shown by dotted blue line in the applet below.

Odd functions have the characteristic that f (x) = - f (-x). They are symmetrical with respect to the origin. A line segment joining the points f (x) and f (-x) always contains the origin , shown by dotted red line in the applet below.

Some of the most common even functions are y = k , where k is a constant, y= x2 , and y = cos(x) . Some of the most common odd functions are y = x3 and y = sin(x) . Some functions that are neither even nor odd include y = x - 4 , y = cos(x) + 1.

Formal tests for symmetry:
1. y – axis : replace x with –x , produces an equivalent equation
2. x - axis : replace y with – y , produces an equivalent equation
3. origin : replace x with –x and y with –y , produces equivalent equation.

In the following applet use check box to select between odd or even functions. Also you can enter any function in the input box to see whether it is odd or even by checking the symmetry about y-axis or origin.

This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com