Graphical interpretation of Even and Odd symmetry in functions.

An [i]even function[/i] is one for which [math]f(x)=f(-x)[/math] is true. Graphically, this suggests that if you pick any [math]x[/math] and get its [math]y[/math] value, then you will get the [i]same[/i] [math]y[/math] value at [math]-x[/math], for all [math]x[/math] in the domain.[br][br]An [i]odd function[/i] is one for which [math]f(x)=-f(-x)[/math] is true. Graphically, this suggests that if you pick any [math]x[/math] and get its [math]y[/math] value, then you will get the [i]opposite[/i] [math]y[/math] value at [math]-x[/math], for all [math]x[/math] in the domain.[br][br]You can test this for the graphed functions by comparing the values of [math]f(x)[/math] with the values of [math]f(-x)[/math] and [math]-f(-x)[/math]. If they match for all [math]x[/math], then [math]f(x)[/math] is even or odd, respectively.[br][br]In simpler terms, an even function's graph is a [i]reflection[/i] of itself over the [math]y[/math]-axis. If you check the "Reflect f(x)" box, [math]f[/math]'s reflection will be drawn in yellow. If the yellow reflection and the original (black) function overlap everywhere, then the function is even.[br][br]An odd function's graph is a 180-degree [i]rotation[/i] of itself around the origin. If you check the "Rotate f(x)" box, a blue copy of the graph will rotate 180 degrees about the origin. If the blue rotation and the original (black) function overlap everywhere, then the function is odd.