04Dtypes

= Types of Relations =

In the following rules, think X-to-Y in each rule


 * One-to-one relations **
 * one X to one Y
 * A one-to-one relation exists if
 * for any x value, there is __only one__ y value
 * and
 * for any y value, there is __only one__ x value


 * Examples of one-to-one relations are
 * straight line graphs (but not vertical or horizontal lines)
 * exponential graphs such as is shown on the right


 * One-to-many relations **
 * One X to many Y
 * A one-to-many relation exists if
 * for some x values, there are __more than one__ y value
 * and
 * for any y value there is __only one__ x value


 * Examples of one-to-many relations are
 * A sideways parabola
 * A sideways cubic such as is shown to the right


 * Many-to-one relations **
 * Many X to one Y
 * A many-to-one relation exists if
 * for any x value there is __only one__ y value
 * and
 * for some y values there are __more than one__ x value


 * Examples of many-to-one relations are
 * an upright parabola
 * a cubic such as is shown on the right


 * Many-to-many relations **
 * Many X to many Y
 * A many-to-many relation exists if
 * for any x value there is __more than one__ y value
 * and
 * for any y value there is __more than one__ x value


 * Examples of many-to-many relations are
 * circles
 * ellipses such as is shown on the right


 * Functions **


 * All relations that are
 * a one-to-one relation
 * or
 * a many-to-one relation
 * are called ** functions **.


 * In other words, a ** function ** is a relation where any x value can only have one y value.


 * Vertical Line Test **


 * A simple way to decide if a graph is showing a function is to use the vertical line test.


 * If you draw a vertical line anywhere,
 * the graph cannot intercept the vertical line more than once.
 * if you can draw a vertical line anywhere that hits the graph more than once, the graph is not showing a function.


 * An upright parabola, y = x 2 ,
 * passes the vertical line test,
 * so a parabola is a function


 * A sideways parabola, y 2 = x,
 * fails the vertical line test,
 * so a sideways parabola is not a function

.