Section 2: The Graph of a Function

# Elementary and Intermediate Algebra: Graphs & Models (3rd Edition)

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Section 2.2 The Graph of a Function 93 Version: Fall 2007 2.2 The Graph of a Function Rene Descartes (1596-1650) was a French philosopher and mathematician who is well known for the famous phrase “cogito ergo sum” (I think, therefore I am), which appears in his Discours de la methode pour bien conduire sa raison, et chercher la verite dans les sciences (Discourse on the Method of Rightly Conducting the Reason, and Seeking Truth in the Sciences). In that same treatise, Descartes introduces his coordinate system, a method for representing points in the plane via pairs of real numbers. Indeed, the Cartesian plane of modern day is so named in honor of Rene Descartes, who some call the “Father of Modern Mathematics.” Descartes’ work, which forever linked geometry and algebra, was continued in an appendix to Discourse on Method , entitled La Geometrie , which some consider the beginning of modern mathematics. Certainly both Newton and Leibniz, in developing the Calculus, built upon the foundation provided in this work by Descartes. A Cartesian Coordinate System consists of a pair of axes, usually drawn at right angles to one another in the plane, one horizontal (labeled x ) and one vertical (labeled y ), as shown in the Figure 1 . The quadrants are numbered I, II, III, and IV, in counterclockwise order, and samples of ordered pairs of the form ( x, y ) are shown in each quadrant of the Cartesian coordinate system in Figure 1 . x y I II III IV x y (3 , 2) ( 3 , 3) ( 4 , 2) (2 , 4) Numbering the quadrants. To the right and up is positive, left and down is negative. Figure 1. The Cartesian coordinate system. Now, suppose that we have a relation R = { (1 , 2) , (3 , 1) , (3 , 4) , (4 , 3) } . Recall that relation is the name given to a collection of ordered pairs. In Figure 2 (b) we’ve plotted each of the ordered pairs in the relation R . This is called the graph of the relation R . Copyrighted material. See: 1

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94 Chapter 2 Functions Version: Fall 2007 Definition 1. The graph of a relation is the collection of all ordered pairs of the relation. These are usually represented as points in a Cartesian coordinate system. 1 3 4 1 2 3 4 R x y 5 5 (1 , 2) (3 , 1) (3 , 4) (4 , 3) (a) (b) Figure 2. A mapping diagram and its graph. In Figure 2 (a), we’ve created a mapping diagram of the ordered pairs. Note that the domain object 3 is paired with two range elements, namely 1 and 4. Hence the relation R is not a function. It is interesting to note that there are two points in the graph of R in Figure 2 (b) that have the same first coordinate, namely (3 , 1) and (3 , 4) . This is a signal that the graph of the relation R is not a function. In the next section we will discuss the Vertical Line Test , which will use this dual use of the first coordinate to determine when a relation is a not a function.
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