Both s1 and s2 are slight modications to the relation

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Unformatted text preview: in the previous example, the relation R2 contained two different points with the same y -coordinates, namely (1, 3) and (2, 3). Remember, in order to say y is a function of x, we just need to ensure the same x-coordinate isn’t used in more than one point.1 To see what the function concept means geometrically, we graph R1 and R2 in the plane. y y 4 4 3 3 2 2 1 1 −2 −1 −1 1 2 3 The graph of R1 x −2 −1 −1 1 2 3 x The graph of R2 The fact that the x-coordinate 1 is matched with two different y -coordinates in R1 presents itself graphically as the points (1, 3) and (1, 4) lying on the same vertical line, x = 1. If we turn our attention to the graph of R2 , we see that no two points of the relation lie on the same vertical line. We can generalize this idea as follows Theorem 1.1. The Vertical Line Test: A set of points in the plane represents y as a function of x if and only if no two points lie on the same vertical line. 1 We will have occasion later in the text to concern ourselves with the concep...
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