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 xcoordinate 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 xcoordinate 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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 Fall '13
 Wong
 Algebra, Trigonometry, Cartesian Coordinate System, The Land, The Waves, René Descartes, Euclidean geometry

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