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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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