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**Unformatted text preview: **ross the graph more than once. However, to fail the Vertical Line Test, all you
need is one vertical line that ﬁts the bill, as the next example illustrates.
Example 1.4.3. Use the Vertical Line Test to determine which of the following relations describes
y as a function of x.
y y 4 4 3 3 2 2 1 1 −1 1
−1 The graph of S1 x −1 1
−1 The graph of S2 x 36 Relations and Functions Solution. Both S1 and S2 are slight modiﬁcations to the relation S in the previous example whose
graph we determined passed the Vertical Line Test. In both S1 and S2 , it is the addition of the
point (1, 2) which threatens to cause trouble. In S1 , there is a point on the curve with x-coordinate
1 just below (1, 2), which means that both (1, 2) and this point on the curve lie on the vertical line
x = 1. (See the picture below.) Hence, the graph of S1 fails the Vertical Line Test, so y is not a
function of x here. However, in S2 notice that the point with x-coordinate 1 on the curve has been
omitted, leaving an ‘open circle’ there. Hence, the vertical line x = 1 crosses the graph of S2 only
at the point (1, 2). Indeed, any vertical line will cross the gr...

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