This is called projecting the curve to the x axis

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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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