Section 2.6 Introduction to Functions 193 x y 2 6 7 12 13 10 1, 6 8, 9 1, 4 3, 10 4, 2 5, 4 0, 0 8, 4 The vertical line test can be demonstrated by graphing the ordered pairs from the relations in Example 1. a. b. 511, 32, 11, 42, 12, 12, 13, 226 511, 42, 12, 12, 13, 226 y 5 4 3 2 1 21 1 Function No vertical line intersects more than once. x 543 1 3 4 5 2 3 4 5 2 Intersects more than once y x 5 4 3 2 1 21 543 1 3 4 5 1 2 3 4 5 Not a Function A vertical line intersects in more than one point. Using the Vertical Line Test Use the vertical line test to determine whether the relations define y as a function of x. a. b. y x y x Example 2 5 4 3 2 1 21 1 Points align vertically Answers 1. Yes 2. Yes 3. No Skill Practice Determine if the relations define y as a function of x. 1. 2. {( ), ( ), ( ), ( )} 3. {( ), ( ), ( ), ( )} 2. Vertical Line Test A relation that is not a function has at least one domain element x paired with more than one range value y. For example, the set {(4, 2), (4, 2)} does not define a function because two different y-values correspond to the same x. These two points are aligned vertically in the xy-plane, and a vertical line drawn through one point also intersects the other point (see Figure 2-32). If a vertical line drawn through a graph of a relation intersects the graph in more than one point, the relation cannot be a function. This idea is stated formally as the vertical line test. The Vertical Line Test Consider a relation defined by a set of points (x, y) in a rectangular coordinate system.The graph defines y as a function of x if no vertical line intersects the graph in more than one point. y x 543 1 3 4 5 2 3 4 5 2 (4, 2) (4, 2) Figure 2-32
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