The relation in Example 1a) is not a function since the x-value of 6 corresponds to two different y-values, 2 and 2. Notice that we can draw a vertical line that intersects the graph in more than one point—the line through (6, 2) and (6, 2). The relation in Example 1b), however, is a function—each x-value corresponds to only one y-value. Anywhere we draw a vertical line through the points on the graph of this relation, the line intersects the graph in exactly one point. This leads us to the vertical line test for a function. Procedure The Vertical Line Test If there is no vertical line that can be drawn through a graph so that it intersects the graph more than once, then the graph represents a function. If a vertical line can be drawn through a graph so that it intersects the graph more than once, then the graph does not represent a function. EXAMPLE 2 Use the vertical line test to determine whether each graph, in blue, represents a function. Identify the domain and range. a) x y 5 5 5 5 b) x y 5 5 5 5 Solution a) Anywhere a vertical line is drawn through the graph, the line will intersect the graph only once. This graph represents a function. The arrows on the graph indicate that the graph continues without bound. The domain of this function is the set of x-values on the graph. Since the graph continues indefi nitely in the x-direction, the domain is the set of all real numbers. The domain is (q, q). The range of this function is the set of y-values on the graph. Since the graph continues indefi nitely in the y-direction, the range is the set of all real numbers. The range is (q, q). b) This graph fails the vertical line test because we can draw a vertical line through the graph that intersects it more than once. This graph does not represent a function. The set of x-values on the graph includes all real numbers from 3 to 3. The domain is 3, 3. The set of y-values on the graph includes all real numbers from 5 to 5. The range is 5, 5. Notice that the graph of a line, except for a vertical line, will always pass the vertical line test. Therefore, every nonvertical line represents a function. 200 CHAPTER 4 Linear Equations in Two Variables and Functions www.mhhe.com/messersmith
messersmith_power_intermediate_algebra_1e_ch4_7_10
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