225 2.3 Functions The graph in Figure 21(a) represents a function because each vertical line intersects the graph in no more than one point. The graph in Figure 21(b) is not the graph of a function because there exists a vertical line that intersects the graph in more than one point. EXAMPLE 4 Using the Vertical Line Test Use the vertical line test to determine whether each relation graphed in Example 3 is a function. SOLUTION We repeat each graph from Example 3, this time with vertical lines drawn through the graphs. (a) x y 0 (4, –3) (–1, 1) (1, 2) (5, 2) (0, –1) (b) x y 0 –4 –6 6 4 (c) x y 0 (d) x y 0 2 –3 • The graphs of the relations in parts (a), (c), and (d) pass the vertical line test because every vertical line intersects each graph no more than once. Thus, these graphs represent functions. • The graph of the relation in part (b) fails the vertical line test because the same x-value corresponds to two different y-values. Therefore, it is not the graph of a function. S Now Try Exercises 27 and 29. Vertical Line Test If every vertical line intersects the graph of a relation in no more than one point, then the relation is a function. Determining Whether Relations Are Functions Because each value of x leads to only one value of y in a function, any vertical line must intersect the graph in at most one point. This is the vertical line test for a function. As another example, the function y = 1 x has the set of all real numbers except 0 as domain because y is undefined if x = 0. In general, the domain of a function defined by an algebraic expression is the set of all real numbers, except those numbers that lead to division by 0 or to an even root of a negative number. (There are also exceptions for logarithmic and trigonometric functions. They are covered in further treatment of precalculus mathematics.) x y x1 0 y2 y1 (x1, y1) (x1, y2) This is not the graph of a function. The same x-value corresponds to two different y-values. Figure 21 x y y3 x1 x2 x3 y2 y1 This is the graph of a function. Each x-value corresponds to only one y-value. (a) (b)
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