78 CHAPTER 2 Functions and Their Graphs THEOREM Vertical-Line Test A set of points in the xy -plane is the graph of a function if and only if every vertical line intersects the graph in at most one point. We can see from the graph that the price of gasoline (adjusted for inflation) was increasing from 2002 to 2008 and was decreasing from 2012 to 2016. The graph also shows that the lowest price occurred in 1998. Look again at Figure 17. The graph shows that for each date on the horizontal axis, there is only one price on the vertical axis. The graph represents a function, although a rule for getting from date to price is not given. When a function is defined by an equation in x and y, the graph of the function is the graph of the equation; that is, it is the set of points x y , ( ) in the xy -plane that satisfy the equation. 1 Identify the Graph of a Function Not every collection of points in the xy -plane represents the graph of y as a function of x. Remember, for a function, each number x in the domain has exactly one image y in the range.This means that the graph of a function cannot contain two points with the same x -coordinate and different y -coordinates.Therefore, the graph of a function must satisfy the following vertical-line test . Identifying the Graph of a Function Which of the graphs in Figure 18 are graphs of functions? EXAMPLE 1 Figure 18 x y (d) x2 1 y2 5 1 1 1 21 21 (c) x 5 y2 x y 6 3 23 (1, 1) (1, 21) (b) y 5 x3 x y 24 4 4 24 (a) y 5 x2 x y 23 3 6 Solution The graphs in Figures 18(a) and 18(b) are graphs of functions, because every vertical line intersects each graph in at most one point. The graphs in Figures 18(c) and 18(d) are not graphs of functions, because there is a vertical line that intersects each graph in more than one point. Notice in Figure 18(c) that the input 1 corresponds to two outputs, 1− and 1. This is why the graph does not represent y as a function of x . Now Work PROBLEMS 15 AND 17 2 Obtain Information from or about the Graph of a Function If x y , ( ) is a point on the graph of a function f, then y is the value of f at x; that is, y f x . ( ) = Also if y f x , ( ) = then x y , ( ) is a point on the graph of f. For example, if 2, 7 ( ) − is on the graph of f, then f 2 7, ( ) − = and if f 5 8, ( ) = then the point 5, 8 ( ) is on the graph of y f x . ( ) = Obtaining Information from the Graph of a Function Let f be the function whose graph is given in Figure 19. (a) What are f f 5, 0, ( ) ( ) − and f 3 ? ( ) (b) What is the domain of f ? EXAMPLE 2 In Words If any vertical line intersects a graph at more than one point, the graph is not the graph of a function.
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