VIEWS: 2 PAGES: 19 POSTED ON: 7/27/2013 Public Domain
Graphs of Functions Text Example Graph f (x) = x2 + 1. To do so, use integer values of x from the set {-3, -2, -1, 0, 1, 2, 3} to obtain seven ordered pairs. Plot each ordered pair and draw a smooth curve through the points. Use the graph to specify the function's domain and range. Solution The graph of f (x) = x2 + 1 is, by definition, the graph of y = x2 + 1. We begin by setting up a partial table of coordinates. 2 x f (x) = x + 1 (x, y) or (x, f (x)) -3 f (-3) = (-3)2 + 1 = 10 (-3, 10) -2 f (-2) = (-2)2 + 1 = 5 (-2, 5) -1 f (-1) = (-1)2 + 1 = 2 (-1, 2) 0 f (0) = 02 + 1 = 1 (0, 1) Text Example cont. Solution x f (x) = x2 + 1 (x, y) or (x, f (x)) 1 f (1) = 12 + 1 = 2 (1, 2) 2 f (2) = (2)2 + 1 = 5 (2, 5) 3 f (3) = (3)2 + 1 = 10 (3, 10) Now, we plot the seven points and draw a smooth curve 10 through them, as shown. The graph of f has a cuplike shape. 9 8 The points on the graph of f have x-coordinates that extend 7 Range: [1, oo) 6 indefinitely far to the left and to the right. Thus, the domain 5 consists of all real numbers, represented by (-oo, oo). By 4 3 contrast, the points on the graph have y-coordinates that start at 2 1 and extend indefinitely upward. Thus, the range consists of all 1 real numbers greater than or equal to 1, represented by [1, oo). -4 -3 -2 -1 1 2 3 4 Domain: all Reals Obtaining Information from Graphs You can obtain information about a function from its graph. At the right or left of a graph, you will find closed dots, open dots, or arrows. • A closed dot indicates that the graph does not extend beyond this point and the point belongs to the graph. • An open dot indicates that the graph does not extend beyond this point and the point does not belong to the graph. • An arrow indicates that the graph extends indefinitely in the direction in which the arrow points. Text Example Use the graph of the function f to answer the following questions. • What are the function values f (-1) and f (1)? • What is the domain of f (x)? 5 4 • What is the range of f (x)? 3 2 1 -5 -4 -3 -2 -1 1 2 3 4 5 Solution -1 -2 a. Because (-1, 2) is a point on the graph of f, the -3 -4 y-coordinate, 2, is the value of the function at -5 the x-coordinate, -1. Thus, f (-l) = 2. Similarly, because (1, 4) is also a point on the graph of f, this indicates that f (1) = 4. Text Example cont. Use the graph of the function f to answer the following questions. • What are the function values f (-1) and f (1)? • What is the domain of f (x)? 5 4 • What is the range of f (x)? 3 2 1 -5 -4 -3 -2 -1 1 2 3 4 5 Solution -1 -2 b. The open dot on the left shows that x = -3 is not in -3 -4 the domain of f. By contrast, the closed dot on the -5 right shows that x = 6 is. We determine the domain of f by noticing that the points on the graph of f have x-coordinates between -3, excluding -3, and 6, including 6. Thus, the domain of f is { x | -3 < x < 6} or the interval (-3, 6]. Text Example cont. Use the graph of the function f to answer the following questions. • What are the function values f (-1) and f (1)? • What is the domain of f (x)? 5 4 • What is the range of f (x)? 3 2 1 -5 -4 -3 -2 -1 1 2 3 4 5 Solution -1 -2 c. The points on the graph all have y-coordinates -3 -4 between -4, not including -4, and 4, including 4. -5 The graph does not extend below y = -4 or above y = 4. Thus, the range of f is { y | -4 < y < 4} or the interval (-4, 4]. The Vertical Line Test for Functions • If any vertical line intersects a graph in more than one point, the graph does not define y as a function of x. Text Example Use the vertical line test to identify graphs in which y is a function of x. a. y b. y c. y d. y x x x x Solution y is a function of x for the graphs in (b) and (c). a. y b. y c. y d. y x x x x y is not a function since y is a function of x. y is a function of x. y is not a function since 2 values of y correspond 2 values of y correspond to an x-value. to an x-value. Increasing, Decreasing, and Constant Functions A function is increasing on an interval if for any x1, and x2 in the interval, where x1 < x2, then f (x1) < f (x2). A function is decreasing on an interval if for any x1, and x2 in the interval, where x1 < x2, then f (x1) > f (x2). A function is constant on an interval if for any x1, and x2 in the interval, where x1 < x2, then f (x1) = f (x2). (x2, f (x2)) (x1, f (x1)) (x1, f (x1)) (x2, f (x2)) (x1, f (x1)) (x2, f (x2)) Increasing Decreasing Constant f (x1) < f (x2) f (x1) < f (x2) f (x1) < f (x2) Text Example Describe the increasing, decreasing, or constant behavior of each function whose graph is shown. a. 5 b. 5 4 4 3 3 2 1 1 -5 -4 -3 -2 -1 1 2 3 4 5 -5 -4 -3 -2 -1 1 2 3 4 5 -1 -1 -2 -2 -3 -3 -4 -4 -5 -5 Solution a. The function is decreasing on the interval (-oo, 0), increasing on the interval (0, 2), and decreasing on the interval (2, oo). Text Example cont. Describe the increasing, decreasing, or constant behavior of each function whose graph is shown. a. 5 b. 5 4 4 3 3 2 1 1 -5 -4 -3 -2 -1 1 2 3 4 5 -5 -4 -3 -2 -1 1 2 3 4 5 -1 -1 -2 -2 -3 -3 -4 -4 -5 -5 Solution b. Although the function's equations are not given, the graph indicates that the function is defined in two pieces. The part of the graph to the left of the y-axis shows that the function is constant on the interval (-oo, 0). The part to the right of the y-axis shows that the function is increasing on the interval (0, oo). Definitions of Relative Maximum and Relative Minimum 1. A function value f(a) is a relative maximum of f if there exists an open interval about a such that f(x) > f(x) for all x in the open interval. 2. A function value f(b) is a relative minimum of f if there exists an open interval about b such that f(b) < f(x) for all x in the open interval. The Average Rate of Change of a Function • Let (x1, f(x1)) and (x2, f(x2)) be distinct points on the graph of a function f. The average rate of change of f from x1 to x2 is Definition of Even and Odd Functions The function f is an even function if f (-x) = f (x) for all x in the domain of f. The right side of the equation of an even function does not change if x is replaced with -x. The function f is an odd function if f (-x) = -f (x) for all x in the domain of f. Every term in the right side of the equation of an odd function changes sign if x is replaced by -x. Example Identify the following function as even, odd, or neither: f(x) = 3x2 - 2. Solution: We use the given function’s equation to find f(-x). f(-x) = 3(-x) 2-2 = 3x2 - 2. The right side of the equation of the given function did not change when we replaced x with -x. Because f(-x) = f(x), f is an even function. Even Functions and y-Axis Symmetry • The graph of an even function in which f (- x) = f (x) is symmetric with respect to the y- axis. Odd Functions and Origin Symmetry • The graph of an odd function in which f (-x) = - f (x) is symmetric with respect to the origin. Graphs of Functions