Example 2 Finding the Domain and Range of a Function

Transcription

1 7_00.qd /7/06 0:9 AM Page 5 Section. Graphs of Functions 5. Graphs of Functions The Graph of a Function In Section., functions were represented graphicall b points on a graph in a coordinate plane in which the input values are represented b the horizontal ais and the output values are represented b the vertical ais. The graph of a function f is the collection of ordered pairs, f such that is in the domain of f. As ou stud this section, remember the geometric interpretations of and f. the directed distance from the -ais f the directed distance from the -ais Eample shows how to use the graph of a function to find the domain and range of the function. Eample Finding the Domain and Range of a Function Use the graph of the function f shown in Figure. to find (a) the domain of f, (b) the function values f and f, and (c) the range of f. What ou should learn Find the domains and ranges of functions and use the Vertical Line Test for functions. Determine intervals on which functions are increasing, decreasing, or constant. Determine relative maimum and relative minimum values of functions. Identif and graph step functions and other piecewise-defined functions. Identif even and odd functions. Wh ou should learn it Graphs of functions provide a visual relationship between two variables. For eample, in Eercise 88 on page 5, ou will use the graph of a step function to model the cost of sending a package. Range (, ) = f ( ) (, 0) 5 6 Stephen Chernin/Gett Images Figure. Domain a. The closed dot at, 5 indicates that is in the domain of f, whereas the open dot at, 0 indicates that is not in the domain. So, the domain of f is all in the interval,. b. Because, 5 is a point on the graph of f, it follows that f 5. Similarl, because, is a point on the graph of f, it follows that f. c. Because the graph does not etend below f 5or above f, the range of f is the interval 5,. Now tr Eercise. STUDY TIP The use of dots (open or closed) at the etreme left and right points of a graph indicates that the graph does not etend beond these points. If no such dots are shown, assume that the graph etends beond these points.

2 7_00.qd /7/06 0:9 AM Page 6 6 Chapter Functions and Their Graphs Eample Finding the Domain and Range of a Function Find the domain and range of f. Algebraic Because the epression under a radical cannot be negative, the domain of f is the set of all real numbers such that 0. Solve this linear inequalit for as follows. (For help with solving linear inequalities, see Appendi D.) 0 Write original inequalit. Add to each side. So, the domain is the set of all real numbers greater than or equal to. Because the value of a radical epression is never negative, the range of f is the set of all nonnegative real numbers. Now tr Eercise 7. Graphical Use a graphing utilit to graph the equation, as shown in Figure.5. Use the trace feature to determine that the -coordinates of points on the graph etend from to the right. When is greater than or equal to, the epression under the radical is nonnegative. So, ou can conclude that the domain is the set of all real numbers greater than or equal to. From the graph, ou can see that the -coordinates of points on the graph etend from 0 upwards. So ou can estimate the range to be the set of all nonnegative real numbers. 5 8 Figure.5 = B the definition of a function, at most one -value corresponds to a given -value. It follows, then, that a vertical line can intersect the graph of a function at most once. This leads to the Vertical Line Test for functions. Vertical Line Test for Functions A set of points in a coordinate plane is the graph of as a function of if and onl if no vertical line intersects the graph at more than one point. 8 Eample Vertical Line Test for Functions Use the Vertical Line Test to decide whether the graphs in Figure.6 represent as a function of. (a) a. This is not a graph of as a function of because ou can find a vertical line that intersects the graph twice. b. This is a graph of as a function of because ever vertical line intersects the graph at most once. Now tr Eercise 7. (b) Figure.6 7

3 7_00.qd /7/06 0:9 AM Page 7 Section. Graphs of Functions 7 Increasing and Decreasing Functions The more ou know about the graph of a function, the more ou know about the function itself. Consider the graph shown in Figure.7. Moving from left to right, this graph falls from to 0, is constant from 0 to, and rises from to. Increasing, Decreasing, and Constant Functions A function f is increasing on an interval if, for an and in the interval, < implies f < f. A function f is decreasing on an interval if, for an and in the interval, < implies f > f. A function f is constant on an interval if, for an and in the interval, f f. TECHNOLOGY TIP Most graphing utilities are designed to graph functions of more easil than other tpes of equations. For instance, the graph shown in Figure.6(a) represents the equation 0. To use a graphing utilit to duplicate this graph ou must first solve the equation for to obtain ±, and then graph the two equations and in the same viewing window. Eample Increasing and Decreasing Functions In Figure.8, determine the open intervals on which each function is increasing, decreasing, or constant. a. Although it might appear that there is an interval in which this function is constant, ou can see that if then < <,, which implies that f < f. So, the function is increasing over the entire real line. b. This function is increasing on the interval,, decreasing on the interval,, and increasing on the interval,. c. This function is increasing on the interval, 0, constant on the interval 0,, and decreasing on the interval,. Decreasing Constant Increasing Figure.7 f() = (, ) f() = f() = +, < 0, 0 + > (0, ) (, ) (, ) (a) Figure.8 (b) (c) Now tr Eercise.

4 7_00.qd /7/06 0:9 AM Page 8 8 Chapter Functions and Their Graphs Relative Minimum and Maimum Values The points at which a function changes its increasing, decreasing, or constant behavior are helpful in determining the relative maimum or relative minimum values of the function. Definitions of Relative Minimum and Relative Maimum A function value f a is called a relative minimum of f if there eists an interval, that contains a such that < < implies f a f. A function value f a is called a relative maimum of f if there eists an interval, that contains a such that < < implies f a f. Relative maima Relative minima Figure.9 shows several different eamples of relative minima and relative maima. In Section., ou will stud a technique for finding the eact points at which a second-degree polnomial function has a relative minimum or relative maimum. For the time being, however, ou can use a graphing utilit to find reasonable approimations of these points. Figure.9 Eample 5 Approimating a Relative Minimum Use a graphing utilit to approimate the relative minimum of the function given b f. The graph of f is shown in Figure.0. B using the zoom and trace features of a graphing utilit, ou can estimate that the function has a relative minimum at the point 0.67,.. See Figure.. Later, in Section., ou will be able to determine that the eact point at which the relative minimum occurs is, 0. f() = Figure.0 Figure. Now tr Eercise..8 TECHNOLOGY TIP When ou use a graphing utilit to estimate the - and -values of a relative minimum or relative maimum, the zoom feature will often produce graphs that are nearl flat, as shown in Figure.. To overcome this problem, ou can manuall change the vertical setting of the viewing window. The graph will stretch verticall if the values of Ymin and Yma are closer together. TECHNOLOGY TIP Some graphing utilities have built-in programs that will find minimum or maimum values. These features are demonstrated in Eample 6.

5 7_00.qd /7/06 0:9 AM Page 9 Section. Graphs of Functions 9 Eample 6 Approimating Relative Minima and Maima Use a graphing utilit to approimate the relative minimum and relative maimum of the function given b f. The graph of f is shown in Figure.. B using the zoom and trace features or the minimum and maimum features of the graphing utilit, ou can estimate that the function has a relative minimum at the point 0.58, 0.8 See Figure.. and a relative maimum at the point 0.58, 0.8. See Figure.. If ou take a course in calculus, ou will learn a technique for finding the eact points at which this function has a relative minimum and a relative maimum. Eample 7 Now tr Eercise. Temperature During a -hour period, the temperature (in degrees Fahrenheit) of a certain cit can be approimated b the model , 0 where represents the time of da, with 0 corresponding to 6 A.M. Approimate the maimum and minimum temperatures during this -hour period. To solve this problem, graph the function as shown in Figure.5. Using the zoom and trace features or the maimum feature of a graphing utilit, ou can determine that the maimum temperature during the -hour period was approimatel 6 F. This temperature occurred at about :6 P.M. 6.6, as shown in Figure.6. Using the zoom and trace features or the minimum feature, ou can determine that the minimum temperature during the -hour period was approimatel F, which occurred at about :8 A.M. 9.8, as shown in Figure.7. f() = + Figure. f() = + Figure. f() = + Figure. TECHNOLOGY SUPPORT For instructions on how to use the minimum and maimum features, see Appendi A; for specific kestrokes, go to this tetbook s Online Stud Center. = Figure.5 Figure.6 Figure.7 Now tr Eercise

6 7_00.qd /7/06 0:9 AM Page 0 0 Chapter Functions and Their Graphs Graphing Step Functions and Piecewise-Defined Functions Librar of Parent Functions: Greatest Integer Function The greatest integer function, denoted b and defined as the greatest integer less than or equal to, has an infinite number of breaks or steps one at each integer value in its domain. The basic characteristics of the greatest integer function are summarized below. A review of the greatest integer function can be found in the Stud Capsules. Graph of f f() = [[ ]] Domain:, Range: the set of integers -intercepts: in the interval 0, -intercept: 0, 0 Constant between each pair of consecutive integers Jumps verticall one unit at each integer value Could ou describe the greatest integer function using a piecewise-defined function? How does the graph of the greatest integer function differ from the graph of a line with a slope of zero? TECHNOLOGY TIP Most graphing utilities displa graphs in connected mode, which means that the graph has no breaks. When ou are sketching graphs that do have breaks, it is better to use dot mode. Graph the greatest integer function [often called Int ] in connected and dot modes, and compare the two results. Because of the vertical jumps described above, the greatest integer function is an eample of a step function whose graph resembles a set of stairsteps. Some values of the greatest integer function are as follows. greatest integer 0 greatest integer greatest integer.5 In Section., ou learned that a piecewise-defined function is a function that is defined b two or more equations over a specified domain. To sketch the graph of a piecewise-defined function, ou need to sketch the graph of each equation on the appropriate portion of the domain. Demonstrate the real-life nature of step functions b discussing Eercises 87 and 88 in this section. If writing is a part of our course, this section provides a good opportunit for students to find other eamples of step functions and write brief essas on their applications. Eample 8 Sketch the graph of Graphing a Piecewise-Defined Function f,, b hand. > This piecewise-defined function is composed of two linear functions. At and to the left of, the graph is the line given b. To the right of, the graph is the line given b (see Figure.8). Notice that the point, 5 is a solid dot and the point, is an open dot. This is because f 5. Now tr Eercise. Figure.8

7 7_00.qd /7/06 0:9 AM Page Even and Odd Functions A graph has smmetr with respect to the -ais if whenever, is on the graph, so is the point,. A graph has smmetr with respect to the origin if whenever, is on the graph, so is the point,. A graph has smmetr with respect to the -ais if whenever, is on the graph, so is the point,. A function whose graph is smmetric with respect to the -ais is an even function. A function whose graph is smmetric with respect to the origin is an odd function. A graph that is smmetric with respect to the -ais is not the graph of a function ecept for the graph of 0. These three tpes of smmetr are illustrated in Figure.9. Section. Graphs of Functions (, ) (, ) (, ) (, ) (, ) (, ) Smmetric to -ais Smmetric to origin Smmetric to -ais Even function Odd function Not a function Figure.9 Test for Even and Odd Functions A function f is even if, for each in the domain of f, f f. A function f is odd if, for each in the domain of f, f f. Eample 9 Is the function given b Algebraic Testing for Evenness and Oddness f This function is even because f f. even, odd, or neither? Graphical Use a graphing utilit to enter in the equation editor, as shown in Figure.50. Then graph the function using a standard viewing window, as shown in Figure.5. You can see that the graph appears to be smmetric about the -ais. So, the function is even. 0 = 0 0 Now tr Eercise 59. Figure.50 Figure.5 0

8 7_00.qd /7/06 0:9 AM Page Chapter Functions and Their Graphs Eample 0 Even and Odd Functions Determine whether each function is even, odd, or neither. a. b. g h c. f Algebraic a. This function is odd because g g. b. This function is even because h h. c. Substituting for produces f. Because f and f, ou can conclude that f f and f f. So, the function is neither even nor odd. Graphical a. In Figure.5, the graph is smmetric with respect to the origin. So, this function is odd. (, ) Figure.5 b. In Figure.5, the graph is smmetric with respect to the -ais. So, this function is even. h() = + Figure.5 (, ) (, ) (, ) g() = c. In Figure.5, the graph is neither smmetric with respect to the origin nor with respect to the -ais. So, this function is neither even nor odd. f() = Now tr Eercise 6. Figure.5 To help visualize smmetr with respect to the origin, place a pin at the origin of a graph and rotate the graph 80. If the result after rotation coincides with the original graph, the graph is smmetric with respect to the origin.

9 7_00.qd /7/06 0:9 AM Page Section. Graphs of Functions. Eercises See for worked-out solutions to odd-numbered eercises. Vocabular Check Fill in the blanks.. The graph of a function f is a collection of, such that is in the domain of f.. The is used to determine whether the graph of an equation is a function of in terms of.. A function f is on an interval if, for an and in the interval, < implies f > f.. A function value f a is a relative of f if there eists an interval, containing a such that < < implies f a f. 5. The function f is called the function, and is an eample of a step function. 6. A function f is if, for each in the domain of f, f f. In Eercises, use the graph of the function to find the domain and range of f. Then find f In Eercises 5 0, use a graphing utilit to graph the function and estimate its domain and range. Then find the domain and range algebraicall. 5. f 6. f 7. f 8. h t t = f() = f() f f 5 In Eercises, use the given function to answer the questions. (a) Determine the domain of the function. (b) Find the value(s) of such that f 0. = f() 5 = f() (c) The values of from part (b) are referred to as what graphicall? (d) Find f 0, if possible. (e) The value from part (d) is referred to as what graphicall? (f) What is the value of f at? What are the coordinates of the point? (g) What is the value of f at? What are the coordinates of the point? (h) The coordinates of the point on the graph of f at which. can be labeled, f or,.. f 6... f() = In Eercises 5 8, use the Vertical Line Test to determine whether is a function of. Describe how ou can use a graphing utilit to produce the given graph f f() = +, 0, > 0

10 7_00.qd /7/06 0:9 AM Page Chapter Functions and Their Graphs In Eercises 9, determine the open intervals over which the function is increasing, decreasing, or constant. In Eercises 0, (a) use a graphing utilit to graph the function and (b) determine the open intervals on which the function is increasing, decreasing, or constant.. f. f 5. f 6. f 7. f In Eercises 6, use a graphing utilit to approimate an relative minimum or relative maimum values of the function.. f 6. f h 6. g f 0. f 6 6. f. f 6 6 f f f In Eercises 7, (a) approimate the relative minimum or relative maimum values of the function b sketching its graph using the point-plotting method, (b) use a graphing utilit to approimate an relative minimum or relative maimum values, and (c) compare our answers from parts (a) and (b). 7. f 5 8. f 9. f 0. f. f 6. f 8 In Eercises 50, sketch the graph of the piecewisedefined function b hand , 0 7. f, 0 <, > 5, 8. g, < < 5, Librar of Parent Functions In Eercises 5 56, sketch the graph of the function b hand. Then use a graphing utilit to verif the graph. 5. f 5. f 5. f 5. f 55. f 56. f In Eercises 57 and 58, use a graphing utilit to graph the function. State the domain and range of the function. Describe the pattern of the graph. 57. f,, f 6,, f,, f,, f,, h,, s < 0 0 > < 0 0 > < 0 0 > 58. g

11 7_00.qd /7/06 0:9 AM Page 5 Section. Graphs of Functions 5 In Eercises 59 66, algebraicall determine whether the function is even, odd, or neither. Verif our answer using a graphing utilit. 59. f t t t 60. f 6 6. g 5 6. h 5 6. f 6. f g s s 66. f s s Think About It In Eercises 67 7, find the coordinates of a second point on the graph of a function f if the given point is on the graph and the function is (a) even and (b) odd. 67., 68. 5, 7 69., , 7., 7. a, c In Eercises 7 8, use a graphing utilit to graph the function and determine whether it is even, odd, or neither. Verif our answer algebraicall. 7. f 5 7. f f 76. f h 78. f f 80. g t t 8. f 8. f 5 In Eercises 8 86, graph the function and determine the interval(s) (if an) on the real ais for which f ~ 0. Use a graphing utilit to verif our results. 8. f 8. f 85. f f 87. Communications The cost of using a telephone calling card is $.05 for the first minute and $0.8 for each additional minute or portion of a minute. (a) A customer needs a model for the cost C of using the calling card for a call lasting t minutes. Which of the following is the appropriate model? C t t C t t (b) Use a graphing utilit to graph the appropriate model. Use the value feature or the zoom and trace features to estimate the cost of a call lasting 8 minutes and 5 seconds. 88. Deliver Charges The cost of sending an overnight package from New York to Atlanta is $9.80 for a package weighing up to but not including pound and $.50 for each additional pound or portion of a pound. Use the greatest integer function to create a model for the cost C of overnight deliver of a package weighing pounds, where > 0. Sketch the graph of the function. In Eercises 89 and 90, write the height h of the rectangle as a function of = + (, ) h h (, ) (, ) = 9. Population During a ear period from 990 to 00, the population P (in thousands) of West Virginia fluctuated according to the model P 0.008t 0.t 0.0t 7.9t 79, 0 t where t represents the ear, with t 0 corresponding to 990. (Source: U.S. Census Bureau) (a) Use a graphing utilit to graph the model over the appropriate domain. (b) Use the graph from part (a) to determine during which ears the population was increasing. During which ears was the population decreasing? (c) Approimate the maimum population between 990 and Fluid Flow The intake pipe of a 00-gallon tank has a flow rate of 0 gallons per minute, and two drain pipes have a flow rate of 5 gallons per minute each. The graph shows the volume V of fluid in the tank as a function of time t. Determine in which pipes the fluid is flowing in specific subintervals of the one-hour interval of time shown on the graph. (There are man correct answers.) Volume (in gallons) V (0, 75) (0, 75) (5, 50) (60, 00) (5, 50) (50, 50) (0, 5) (0, 5) (0, 0) Time (in minutes) t

12 7_00.qd /7/06 0:9 AM Page 6 6 Chapter Functions and Their Graphs Snthesis True or False? In Eercises 9 and 9, determine whether the statement is true or false. Justif our answer. 9. A function with a square root cannot have a domain that is the set of all real numbers. 9. It is possible for an odd function to have the interval 0, as its domain. Think About It In Eercises 95 00, match the graph of the function with the best choice that describes the situation. (a) The air temperature at a beach on a sunn da (b) The height of a football kicked in a field goal attempt (c) The number of children in a famil over time (d) The population of California as a function of time (e) The depth of the tide at a beach over a -hour period (f) The number of cupcakes on a tra at a part If f is an even function, determine if g is even, odd, or neither. Eplain. (a) g f (b) g f (c) g f (d) g f 0. Think About It Does the graph in Eercise 6 represent as a function of? Eplain. 05. Think About It Does the graph in Eercise 7 represent as a function of? Eplain. 06. Writing Write a short paragraph describing three different functions that represent the behaviors of quantities between 995 and 006. Describe one quantit that decreased during this time, one that increased, and one that was constant. Present our results graphicall. Skills Review In Eercises 07 0, identif the terms. Then identif the coefficients of the variable terms of the epression In Eercises, find (a) the distance between the two points and (b) the midpoint of the line segment joining the points.., 7, 6,... 5, 0,, 6 5,,, 6,,, 6 In Eercises 5 8, evaluate the function at each specified value of the independent variable and simplif f 5 (a) f 6 (b) f (c) f f (a) f (b) f (c) f f (a) f (b) f (c) f 6 f (a) f (b) f 0 (c) f 0. Proof Prove that a function of the following form is odd. a n n a n n... a a 0. Proof Prove that a function of the following form is even. a n n a n n... a a 0 In Eercises 9 and 0, find the difference quotient and simplif our answer. 9. f 9, f h f, h h 0 f 6 h f 6 0. f 5 6,, h 0 h