3.2 Introduction to Functions
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1 8 CHAPTER Graphs and Functions Write each statement as an equation in two variables. Then graph each equation. 97. The -value is more than three times the -value. 98. The -value is - decreased b twice the -value. 99. The -value is more than the square of the -value. 00. The -value is decreased b the square of the -value. Use a graphing calculator to verif the graphs of the following eercises. 0. Eercise 9 0. Eercise 0 0. Eercise 7 0. Eercise 8. Introduction to Functions S Define Relation, Domain, and Range. Identif Functions. Use the Vertical Line Test for Functions. Find the Domain and Range of a Function. Use Function Notation. Defining Relation, Domain, and Range Recall our eample from the last section about products sold and monthl salar. We modeled the data given b the equation = This equation describes a relationship between -values and -values. For eample, if = 000, then this equation describes how to find the -value related to = 000. In words, the equation = sas that 000 plus of the -value gives the corresponding -value. The -value of 000 corresponds to the -value of # 000 = 00 for this equation, and we have the ordered pair (000, 00). There are other was of describing relations or correspondences between two numbers or, in general, a first set (sometimes called the set of inputs) and a second set (sometimes called the set of outputs). For eample, First Set: Correspondence Second Set: People in a certain cit Each person s age, to The set of nonnegative the nearest ear integers A few eamples of ordered pairs from this relation might be (Ana, ), (Bob, 6), (Tre, ), and so on. Below are just a few other was of describing relations between two sets and the ordered pairs that the generate. Correspondence First Set: a c e Second Set: Ordered Pairs a,, c,, e, Ordered Pairs -, -, (, ), (, ),, - Some Ordered Pairs (, ), (0, ), and so on Relation, Domain, and Range A relation is a set of ordered pairs. The domain of the relation is the set of all first components of the ordered pairs. The range of the relation is the set of all second components of the ordered pairs. For eample, the domain for our relation on the left above is a, c, e6 and the range is, 6. Notice that the range does not include the element of the second set.
2 Section. Introduction to Functions 9 This is because no element of the first set is assigned to this element. If a relation is defined in terms of - and -values, we will agree that the domain corresponds to -values and that the range corresponds to -values that have -values assigned to them. Helpful Hint Remember that the range includes onl elements that are paired with domain values. For the correspondence to the right, the range is a6. First Set: Correspondence Second Set: a b c EXAMPLE Determine the domain and range of each relation. a.,,,, 0, -,, -6 b. c. Cities Lubbock Colorado Springs Omaha Yonkers Sacramento Populations (in thousands) Solution a. The domain is the set of all first coordinates of the ordered pairs,, 0, 6. The range is the set of all second coordinates,,, -6. b. Ordered pairs are not listed here but are given in graph form. The relation is -,, -,, -,, -,, 0,,,,,,, 6. The domain is -, -, -, -, 0,,, 6. The range is 6. c. The domain is the set of inputs, {Lubbock, Colorado Springs, Omaha, Yonkers, Sacramento}. The range is the numbers in the set of outputs that correspond to elements in the set of inputs {70, 0,, 06, 97}. Helpful Hint Domain or range elements that occur more than once need to be listed onl once. Determine the domain and range of each relation. a.,,, -,, -,, 66 b. c. Career Administrative Secretar Game Developer Engineer Restaurant Manager Marketing Average Starting Salar (in thousands)
3 0 CHAPTER Graphs and Functions Identifing Functions Now we consider a special kind of relation called a function. Function A function is a relation in which each first component in the ordered pairs corresponds to eactl one second component. Helpful Hint A function is a special tpe of relation, so all functions are relations, but not all relations are functions. EXAMPLE Determine whether the following relations are also functions. a. -,,, 7, -,, 9, 96 b. c. Correspondence (6, 6) People in a Each person s The set of (, ) certain cit age nonnegative integers (, ) 6 (0, ) (0, ) 6 Solution a. Although the ordered pairs -, and -, have the same -value, each -value is assigned to onl one -value, so this set of ordered pairs is a function. b. The -value 0 is assigned to two -values, - and, in this graph, so this relation does not define a function. c. This relation is a function because although two people ma have the same age, each person has onl one age. This means that each element in the first set is assigned to onl one element in the second set. Determine whether the following relations are also functions. a.,, -, -, 8,, 9, 6 b. (, ) (, ) (, 0) (, ) Answer to Concept Check: Two different ordered pairs can have the same -value but not the same -value in a function. c. Correspondence People in Birth date Set of nonnegative a certain cit (da of month) integers CONCEPT CHECK Eplain wh a function can contain both the ordered pairs, and, but not both, and,.
4 Section. Introduction to Functions We will call an equation such as = + a relation since this equation defines a set of ordered pair solutions. EXAMPLE Is the relation = + also a function? * Solution The relation = + is a function if each -value corresponds to just one -value. For each -value substituted in the equation = +, the multiplication and addition performed on each gives a single result, so onl one -value will be associated with each -value. Thus, = + is a function. * For further discussion including the graph, see Objective. Is the relation = - + also a function? EXAMPLE Is the relation = also a function? * Solution In =, if =, then = 9. Also, if = -, then = 9. In other words, we have the ordered pairs 9, and 9, -. Since the -value 9 corresponds to two -values, and -, = is not a function. * For further discussion including the graph, see Objective. Is the relation = - also a function? Using the Vertical Line Test As we have seen so far, not all relations are functions. Consider the graphs of = + and = shown net. For the graph of = +, notice that each -value corresponds to onl one -value. Recall from Eample that = + is a function. Graph of Eample : Graph of Eample : = + = (, ) (9, ) (9, ) For the graph of = the -value 9, for eample, corresponds to two -values, and -, as shown b the vertical line. Recall from Eample that = is not a function. Graphs can be used to help determine whether a relation is also a function b the following vertical line test. Vertical Line Test If no vertical line can be drawn so that it intersects a graph more than once, the graph is the graph of a function.
5 CHAPTER Graphs and Functions EXAMPLE Determine whether the following graphs are graphs of functions. a. b. c. Solution Yes, this is the graph of a function since no vertical line will intersect this graph more than once. d. e. Yes, this is the graph of a function. No, this is not the graph of a function. Note that vertical lines can be drawn that intersect the graph in two points. Solution Yes, this is the graph of a function. No, this is not the graph of a function. A vertical line can be drawn that intersects this line at ever point. Determine whether the following graphs are graphs of functions. a. b. c. d. e.
6 Section. Introduction to Functions Recall that the graph of a linear equation in two variables is a line, and a line that is not vertical will pass the vertical line test. Thus, all linear equations are functions ecept those whose graph is a vertical line. CONCEPT CHECK Determine which equations represent functions. Eplain our answer. a. = b. = c. + = 6 Finding the Domain and Range of a Function Net, we practice finding the domain and range of a relation from its graph. EXAMPLE 6 Find the domain and range of each relation. Determine whether the relation is also a function. a. (, ) (, ) (, ) b. c. d. Helpful Hint In Eample 6, Part a, notice that the graph contains the end points -, and, - whereas the graphs in Parts c and d contain arrows that indicate that the continue forever. Answer to Concept Check: a, b, c; answers ma var Solution B the vertical line test, graphs a, c, and d are graphs of functions. The domain is the set of values of and the range is the set of values of. We read these values from each graph. a. b. (, ) (, ) (, ) Domain: The -values graphed are from to, or [, ]. Range: The -values graphed are from to, or [, ]. Domain: [, ] Range: [, ]
7 CHAPTER Graphs and Functions c. d. Range: [0, ) Range: (, ) Domain: (, ) Domain: (, ) 6 Find the domain and range of each relation. Determine whether each relation is also a function. a. b. (, 9) (, 0) (0, ) c. d. Using Function Notation Man times letters such as f, g, and h are used to name functions. Function Notation To denote that is a function of, we can write = f Read f of. ()* Function Notation This notation means that is a function of or that depends on. For this reason, is called the dependent variable and the independent variable. For eample, to use function notation with the function = +, we write f = +. The notation f means to replace with and find the resulting or function value. Since f = + then f = + = 7
8 Section. Introduction to Functions Helpful Hint Make sure ou remember that f = corresponds to the ordered pair (, ). This means that when =, or f = 7. The corresponding ordered pair is (, 7). Here, the input is and the output is f or 7. Now let s find f, f0, and f -. f = + f = + f = + f = + f 0 = 0 + f - = - + = 8 + = 0 + = - + = = = - Ordered Pairs:, 0, -, - Helpful Hint Note that f() is a special smbol in mathematics used to denote a function. The smbol f() is read f of. It does not mean f # (f times ). EXAMPLE 7 If f = and g = 8 -, find the following. a. f b. g c. f - d. g 0 Solution a. Substitute for in f = and simplif. f = f = = b. g = 8 - g = 8 - = 6 c. f = f - = = d. g = 8 - g 0 = 80 - = - 7 If f = - and g = + -, find the following. a. f b. g c. f 0 d. g - CONCEPT CHECK Suppose = f and we are told that f = 9. Which is not true? a. When =, = 9. b. A possible function is f =. c. A point on the graph of the function is, 9. d. A possible function is f = +. If it helps, think of a function, f, as a machine that has been programmed with a certain correspondence or rule. An input value (a member of the domain) is then fed into the machine, the machine does the correspondence or rule, and the result is the output (a member of the range). f f() g g() Answer to Concept Check: d
9 6 CHAPTER Graphs and Functions EXAMPLE 8 Given the graphs of the functions f and g, find each function value b inspecting the graphs. or f() (, ) or g() (, f()) f g a. f b. f - c. g d. g 0 e. Find all -values such that f =. f. Find all -values such that g = 0. Solution a. To find f, find the -value when =. We see from the graph that when =, or f =. Thus, f =. b. f - = from the ordered pair -,. c. g = from the ordered pair,. d. g 0 = 0 from the ordered pair 0, 0. e. To find -values such that f =, we are looking for an ordered pairs on the graph of f whose f or -value is. The are, and -,. Thus f = and f - =. The -values are and -. f. Find ordered pairs on the graph of g whose g or -value is 0. The are, 0, 0, 0, and -, 0. Thus g = 0, g 0 = 0, and g - = 0. The -values are, 0, and -. 8 Given the graphs of the functions f and g, find each function value b inspecting the graphs. or f() or g() f g a. f b. f 0 c. g - d. g 0 e. Find all -values such that f =. f. Find all -values such that g = -. Man tpes of real-world paired data form functions. The broken-line graphs on the net page show the total and online enrollment in postsecondar institutions. EXAMPLE 9 The following graph shows the total and online enrollments in postsecondar institutions as functions of time.
10 Section. Introduction to Functions 7 Number of Students (in millions) Total and Online Enrollment in Degree-granting Postsecondar Institutions Online Enrollment Total Enrollment Semester Source: Projections of Education Statistics to 08, National Center for Education Statistics a. Approimate the total enrollment in fall 009. b. In fall 00, the total enrollment was 6.6 million students. Find the increase in total enrollment from fall 00 to fall 009. Solution a. Find the semester 009 and move upward until ou reach the top broken-line graph. From the point on the graph, move horizontall to the left until the vertical ais is reached. In fall 009, approimatel 9 million students, or 9,000,000 students, were enrolled in degree-granting postsecondar institutions. b. The increase from fall 00 to fall 009 is 9 million million =. million or,00,000 students. 9 Use the graph in Eample 9 and approimate the online enrollment in fall 009. Notice that each graph separatel in Eample 9 is the graph of a function since for each ear there is onl one total enrollment and onl one online enrollment. Also notice that each graph resembles the graph of a line. Often, businesses depend on equations that closel fit data-defined functions like this one to model the data and predict future trends. For eample, b a method called least squares, the function f = approimates the data for the red graph, and the function f = approimates the data for the blue graph. For each function, is the number of ears since 000, and f is the number of students (in millions). The graphs and the data functions are shown net. Helpful Hint Each function graphed is the graph of a function and passes the vertical line test. Number of Students (in millions) Total and Online Enrollment in Degree-granting Postsecondar Institutions f() 0. 6 Online Enrollment Total Enrollment f() Semester Source: Projections of Education Statistics to 08, National Center for Education Statistics
11 8 CHAPTER Graphs and Functions EXAMPLE 0 Use the function f = and the discussion following Eample 9 to predict the total enrollment in degree-granting postsecondar institutions for fall 00. Solution To predict the total enrollment in fall 00, remember that represents the number of ears since 000, so = = 0. Use f = and find f 0. f = f 0 = = 9. We predict that in the semester fall 00, the total enrollment was 9. million, or 9,00,000 students. 0 Use f = to approimate the online enrollment in fall 00. Graphing Calculator Eplorations It is possible to use a graphing calculator to sketch the graph of more than one equation on the same set of aes. For eample, graph the functions f = and g = + on the same set of aes. To graph on the same set of aes, press the Y = ke and enter the equations on the first two lines. Y = Y = + Then press the GRAPH ke as usual. The screen should look like this Notice that the graph of or g = + is the graph of = moved units upward. Graph each pair of functions on the same set of aes. Describe the similarities and differences in their graphs.. f = 0 0. f = g = h = -. f =. f = 0 0 H = - 6 G = f = - 6. f = F = F = +
12 Section. Introduction to Functions 9 Vocabular, Readiness & Video Check Use the choices below to fill in each blank. Some choices ma not be used. domain vertical relation.7, - range horizontal function -,.7. A is a set of ordered pairs.. The of a relation is the set of all second components of the ordered pairs.. The of a relation is the set of all first components of the ordered pairs.. A is a relation in which each first component in the ordered pairs corresponds to eactl one second component.. B the vertical line test, all linear equations are functions ecept those whose graphs are lines. 6. If f - =.7, the corresponding ordered pair is. Martin-Ga Interactive Videos See Video. Watch the section lecture video and answer the following questions. 7. Based on the lecture before Eample, wh can an equation in two variables define a relation? 8. Based on the lecture before Eample, can a relation in which a second component corresponds to more than one first component be a function? 9. Based on Eample and the lecture before, eplain wh the vertical line test works. 0. From Eample 8, do all linear equations in two variables define functions? Eplain.. From Eamples 9 and 0, what is the connection between function notation to evaluate a function at certain values and ordered pair solutions of the function?. Eercise Set Find the domain and the range of each relation. Also determine whether the relation is a function. See Eamples and.. -, 7, 0, 6, -,,, 66., 9, -, 9,,, 0, -6. -,, 6,, -, -, -7, p, 0, 0, p, -,,, , -, 0, 0,, 6 0. ea, b, a0, 7 b, 0., pf 8. 6, 6,, 6,, -, 7, 66..,,,,,,, 6 6.,,,,,,, 6 7. ea, b, a, -7b, a0, bf
13 0 CHAPTER Graphs and Functions Year of Winter Olmpics Animal Polar Bear Cow Chimpanzee Giraffe Gorilla Kangaroo Number of Gold Medals won b U.S Average Life Span (in ears) 8. 0 A B C 00 In Eercises 9 through, determine whether the relation is a function. See Eample First Set: Correspondence Second Set: Class of algebra students People who live in Cincinnati, Ohio blue, green, brown Whole numbers from 0 to Final grade average Birth date Ee color Number of children nonnegative numbers das of the ear People who live in Cincinnati, Ohio 0 women in a water aerobics class Use the vertical line test to determine whether each graph is the graph of a function. See Eample... Red Fo. Degrees Fahrenheit Degrees Celsius Words Number of Letters Cat Dog To Of Given 7 6
14 Section. Introduction to Functions Find the domain and the range of each relation. Use the vertical line test to determine whether each graph is the graph of a function. See Eample MIXED Decide whether each is a function. See Eamples through 6.. = +. = -. =. =. - = = 9 7. = 8. =. =. = = - 0. = + If f = +, g = - 6 +, and h = - 7, find the following. See Eample 7.. f(). f -. h - 6. h(0) 7. g() 8. g() 9. g(0) 60. h - Given the following functions, find the indicated values. See Eample f = ; a. f(0) b. f() c. f - 6. g = - ; a. g(0) b. g - c. g() 6. g = + ; a. g - b. g - c. g a b
15 CHAPTER Graphs and Functions 6. h = - ; a. h - b. ha - b c. ha b 6. f = -; a. f() b. f(0) c. f(606) 66. h = 7; a. h(7) b. h() c. ha - b 67. f = a. f() b. f - c. f(.) 68. g = a. g() b. g - c. g(7.) Use the graph of the functions below to answer Eercises 69 through 80. See Eample f() If f = -0, write the corresponding ordered pair. 70. If f - = -0, write the corresponding ordered pair. 7. If g = 6, write the corresponding ordered pair. 7. If g - = 8, write the corresponding ordered pair. 7. Find f Find f Find g(). 76. Find g Find all values of such that f = Find all values of such that f = Find all positive values of such that g =. 80. Find all values of such that g = 0. g() From the Chapter opener, we have two functions to describe the percent of college students taking at least one online course. For both functions, is the number of ears since 000 and (or f() or g()) is the percent of students taking at least one online course. f =.7 +. or g = Use this for Eercises See Eamples 9 and Find f(9) and describe in words what this means. 8. Find g(9) and describe in words what this means. 8. Assume the trend of g() continues. Find g(6) and describe in words what this means. 8. Assume the trend of f() continues. Find f(6) and describe in words what this means. 8. Use Eercises 8 8 and compare f(9) and g(9), then f(6) and g(6). As increases, are the function values staing about the same or not? Eplain our answer. 86. Use the Chapter opener graph and stud the graphs of f() and g(). Use these graphs to answer Eercise 8. Eplain our answer. Use this information to answer Eercises 87 and 88. The function f = , can be used to predict diamond production. For this function, is the number of ears after 000, and f is the value (in billions of dollars) of the ears diamond production. 87. Use the function to predict diamond production in Use the function to predict diamond production in Since = + 7 describes a function, rewrite the equation using function notation. 90. In our own words, eplain how to find the domain of a function given its graph. The function Ar = pr ma be used to find the area of a circle if we are given its radius. 9. Find the area of a circle whose radius is centimeters. (Do not approimate p.) 9. Find the area of a circular garden whose radius is 8 feet. (Do not approimate p.) The function V = ma be used to find the volume of a cube if we are given the length of a side. 9. Find the volume of a cube whose side is inches. 9. Find the volume of a die whose side is.7 centimeters. r
16 Section. Introduction to Functions Forensic scientists use the following functions to find the height of a woman if the are given the length of her femur bone f or her tibia bone t in centimeters. Hf =.9f Ht =.7t cm Femur cm Tibia 0. = = Is it possible to find the perimeter of the following geometric figure? If so, find the perimeter. meters 9. Find the height of a woman whose femur measures 6 centimeters. 96. Find the height of a woman whose tibia measures centimeters. The dosage in milligrams D of Ivermectin, a heartworm preventive, for a dog who weighs pounds is given b D = Find the proper dosage for a dog that weighs 0 pounds. 98. Find the proper dosage for a dog that weighs 0 pounds. 99. The per capita consumption (in pounds) of all beef in the United States is given b the function C = , where is the number of ears since 000. (Source: U.S. Department of Agriculture and Cattle Network) a. Find and interpret C(). b. Estimate the per capita consumption of beef in the United States in The amount of mone (in billions of dollars) spent b the Boeing Compan and subsidiaries on research and development annuall is represented b the function R = , where is the number of ears after 00. (Source: Boeing Corporation) a. Find and interpret R(). b. Estimate the amount of mone spent on research and development b Boeing in 0. REVIEW AND PREVIEW Complete the given table and use the table to graph the linear equation. See Section = = = = - 0 meters 08. Is it possible to find the area of the figure in Eercise 07? If so, find the area. CONCEPT EXTENSIONS For Eercises 09 through, suppose that = f and it is true that f 7 = 0. Determine whether each is true or false. See the third Concept Check in this section. 09. An ordered pair solution of the function is (7, 0). 0. When is 0, is 7.. A possible function is f = +. A possible function is f = 0-0. Given the following functions, find the indicated values.. h = + 7; a. h() b. h(a). f = - ; a. f() b. f(a). f = - ; 6. f = + 7 a. f() b. f a c. f - d. f + h a. f b. f a c. f - d. f + h 7. What is the greatest number of -intercepts that a function ma have? Eplain our answer. 8. What is the greatest number of -intercepts that a function ma have? Eplain our answer. 9. In our own words, eplain how to find the domain of a function given its graph. 0. Eplain the vertical line test and how it is used.. Describe a function whose domain is the set of people in our hometown.. Describe a function whose domain is the set of people in our algebra class
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