Represent Relations and Functions

Transcription

1 TEKS. a., a., a.5, A..A Represent Relations and Functions Before You solved linear equations. Now You will represent relations and graph linear functions. Wh? So ou can model changes in elevation, as in E. 48. Ke Vocabular relation domain range function equation in two variables linear function A relation is a mapping, or pairing, of input values with output values. The set of input values is the domain, and the set of output values is the range. KEY CONCEPT Representing Relations A relation can be represented in the following was. For Your Notebook Ordered Pairs Table Graph Mapping Diagram (, ) Input Output (, ) (0, ) 0 (, ) 0 E XAMPLE Represent relations Consider the relation given b the ordered pairs (, ), (, ), (, ), (, ), and (, ). a. Identif the domain and range. b. Represent the relation using a graph and a mapping diagram. REVIEW GRAPHING For help with plotting points in a coordinate plane, see p a. The domain consists of all the -coordinates:,,,, and. The range consists of all the -coordinates:,,, and. b. Graph Mapping Diagram Input Output 7 Chapter Linear Equations and Functions

2 FUNCTIONS A function is a relation for which each input has eactl one output. If an input of a relation has more than one output, the relation is not a function. E XAMPLE Identif functions Tell whether the relation is a function. Eplain. AVOID ERRORS A relation can map more than one input onto the same output and still be a function. a. Input Output b. Input Output a. The relation is a function because each input is mapped onto eactl one output. b. The relation is not a function because the input is mapped onto both and. at classzone.com GUIDED PRACTICE for Eamples and. Consider the relation given b the ordered pairs (4, ), (, ), (0, ), (, ), and (, 4). a. Identif the domain and range. b. Represent the relation using a table and a mapping diagram.. Tell whether the relation is a function. Eplain VERTICAL LINE TEST You can use the graph of a relation to determine whether it is a function b appling the vertical line test. KEY CONCEPT For Your Notebook REVIEW LOGICAL STATEMENTS For help with if and onl if statements, see p. 00. Vertical Line Test A relation is a function if and onl if no vertical line intersects the graph of the relation at more than one point. Function Not a function. Represent Relations and Functions 7

3 E XAMPLE Use the vertical line test BASKETBALL The first graph below plots average points per game versus age at the end of the NBA regular season for the 8 members of the Minnesota Timberwolves with the highest averages. The second graph plots average points per game versus age for one team member, Kevin Garnett, over his first 9 seasons. Are the relations shown b the graphs functions? Eplain. READING GRAPHS The zigzag smbol on the horizontal ais of each graph indicates that values of were skipped. Average points Timberwolves Age (ears) Average points 0 The team graph does not represent a function because vertical lines at 5 8 and 5 9 each intersect the graph at more than one point. The graph for Kevin Garnett does represent a function because no vertical line intersects the graph at more than one point. 0 0 Kevin Garnett Age (ears) GUIDED PRACTICE for Eample. WHAT IF? In Eample, suppose that Kevin Garnett averages 4. points per game in his tenth season as he did in his ninth. If the relation given b the second graph is revised to include the tenth season, is the relation still a function? Eplain. EQUATIONS IN TWO VARIABLES Man functions can be described b an equation in two variables, such as 5 5. The input variable (in this case, ) is called the independent variable. The output variable (in this case, ) is called the dependent variable because its value depends on the value of the input variable. An ordered pair (, ) is a solution of an equation in two variables if substituting and in the equation produces a true statement. For eample, (, ) is a solution of 5 5 because 5 () 5 is true. The graph of an equation in two variables is the set of all points (, ) that represent solutions of the equation. KEY CONCEPT For Your Notebook Graphing Equations in Two Variables To graph an equation in two variables, follow these steps: STEP Construct a table of values. STEP Plot enough points from the table to recognize a pattern. STEP Connect the points with a line or a curve. 74 Chapter Linear Equations and Functions

4 E XAMPLE 4 Graph an equation in two variables Graph the equation 5. STEP Construct a table of values. 0 5 STEP STEP Plot the points. Notice that the all lie on a line. Connect the points with a line. READING The parentheses in f() do not indicate multiplication. The smbol f() does not mean f times. LINEAR FUNCTIONS The function 5 in Eample 4 is a linear function because it can be written in the form 5 m b where m and b are constants. The graph of a linear function is a line. B renaming as f(), ou can write 5 m b using function notation. 5 m b f() 5 m b Linear function in - notation Linear function in function notation The notation f() is read the value of f at, or simpl f of, and identifies as the independent variable. The domain consists of all values of for which f() is defined. The range consists of all values of f() where is in the domain of f. E XAMPLE 5 Classif and evaluate functions Tell whether the function is linear. Then evaluate the function when 54. a. f() 5 7 b. g() a. The function f is not linear because it has an -term. f() 5 7 Write function. f(4) 5(4) (4) 7 Substitute 4 for. 5 Simplif. REPRESENT FUNCTIONS Letters other than f, such as g or h, can also name functions. b. The function g is linear because it has the form g() 5 m b. g() Write function. g(4) 5 5(4) 8 Substitute 4 for. 5 Simplif. GUIDED PRACTICE for Eamples 4 and 5 4. Graph the equation 5. Tell whether the function is linear. Then evaluate the function when f() 5 6. g() 54. Represent Relations and Functions 75

5 DOMAINS IN REAL LIFE In Eample 5, the domain of each function is all real numbers because there is an output for ever real number. In real life, ou ma need to restrict the domain so that it is reasonable in the given situation. E XAMPLE 6 Use a function in real life DIVING A diver using a Diver Propulsion Vehicle (DPV) descends to a depth of 0 feet. The pressure P (in atmospheres) on the diver is given b P(d) 5 0.0d where d is the depth (in feet). Graph the function, and determine a reasonable domain and range. What is the pressure on the diver at a depth of feet? The graph of P(d) is shown. Because the depth varies from 0 feet to 0 feet, a reasonable domain is 0 d 0. The minimum value of P(d) is P(0) 5, and the maimum value of P(d) is P(0) So, a reasonable range is P(d) 4.9. c At a depth of feet, the pressure on the diver is P() 5 0.0() < atmospheres, which ou can verif from the graph. Pressure (atmospheres) Pressure on a Diver P(d) (, ) Depth (ft) d GUIDED PRACTICE for Eample 6 7. OCEAN EXPLORATION In 960, the deep-sea vessel Trieste descended to an estimated depth of 5,800 feet. Determine a reasonable domain and range of the function P(d) in Eample 6 for this trip.. EXERCISES SKILL PRACTICE HOMEWORK KEY 5 WORKED-OUT SOLUTIONS on p. WS for Es. 7, 7, and 45 5 TAKS PRACTICE AND REASONING Es. 9, 0, 4, 40, 46, 49, 5, and 5. VOCABULARY Cop and complete: In the equation 5 5, is the? variable and is the? variable.. WRITING Describe how to find the domain and range of a relation given b a set of ordered pairs. EXAMPLE on p. 7 for Es. 9 REPRESENTING RELATIONS Identif the domain and range of the given relation. Then represent the relation using a graph and a mapping diagram.. (, ), (, ), (, ), (4, ) 4. (5, ), (, ), (, ), (, ) 5. (6, ), (, ), (, 8), (, 5) 6. (7, 4), (, 5), (, ), (, 6) 7. (5, 0), (0, 0), (5, 0), (0, 0) 8. (4, ), (4, ), (6, 4), (6, 4) 76 Chapter Linear Equations and Functions

6 9. TAKS REASONING What is the domain of the relation given b the ordered pairs (4, ), (, ), (, 4), (, ), and (, )? A,,, and 4 B 4,,, and C 4,,, and D 4,,,,, and 4 EXAMPLE on p. 7 for Es. 0 0 IDENTIFYING FUNCTIONS Tell whether the relation is a function. Eplain. 0. Input Output. Input Output. Input Output Input Output 0 ERROR ANALYSIS Describe and correct the error in the student s work. 4. The relation given b the ordered pairs (4, ), (, 5), (, 6), and (7, ) is not a function because the inputs 4 and 7 are both mapped to the output The relation given b the table is a function because there is onl one value of for each value of. IDENTIFYING FUNCTIONS Tell whether the relation is a function. Eplain. 6. (, ), (0, ), (, 0), (, ), (, ) 7. (, 5), (, 5), (, 4), (, 0), (, 4) 8. (0, ), (, 0), (, ), (, ), (4, 4) 9. (, ), (, 5), (4, 8), (5, 9), (, 5) 0. TAKS REASONING The relation given b the ordered pairs (6, ), (, 4), (, 5), and (4, 0) is a function. Which ordered pair can be included with this relation to form a new relation that is also a function? A (, 5) B (6, ) C (, 9) D (4, 4) EXAMPLE on p. 74 for Es. VERTICAL LINE TEST Use the vertical line test to tell whether the relation is a function TAKS RESPONSE Eplain wh a relation is not a function if a vertical line intersects the graph of the relation more than once. EXAMPLE 4 on p. 75 for Es. 5 GRAPHING EQUATIONS Graph the equation }. 5 } 4. Represent Relations and Functions 77

7 EXAMPLE 5 on p. 75 for Es. 4 9 EVALUATING FUNCTIONS Tell whether the function is linear. Then evaluate the function for the given value of. 4. f() 5 5; f(8) 5. f() 5 ; f() 6. f() 5 0; f(4) 7. f() 5 6; f() 8. g() 5 5 8; g(5) 9. h() 5 7 } ; h(5) 40. TAKS RESPONSE Which, if an, of the relations described b the equations 5, 5, and 5 represent functions? Eplain. 4. CHALLENGE Let f be a function such that f(a b) 5 f(a) f(b) for all real numbers a and b. Show that f(a) 5 p f(a) and that f(0) 5 0. PROBLEM SOLVING EXAMPLE on p. 74 for Es BICYCLING The graph shows the ages of the top three finishers in the Mt. Washington Auto Road Biccle Hillclimb each ear from 00 through 004. Do the ordered pairs (age, finishing place) represent a function? Eplain. 4. BASEBALL The graph shows the number of games started and the number of wins for each starting pitcher on a baseball team during a regular season. Do the ordered pairs (starts, wins) represent a function? Eplain. Finishing place Wins Age (ears) Starts 44. GEOMETRY The volume V of a cube with edge length s is given b the function V(s) 5 s. Find V(4). Eplain what V(4) represents. 45. GEOMETRY The volume V of a sphere with radius r is given b the function V(r) 5 } 4 πr. Find V(6). Eplain what V(6) represents. EXAMPLE 6 on p. 76 for Es TAKS RESPONSE For the period , the average price p (in dollars) of a theater ticket in the United States can be modeled b the function p(t) t.89 where t is the number of ears since 974. Determine a reasonable domain and range for p(t). Eplain the meaning of the range. 47. MULTI-STEP PROBLEM Anthropologists can estimate a person s height from the length of certain bones. The height h (in inches) of an adult human female can be modeled b the function h(l) 5.95l 8.7 where l is the length (in inches) of the femur, or thigh bone. The function is valid for femur lengths between 5 inches and 4 inches, inclusive. a. Graph the function, and determine a reasonable domain and range. b. Suppose a female s femur is 5.5 inches long. About how tall was she? c. If an anthropologist estimates a female s height as 5 feet inches, about how long is her femur? 78 5 WORKED-OUT SOLUTIONS on p. WS 5 TAKS PRACTICE AND REASONING

8 48. MOUNTAIN CLIMBING A climber on Mount Rainier in Washington hikes from an elevation of 5400 feet above sea level to Camp Muir, which has an elevation of 0,00 feet. The elevation h (in feet) as the climber ascends can be modeled b h(t) 5 000t 5400 where t is the time (in hours). Graph the function, and determine a reasonable domain and range. What is the climber s elevation after hiking.5 hours? 49. EXTENDED TAKS REASONING RESPONSE The table shows the populations of several states and their electoral votes in the 004 and 008 U.S. presidential elections. The figures are based on U.S. census data for the ear 000. a. Identif the domain and range of the relation given b the ordered pairs (p, v). b. Is the relation from part (a) a function? Eplain. c. Is the relation given b the ordered pairs (v, p) a function? Eplain. 50. CHALLENGE The table shows ground shipping charges for an online retail store. a. Is the shipping cost a function of the merchandise cost? Eplain. b. Is the merchandise cost a function of the shipping cost? Eplain. State Population (millions), p Electoral votes, v California Florida Illinois.4 New York 8.98 Ohio.5 0 Pennslvania.8 Teas Merchandise cost Shipping cost $.0 $0.00 $4.50 $0.0 $60.00 $7.5 $60.0 $00.00 $9.50 Over $00.00 $.50 MIXED REVIEW FOR TAKS TAKS PRACTICE at classzone.com REVIEW Lesson.5; TAKS Workbook 5. TAKS PRACTICE Kate is studing a bacteria culture in biolog class. The table shows the number of bacteria, b, in the culture after t hours. How man bacteria are there after 0 hours? TAKS Obj. 0 Time (hours), t Bacteria (billions), b A 64 billion B 8 billion C 56 billion D 04 billion REVIEW TAKS Preparation p. 470; TAKS Workbook 5. TAKS PRACTICE What is the area of the composite figure? TAKS Obj. 8 F 8 cm G 4 cm H 6 cm J 0 cm 6 cm 6 cm 5 cm cm 7 cm 7 cm EXTRA PRACTICE for Lesson., p. 0 ONLINE. Represent QUIZ Relations classzone.com and Functions 79

9 Etension Use after Lesson. Ke Vocabular discrete function continuous function Use Discrete and Continuous TEKS A..A Functions GOAL Graph and classif discrete and continuous functions. The graph of a function ma consist of discrete, or separate and unconnected, points in a plane. The graph of a function ma also be a continuous, or unbroken, line or curve or part of a line or curve. KEY CONCEPT For Your Notebook Discrete and Continuous Functions The graph of a discrete function consists of separate points. The graph of a continuous function is unbroken. E XAMPLE Graph and classif functions Graph the function f() for the given domain. Classif the function as discrete or continuous for the domain. Then identif the range. a. Domain: 5, 0,, 4 b. Domain: a. Make a table using the -values in the domain b. Note that f() is a linear function defined for, and that f() So, the graph is the ra with endpoint (, 0.5) that passes through all the points from the table in part (a). (, 0.5) The graph consists of separate points, so the function is discrete. Its range is 0,,,. The graph is unbroken, so the function is continuous. Its range is Chapter Linear Equations and Functions

10 E XAMPLE Graph and classif real-world functions Write and graph the function described. Determine the domain and range. Then tell whether the function is discrete or continuous. a. A student group is selling chocolate bars for $ each. The function f() gives the amount of mone collected after selling chocolate bars. b. A low-flow shower head releases.8 gallons of water per minute. The function V() gives the volume of water released after minutes. a. The function is f() 5. The first four points of the graph of f() are shown. Onl whole chocolate bars can be sold, so the domain is the set of whole numbers 0,,,,.... From the graph, ou can see that the range is 0,, 4, 6,.... The graph consists of separate points, so the function is discrete. b. The function is V() 5.8. You can run the shower an nonnegative amount of time, so the domain is 0. From the graph, ou can see that the range is 0. The graph is unbroken, so the function is continuous. PRACTICE EXAMPLE on p. 80 for Es. 4 Graph the function for the given domain. Classif the function as discrete or continuous. Then identif the range of the function.. 5 ; domain:,, 0,,. f() ; domain: 4,, 0,, ; domain: < 5 4. f() 5 } 6; domain: 6 EXAMPLE on p. 8 for Es. 5 8 Write and graph the function described. Determine the domain and range. Then tell whether the function is discrete or continuous. 5. Amanda walks at an average speed of.5 miles per hour. The function d() gives the distance (in miles) Amanda walks in hours. 6. A token to ride a subwa costs $.5. The function s() gives the cost of riding the subwa times. 7. A famil has gallons of milk delivered ever Thursda. The function m() gives the total amount of milk that is delivered to the famil after weeks. 8. Steel cable that is } inch in diameter weighs 0.4 pound per foot. The 8 function w() gives the weight of feet of steel cable. 9. On a number line, the signed distance from a number a to a number b is given b b a. The function d() gives the signed distance from to an number. Etension: Use Discrete and Continuous Functions 8