0. Lesson Ke Vocabular function domain range function form Functions A function is a relationship that pairs each input with eactl one output. The domain is the set of all possible input values. The range is the set of all possible output values. Remember The ordered pair (, ) shows the output for an input. Input Output EXAMPLE Finding Domain and Range from a Graph Find the domain and range of the function represented b the graph. Write the ordered pairs. Identif the inputs and outputs. inputs (, ), (, 0), (, ), (, ) outputs The domain is,,, and. The range is, 0,, and. Eercises Find the domain and range of the function represented b the graph... An equation is in function form if it is solved for. + = = + not in function form in function form Chapter 0 Functions
EXAMPLE Finding the Range of a Function Input, + 8 Output, ( ) + 8 0 (0) + 8 8 () + 8 () + 8 0 () + 8 The domain of the function represented b + = 8 is, 0,,, and. What is the range of the function represented b the table? Write the function in function form. + = 8 = + 8 Use this form to make an input-output table. The range is, 8,, 0, and. EXAMPLE Real-Life Application The table shows the percent (in decimal form) of the moon that was visible at midnight das after Januar, 0. (a) Interpret the domain and range. (b) What percent of the moon was visible on Januar, 0? a. Zero das after Januar is Januar. One da after Januar is Januar. So, the domain of 0. 0,,,, and represents Januar,,, 7, and 8. The range is 0.7, 0., 0., 0., and 0.. These amounts are decreasing, so the moon was less visible each da. b. Januar, 0 corresponds to the input =. When =, = 0.. So, 0., or % of the moon was visible on Januar, 0. 0 0.7 0. 0. 0. Eercises 9 Cop and complete the input-output table for the function. Then find the domain and range of the function represented b the table.. =. + = 0 0. The table shows the percent (in decimal form) of the moon that was visible at midnight das after December 7, 0. (a) Interpret the domain and range. 0 0. 0. 0. 0. 0. (b) What percent of the moon was visible on December, 0? Section 0. Domain and Range of a Function
0. Eercises. VOCABULARY Is the equation = in function form? Eplain.. DIFFERENT WORDS, SAME QUESTION Which is different? Find both answers. Find the range of the function represented b the table. Find the inputs of the function represented b the table. Find the -values of the function represented b (, 7), (, ), and (, ). Find the domain of the function represented b (, 7), (, ), and (, ). 7 9+(-)= +(-)= +(-9)= 9+(-)=. The number of earrings and headbands ou can bu with $ is represented b the equation 8 + =. The table shows the number of earrings and headbands. a. Write the equation in function form. b. Find the domain and range. c. Wh is = not in the domain of the function? Earrings, 0 Headbands, 0 Find the domain and range of the function represented b the graph.... The domain is, 0,, and. The range is,,,. 7. ERROR ANALYSIS Describe and correct the error in finding the domain and range of the function represented b the graph. 8. REASONING Find the domain and range of the function represented b the table. Tickets, 8 Cost, $ $ $ $ Chapter 0 Functions
Cop and complete the input-output table for the function. Then find the domain and range of the function represented b the table. 9. = + 0. =. =. + 0 0 8 0. VAULTING In the sport of vaulting, a vaulter performs a routine while on a moving horse. For each round of competition, the vaulter receives a score from to 0. a. Find the domain and range of the function represented b the table. b. Interpret the domain and range. c. What is the mean score of the vaulter?.8 7.9 8.. MANATEE Florida s state marine mammal is the manatee. A manatee eats about % of its bod weight each da. a. Write an equation in function form that represents the amount (in pounds) of food a manatee eats each da for its weight. b. Create an input-output table for the equation in part (a). Use the inputs 0, 00, 0, 00, 70, and 900. c. Find the domain and range of the function represented b the table. d. An aquatic center has manatees that weigh 00 pounds, 70 pounds, and 00 pounds. How man pounds of food do all three manatees eat in a da? in a week?. Describe the domain and range of the function. a. = b. = c. = d. = + Graph the linear equation.. = + 8. + = 7. = 8. = 7 9. MULTIPLE CHOICE The minimum number of people needed for a group rate at an amusement park is 8. Which inequalit represents the number of people needed to get the group rate? A 8 B > 8 C < 8 D 8 Section 0. Domain and Range of a Function
0. Lesson Ke Vocabular discrete domain continuous domain Discrete and Continuous Domains A discrete domain is a set of input values that consists of onl certain numbers in an interval. Eample: Integers from to 0 A continuous domain is a set of input values that consists of all numbers in an interval. Eample: All numbers from to. 0 Total cost (dollars) EXAMPLE Museum Tickets 70 (,.8) 0 0 (, 7.8) 0 0 (,.9) 0 (,.9) 0 (0, 0) 0 0 Number of tickets Graphing Discrete Data The function =.9 represents the cost (in dollars) of tickets for the South Florida Museum. Graph the function using a domain of 0,,,, and. Is the domain of the graph discrete or continuous? Eplain. Make an input-output table. Input,.9 Output, Ordered Pair, (, ) 0.9(0) 0 (0, 0).9().9 (,.9).9().9 (,.9).9() 7.8 (, 7.8).9().8 (,.8) Plot the ordered pairs. Because ou cannot bu part of a ticket, the graph consists of individual points. So, the domain is discrete.. The function m = 0 9d represents the amount of mone m (in dollars) ou have after buing d DVDs. Graph the function. Is the domain discrete or continuous? Eplain. Chapter 0 Functions
Calories consumed EXAMPLE Cereal Bar Calories c 700 00 00 00 00 00 00 0 0 (, 0) (, 90) (, 0) (, 0) (0, 0) Number of bars eaten b Graphing Continuous Data A cereal bar contains 0 calories. The number c of calories consumed is a function of the number b of bars eaten. Graph the function. Is the domain of the graph discrete or continuous? Make an input-output table. Input, b Output, c Ordered Pair, ( b, c ) 0 0 (0, 0) 0 (, 0) 0 (, 0) 90 (, 90) 0 (, 0) Plot the ordered pairs. Because ou can eat part of a cereal bar, b can be an value greater than or equal to 0. Draw a line through the points. So, the domain is continuous. EXAMPLE Standardized Test Practice Input Time, t (seconds) Output Distance, d (miles) 0. 0.88.0 8.7 0.70 You conduct an eperiment on the speed of sound waves in dr air at 8 F. You record our data in a table. Which of the following is true? A The domain is t and t 0 and it is discrete. C The domain is d 0. and d.7 and it is discrete. B The domain is t and t 0 and it is continuous. D The domain is d 0. and d.7 and it is continuous. The domain is the set of possible input values, or the time t. The time t can be an value from to 0. So, the domain is continuous. The correct answer is B. Eercises 8. A 0-gallon bathtub is draining at a rate of. gallons per minute. The number g of gallons remaining is a function of the number m of minutes. Graph the function. Is the domain discrete or continuous?. Are the data shown in the table discrete or continuous? Eplain. Number of Stories Height of Building (feet) 8 0 Section 0. Discrete and Continuous Domains 7
0. Eercises. VOCABULARY Describe the difference between a discrete domain and a continuous domain.. WRITING Describe how ou can use a graph to determine whether a domain is discrete or continuous. 9+(-)= +(-)= +(-9)= 9+(-)= Describe the domain and range of the function. Is the domain discrete or continuous?. 7. 7 7 7 Graph the function. Is the domain of the graph discrete or continuous?. Input Bags, Output Marbles,. Input Years, Output Height of a Tree, (feet) 0 0 0 0 9 7. Input Width, (inches) Output Volume, (cubic inches) 0 0 00 0 8. Input Hats, Output Cost, (dollars) 0 0 8..9 9. ERROR ANALYSIS Describe and correct the error in classifing the domain. 0. YARN The function m = 0 8.b represents the amount m of mone (in dollars) that ou have after buing b balls of arn. Graph the function using a domain of 0,,, and. Is the domain discrete or continuous? The domain is continuous. 0 0 8 Chapter 0 Functions
. REASONING The input of one function is length. The input of another function is number of shirts. Which function has a continuous domain? Eplain.. DISTANCE The function =.8 converts length from meters to feet. Graph the function. Is the domain discrete or continuous?. AREA The area A of the triangle is a function of the height h. Graph the function. Is the domain discrete or continuous? h 8 in.. PACKING You are packing books into boes. The function = 0 represents the number of books that will fit into boes. a. Is in the domain? Eplain. b. Is 0 in the range? Eplain.. You want to fill a -foot shelf with framed pictures. There are pictures in -inch frames and pictures in 8-inch frames. a. Write a function for this situation. b. Graph the function. c. Is the domain discrete or continuous? in. 8 in. Find the slope of the line.. 7. 8. 9. MULTIPLE CHOICE What is the -intercept of the graph of the linear equation? A B C D 7 7 Section 0. Discrete and Continuous Domains 9
0 Stud Help You can use a comparison chart to compare two topics. Here is an eample of a comparison chart for domain and range. Definition Algebra Eample: = m + b Domain the set of all possible input values -values Range the set of all possible output values corresponding -values Ordered pairs Eample: (, 0), (, ), (, ), (, ) Table Eample: 0 0 Graph Eample: 9,,,, 0,,,,, 0,,, 0,,, 9,, Make a comparison chart to help ou stud and compare these topics.. discrete data and continuous data After ou complete this chapter, make comparison charts for the following topics.. linear functions with positive slopes and linear functions with negative slopes. linear functions and nonlinear functions Creating a comparison chart causes canines to crstalize concepts. 0 Chapter 0 Functions
0. 0. Quiz Find the domain and range of the function represented b the graph.... Cop and complete the input-output table for the function. Then find the domain and range of the function represented b the table.. =. + = 0 0 Graph the function. Is the domain of the graph discrete or continuous?. Rulers, Cost, 0 0.. 7. Gallons, Miles Remaining, 0 00 0 9 8. Minutes, 0 0 0 0 Height, 0 0 9. Rela Teams, 8 Athletes, 8 0. VIDEO GAME The function m = 0 r represents ents the amount m (in dollars) of mone ou have after renting r video games. Graph the function using a domain of 0,,,, and. Is the domain of the graph discrete or continuous?. WATER Water accounts for about 0% of a person s bod weight. a. Write an equation in function form that represents the water weight of a person that weighs pounds. b. Make an input-output table for the function in part (a). Use the inputs 00, 0, 0, and 0. Sections 0. 0. Quiz
0. Lesson Ke Vocabular linear function A linear function is a function whose graph is a line. EXAMPLE Finding a Linear Function Using a Graph Use the graph to write a linear function that relates to. The points lie on a line. Find the slope and -intercept of the line. rise slope = run = Because the line crosses the -ais at (0, ), the -intercept is. So, the linear function is =. (, ) (, ) (, 0) (0, ) EXAMPLE Finding a Linear Function Using a Table (, 9) 9 (, 7) 8 7 (, ) (0, ) Use the table to write a linear function that relates to. 0 9 7 Plot the points in the table. The points lie on a line. Find the slope and -intercept of the line. rise slope = run = = Because the line crosses the -ais at (0, ), the -intercept is. So, the linear function is = +. Eercises 0 Use the graph or table to write a linear function that relates to... 0 Chapter 0 Functions
EXAMPLE Real-Life Application Hours Kaaking, Calories Burned, 00 00 800 8 00 Graph the data in the table. (a) Is the domain discrete or continuous? (b) Write a linear function that relates to. (c) How man calories do ou burn in. hours? a. Plot the points. Time can represent an value greater than or equal to 0, so the domain is continuous. Draw a line through the points. b. The slope is 00 = 00 and the -intercept is 0. Calories burned Kaaking 00 00 800 (8, 00) (, 800) 00 00 (, 00) 900 00 00 (, 00) 0 0 7 8 Hours So, the linear function is = 00. c. Find the value of when =.. = 00 Write the equation. = 00(.) Substitute. for. = 0 Multipl. You burn 0 calories in. hours of kaaking. Hours Rock Climbing, Calories Burned, 90 900 9 80 7800. Graph the data in the table. a. Is the domain discrete or continuous? b. Write a linear function that relates to. c. How man calories do ou burn in. hours? Representing a Function Words An output is more than the input. Equation = + Input-Output Table Input, 0 Output, Graph Section 0. Linear Function Patterns
0. Eercises. VOCABULARY Describe four was to represent a function.. VOCABULARY Is the function represented b the graph a linear function? Eplain. 7 0 0 7 9+(-)= +(-)= +(-9)= 9+(-)= The table shows a familiar linear pattern from geometr. Write a linear function that relates to. What do the variables and represent? Graph the linear function... π π π π π 8 0 Use the graph or table to write a linear function that relates to.. 0 8 8 0. 8 8 8 8 7. 8. 0 9. 8 0 0. 0 0 0 7 9. MOVIES The table shows the cost (in dollars) of renting movies. a. Graph the data. Is the domain of the graph discrete or continuous? b. Write a linear function that relates to. c. How much does it cost to rent three movies? Number of Movies, 0 Cost, 0 Chapter 0 Functions
. BIKE JUMPS A bunn hop is a bike trick in which the rider brings both tires off the ground without using a ramp. The table shows the height (in inches) of a bunn hop on a bike that weighs pounds. Weight, 9 Height, 0. 9.8 9. a. Graph the data. Then describe the pattern. b. Write a linear function that relates the height of a bunn hop to the weight of the bike. c. What is the height of a bunn hop on a bike that weighs. pounds? Years of Education, Annual Salar, 0 8 0 0 88. SALARY The table shows a person s annual salar (in thousands of dollars) after ears of education beond high school. a. Graph the data. b. Write a linear function that relates the person s annual salar to the number of ears of education beond high school. c. What is the annual salar of the person after 8 ears of education beond high school?. The Heat Inde is calculated using the relative humidit and the temperature. For ever degree increase in the temperature from 9 F to 98 F at 7% relative humidit, the Heat Inde rises F. a. On a summer da in Pensacola, the relative humidit is 7%, the temperature is 9 F, and the Heat Inde is F. Construct a table that relates the temperature t to the Heat Inde H. Start the table at 9 F and end it at 98 F. b. Write a linear function that represents this situation. c. Estimate the Heat Inde when the temperature is 00 F. Find the annual simple interest rate.. I = $0, P = $00, t = ears. I = $, P = $000, t = 8 months 7. MULTIPLE CHOICE You bu a pair of gardening gloves for $. and packets of seeds for $0.88 each. Which equation represents the total cost? A = 0.88. B = 0.88 +. C =. 0.88 D =. + 0.88 Section 0. Linear Function Patterns
0. Lesson Ke Vocabular nonlinear function The graph of a linear function shows a constant rate of change. A nonlinear function does not have a constant rate of change. So, its graph is not a line. EXAMPLE Identifing Functions from Tables Does the table represent a linear or nonlinear function? Eplain. a. + + + b. + + + 9 0 8 8 8 As increases b, decreases b 8. The rate of change is constant. So, the function is linear. 7 88 +9 + + As increases b, increases b different amounts. The rate of change is not constant. So, the function is nonlinear. EXAMPLE Identifing Functions from Graphs Does the graph represent a linear or nonlinear function? Eplain. a. b. The graph is not a line. So, the function is nonlinear. The graph is a line. So, the function is linear. Eercises Does the table or graph represent a linear or nonlinear function? Eplain.... 0 7 0 0 8 0 8 Chapter 0 Functions
EXAMPLE Standardized Test Practice Which equation represents a nonlinear function? A =.7 B = π C = D = ( ) The equations =.7, = π, and = ( ) can be rewritten in slope-intercept form. So, the are linear functions. The equation = cannot be rewritten in slope-intercept form. So, it is a nonlinear function. The correct answer is C. EXAMPLE Real-Life Application Account A earns simple interest. Account B earns compound interest. The table shows the balances for ears. Graph the data and compare the graphs. Remember The simple interest formula is given b I = Prt. I is the simple interest P is the principal r is the annual interest rate t is the time in ears Year, t Account A Balance Account B Balance 0 $00 $00 $0 $0 $0 $ $0 $.0 $0 $. $0 $.0 Balance (dollars) 70 0 0 0 0 0 0 00 Savings Account Account B Account A 0 0 7 t Year The balance of Account A has a constant rate of change of $0. So, the function representing the balance of Account A is linear. The balance of Account B increases b different amounts each ear. Because the rate of change is not constant, the function representing the balance of Account B is nonlinear. Eercises Does the equation represent a linear or nonlinear function? Eplain.. = +. =. = Section 0. Comparing Linear and Nonlinear Functions 7
0. Eercises. VOCABULARY Describe the difference between a linear function and a nonlinear function.. WHICH ONE DOESN T BELONG? Which equation does not belong with the other three? Eplain our reasoning. = = 0 = = 9+(-)= +(-)= +(-9)= 9+(-)= Graph the data in the table. Decide whether the function is linear or nonlinear.. 0. 8. 0. 0 7 Does the table or graph represent a linear or nonlinear function? Eplain. 7. 8. 9. 7 7 9 0. 9 9. VOLUME The table shows the volume V (in cubic feet) of a cube with a side length of feet. Does the table represent a linear or nonlinear function? Eplain. Side Length, 7 8 Volume, V 8 7 8 Chapter 0 Functions
Does the equation represent a linear or nonlinear function? Eplain. 8. + = 7. + = +. = Pounds, Cost,.80?.0. SUNFLOWER SEEDS The table shows the cost (in dollars) of pounds of sunflower seeds. a. What is the missing -value that makes the table represent a linear function? b. Write a linear function that represents the cost of pounds of seeds.. LIGHT The frequenc (in terahertz) of a light wave is a function of its wavelength (in nanometers). Does the table represent a linear or nonlinear function? Eplain. Color Red Yellow Green Blue Violet Wavelength, 0 9 0 00 Frequenc, 0 79 7. LIGHTHOUSES The table shows the heights (in feet) of four Florida lighthouses and the number of steps in each. Does the table represent a linear or nonlinear function? Eplain. Lighthouse Height, Steps, Ponce de Leon Inlet 7 St. Augustine 7 9 Cape Canaveral 79 Ke West 8 98 8. PROJECT The wooden bars of a lophone produce different musical notes when struck. The pitch of a note is determined b the length of the bar. Use the Internet or some other reference to decide whether the pitch of a note is a linear function of the length of the bar. 9. The radius of the base of a clinder is feet. Is the volume of the clinder a linear or nonlinear function of the height of the clinder? Classif the angle as acute, obtuse, right, or straight. 0..... MULTIPLE CHOICE What is the value of? A 0 B 0 C 90 D 80 0 Section 0. Comparing Linear and Nonlinear Functions 9
0. 0. Quiz Use the graph or table to write a linear function that relates to..... 0 0. 0 Does the table or graph represent a linear or nonlinear function? Eplain.. 7. 8. 9. 0.. 0 0 0 7 0 7 7 9. ADVERTISING The table shows the revenue R (in millions of dollars) of a compan when it spends A (in millions of dollars) on advertising. a. Write a linear function that relates the revenue to the advertising cost. b. What is the revenue of the compan when it spends $0 million on advertising? Advertising, A Revenue, R 0 0 8 8. CHICKEN SALAD The equation = 7.9 represents the cost (in dollars) of buing pounds of chicken salad. Does this equation represent a linear or nonlinear function? Eplain. 0 Chapter 0 Functions
0 Chapter Test. Find the domain and range of the. Cop and complete the input-output function represented b the graph. table for the function = 7. Then find the domain and range of the function represented b the table. 0 Graph the function. Is the domain of the graph discrete or continuous?. Hair Clips, Cost,. Minutes, Gallons, 0 0 0 0. 0 0.. Write a linear function that. Does the table represent a linear relates to. or nonlinear function? Eplain. 0 8 0 8 7. SAVINGS You save % of our monthl earnings (in dollars). a. Write an equation in function form that represents the amount (in dollars) ou save each month. b. Create an input-output table for the equation in part (a). Use the inputs, 0,, and 0. c. What is the total amount saved during those months? 8. FOOD DRIVE You are putting cans of food into boes for a food drive. One bo holds 0 cans of food. Write a linear function that represents the number of cans of food that will fit in boes. 9. SURFACE AREA The table shows the surface area S (in square inches) of a cube with a side length of feet. Does the table represent a linear or nonlinear function? Eplain. Side Length, Surface Area, S 9 Chapter Test