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4.7Piecewise FunctionsEssential QuestionHow can you describe a function that isrepresented by more than one equation?Writing Equations for a FunctionWork with a partner.a. Does the graph represent y as a functionof x? Justify your conclusion.CONSTRUCTINGVIABLEARGUMENTSTo be proficient in math,you need to justify yourconclusions andcommunicate themto others.64b. What is the value of the function whenx 0? How can you tell?c. Write an equation that represents the valuesof the function when x 0.f(x) 2 6 4 2246x 2, if x 0d. Write an equation that represents the valuesof the function when x 0.f(x) y 4 6, if x 0e. Combine the results of parts (c) and (d) to write a single description of the function.f(x) , if x 0, if x 0Writing Equations for a FunctionWork with a partner.a. Does the graph represent y as a functionof x? Justify your conclusion.6b. Describe the values of the function for thefollowing intervals.f(x) , if 6 x 3, if 3 x 0, if 0 x 3, if 3 x 6y42 6 4 2246x246x 2 4 6Communicate Your Answer63. How can you describe a functionthat is represented by more thanone equation?y424. Use two equations to describe 6the function represented bythe graph. 4 2 2 4 6Section 4.7hsnb alg1 pe 0407.indd 217Piecewise Functions2172/4/15 4:04 PM
4.7 LessonWhat You Will LearnEvaluate piecewise functions.Graph and write piecewise functions.Core VocabulVocabularylarryGraph and write step functions.piecewise function, p. 218step function, p. 220Previousabsolute value functionvertex formvertexWrite absolute value functions.Evaluating Piecewise FunctionsCore ConceptPiecewise FunctionA piecewise function is a function defined by two or more equations. Each“piece” of the function applies to a different part of its domain. An example isshown below.y4x 2,if x 0f(x) 2x 1, if x 02 The expression x 2 representsf(x) 2x 1, x 0the value of f when x is less thanor equal to 0.4 x 4 22{ The expression 2x 1represents the value of f whenx is greater than 0.f(x) x 2, x 0 4Evaluating a Piecewise FunctionEvaluate the function f above when (a) x 0 and (b) x 4.SOLUTIONa. f(x) x 2Because 0 0, use the first equation.f(0) 0 2Substitute 0 for x.f(0) 2Simplify.The value of f is 2 when x 0.b. f(x) 2x 1Because 4 0, use the second equation.f(4) 2(4) 1Substitute 4 for x.f(4) 9Simplify.The value of f is 9 when x 4.Monitoring ProgressHelp in English and Spanish at BigIdeasMath.comEvaluate the function.{3,if x 2f(x) x 2, if 2 x 54x,if x 5218Chapter 4hsnb alg1 pe 0407.indd 2181. f( 8)2. f( 2)3. f(0)4. f(3)5. f(5)6. f(10)Writing Linear Functions2/4/15 4:04 PM
Graphing and Writing Piecewise FunctionsGraphing a Piecewise FunctionGraph y { x 4, if x 0. Describe the domain and range.x,if x 0SOLUTIONyStep 1 Graph y x 4 for x 0. Becausex is not equal to 0, use an open circleat (0, 4).42y x, x 0Step 2 Graph y x for x 0. Because x isgreater than or equal to 0, use a closedcircle at (0, 0). 4 224 x 2The domain is all real numbers.The range is y 4.Monitoring Progressy x 4, x 0Help in English and Spanish at BigIdeasMath.comGraph the function. Describe the domain and range.x 1, if x 0x 2, if x 07. y 8. y x,if x 0if x 04x,{{Writing a Piecewise FunctionyWrite a piecewise function for the graph.4SOLUTION2Each “piece” of the function is linear. 4Left Piece When x 0, the graph is the linegiven by y x 3. 224 x 2Right Piece When x 0, the graph is the linegiven by y 2x 1. 4So, a piecewise function for the graph isf(x) {x 3,if x 0.2x 1, if x 0Monitoring ProgressHelp in English and Spanish at BigIdeasMath.comWrite a piecewise function for the graph.y9.y10.4321 4 224 x 424 x 2 2Section 4.7hsnb alg1 pe 0407.indd 219 2Piecewise Functions2192/4/15 4:04 PM
Graphing and Writing Step FunctionsSTUDY TIPThe graph of a stepfunction looks likea staircase.A step function is a piecewise function defined by a constant value over each part ofits domain. The graph of a step function consists of a series of line segments.y420if 0 x 2if 2 x 4if 4 x 6if 6 x 8if 8 x 10if 10 x 122,3,4,f(x) 5,6,7,60246812 x10Graphing and Writing a Step FunctionYou rent a karaoke machine for 5 days. The rental company charges 50 for the firstday and 25 for each additional day. Write and graph a step function that representsthe relationship between the number x of days and the total cost y (in dollars) ofrenting the karaoke machine.SOLUTIONStep 1 Use a table to organizethe information.Step 2 Write the step function.Numberof daysTotal cost(dollars)0 x 1501 x 2752 x 31003 x 41254 x 5150{50,75,f(x) 100,125,150,if 0if 1if 2if 3if 4 xxxxx 12345Step 3 Graph the step function.Total cost (dollars)Karaoke Machine Rentaly1751501251007550250012345 xNumber of daysMonitoring ProgressHelp in English and Spanish at BigIdeasMath.com11. A landscaper rents a wood chipper for 4 days. The rental company charges 100 for the first day and 50 for each additional day. Write and graph a stepfunction that represents the relationship between the number x of days and thetotal cost y (in dollars) of renting the chipper.220Chapter 4hsnb alg1 pe 0407.indd 220Writing Linear Functions2/4/15 4:04 PM
Writing Absolute Value FunctionsThe absolute value function f(x) x can be written as a piecewise function.{ x,x,f(x) REMEMBERThe vertex form of anabsolute value function isg(x) a x h k, wherea 0. The vertex of thegraph of g is (h, k).if x 0if x 0Similarly, the vertex form of an absolute value function g(x) a x h k can bewritten as a piecewise function.g(x) {a[ (x h)] k, if x h 0a(x h) k,if x h 0Writing an Absolute Value FunctionIn holography, light from a laser beam issplit into two beams, a reference beam andan object beam. Light from the object beamreflects off an object and is recombinedwith the reference beam to form imageson film that can be used to createthree-dimensional images.a. Write an absolute value function thatrepresents the path of the reference beam.y8(5, 8) r4filmplateobjectbeam2mirror(0, 0)b. Write the function in part (a) as apiecewise function.laser2468xSOLUTIONa. The vertex of the path of the reference beam is (5, 8). So, the function has theform g(x) a x 5 8. Substitute the coordinates of the point (0, 0) intothe equation and solve for a.g(x) a x 5 80 a 0 5 8 1.6 aSTUDY TIPRecall that the graph ofan absolute value functionis symmetric about theline x h. So, it makessense that the piecewisedefinition “splits” thefunction at x 5.Vertex form of the functionSubstitute 0 for x and 0 for g(x).Solve for a.So, the function g(x) 1.6 x 5 8 represents the path of thereference beam.b. Write g(x) 1.6 x 5 8 as a piecewise function.g(x) { 1.6[ (x 5)] 8, if x 5 0 1.6(x 5) 8,if x 5 0Simplify each expression and solve the inequalities.So, a piecewise function for g(x) 1.6 x 5 8 is1.6x,if x 5g(x) . 1.6x 16, if x 5{Monitoring ProgressHelp in English and Spanish at BigIdeasMath.com12. WHAT IF? The reference beam originates at (3, 0) and reflects off a mirrorat (5, 4).a. Write an absolute value function that represents the path of thereference beam.b. Write the function in part (a) as a piecewise function.Section 4.7hsnb alg1 pe 0407.indd 221Piecewise Functions2212/4/15 4:04 PM
4.7ExercisesDynamic Solutions available at BigIdeasMath.comVocabulary and Core Concept Check1. VOCABULARY Compare piecewise functions and step functions.2. WRITING Use a graph to explain why you can write the absolute value function y x asa piecewise function.Monitoring Progress and Modeling with MathematicsIn Exercises 3–12, evaluate the function. (See Example 1.)f(x) {5xx 3,1,{if x 2if x 2 x 4, if x 1g(x) 3,if 1 x 22x 5, if x 23. f( 3)4. f( 2)5. f(0)6. f(5)7. g( 4)8. g( 1)9. g(0)10. g(1)11. g(2)12. g(5)13. MODELING WITH MATHEMATICS On a trip, the totaldistance (in miles) you travel in x hours is representedby the piecewise functiond(x) {55x,if 0 x 2.65x 20, if 2 x 5How far do you travel in 4 hours?14. MODELING WITH MATHEMATICS The total cost(in dollars) of ordering x custom shirts is representedby the piecewise function{17x 20,if 0 x 25c(x) 15.80x 20, if 25 x 50.14x 20,if x 50Determine the total cost of ordering 26 shirts.In Exercises 15–20, graph the function. Describe thedomain and range. (See Example 2.) x,if x 215. y x 6, if x 2{16. y 17. y 18. y {{{{{2x,if x 3 2x, if x 3 3x 2, if x 1x 2,if x 1x 8,if x 44x 4, if x 4if x 3if 3 x 3 2x 4, if x 31,19. y x 1,2x 1,20. y x 2, 3,if x 1if 1 x 2if x 221. ERROR ANALYSIS Describe and correct the error infinding f(5) when f(x) {2x 3, if x 5.x 8, if x 5f(5) 2(5) 3 722. ERROR ANALYSIS Describe and correct the error ingraphing y {x 6, if x 2.1,if x 2 y42 5222Chapter 4hsnb alg1 pe 0407.indd 222 3 11xWriting Linear Functions2/4/15 4:04 PM
In Exercises 23–30, write a piecewise function for thegraph. (See Example 3.)23.324.yy21 1 23x12x 2 325.135. MODELING WITH MATHEMATICS The cost to join anintramural sports league is 180 per team and includesthe first five team members. For each additional teammember, there is a 30 fee. You plan to have ninepeople on your team. Write and graph a step functionthat represents the relationship between the numberp of people on your team and the total cost of joiningthe league. (See Example 4.)36. MODELING WITH MATHEMATICS The rates for a26.yy 42 26x42x 2parking garage are shown. Write and graph a stepfunction that represents the relationship between thenumber x of hours a car is parked in the garage andthe total cost of parking in the garage for 1 day. 4 627.228.yy2 2x2 24x 2In Exercises 37–46, write the absolute value function asa piecewise function. 429.1 437. y x 138. y x 3439. y x 2 40. y x 5 241. y 2 x 3 42. y 4 x 1 43. y 5 x 8 44. y 3 x 6 45. y x 3 246. y 7 x 1 530.y 2yx 2 424xIn Exercises 31–34, graph the step function. Describethe domain and range.{{{{ xxxx 2468 4,if 1 6, if 2 32. f(x) 8, if 3 10, if 4 xxxx 3,4,31. f(x) 5,6,if 0if 2if 4if 69, if 16, if 233. f(x) 5, if 41, if 9 xxxx 2, if 6 1, if 5 34. f(x) 0,if 3 1,if 2 on a boat on a lake. You can get a sunburn fromthe sunlighsunlight that hits you directly and also from thesunlight ththat reflects off the water. (See Example 5.)y234524912xxxx47. MOMODELING WITH MATHEMATICS You are sitting5311 5 3 20xa. Write an absolute value function that representsthe path of the sunlight that reflects off the water.b. Write the function in part (a) as a piecewisefunction.Section 4.7hsnb alg1 pe 0407.indd 2233Piecewise Functions2232/4/15 4:04 PM
48. MODELING WITH MATHEMATICS You are trying to52. HOW DO YOU SEE IT? The graph shows the total costC of making x photocopies at a copy shop.make a hole in one on the miniature golf green.yMaking Photocopies5Total cost (dollars)311359 x7C4035302520151050a. Write an absolute value function that representsthe path of the golf ball.0100 200 300 400 500xNumber of copiesb. Write the function in part (a) as a piecewisefunction.a. Does it cost more money to make 100 photocopiesor 101 photocopies? Explain.49. REASONING The piecewise function f consists of twolinear “pieces.” The graph of f is shown.b. You have 40 to make photocopies. Can you buymore than 500 photocopies? Explain.y4253. USING STRUCTURE The output y of the greatest24integer function is the greatest integer less than orequal to the input value x. This function is written asf(x) ⟨x⟩. Graph the function for 4 x 4. Is it apiecewise function? a step function? Explain.xa. What is the value of f( 10)?54. THOUGHT PROVOKING Explain whyb. What is the value of f(8)?y 50. CRITICAL THINKING Describe how the graph of eachpiecewise function changes when is replaced with and is replaced with . Do the domain and rangechange? Explain.a. f(x) b. f(x) {{13if x 1 x 3,if x 155. MAKING AN ARGUMENT During a 9-hour snowstorm,it snows at a rate of 1 inch per hour for the first2 hours, 2 inches per hour for the next 6 hours, and1 inch per hour for the final hour.51. USING STRUCTURE Graphy 2x 2, if x 3 3,if x 3does not represent a function. How can you redefine yso that it does represent a function?x 2,if x 2 x 1, if x 2—2 x —2 ,{a. Write and graph a piecewise function thatrepresents the depth of the snow during thesnowstorm.{ x 2, if x 2.x,if x 2b. Your friend says 12 inches of snow accumulatedduring the storm. Is your friend correct? Explain.Describe the domain and range.Maintaining Mathematical ProficiencyReviewing what you learned in previous grades and lessonsWrite the sentence as an inequality. Graph the inequality. (Section 2.5)56. A number r is greater than 12 andno more than 13.57. A number t is less than or equalto 4 or no less than 18.Graph f and h. Describe the transformations from the graph of f to the graph of h. (Section 3.6)58. f(x) x; h(x) 4x 3224Chapter 4hsnb alg1 pe 0407.indd 22459. f(x) x; h(x) x 8160. f(x) x; h(x) —2 x 5Writing Linear Functions2/4/15 4:04 PM
Section 4.7 Piecewise Functions 221 Writing Absolute Value Functions The absolute value function f(x) x can be written as a piecewise function. f(x) { x, x, if x 0 if x 0 Similarly, the vertex form of an absolute value function g(x) a x h k can be written as a piecewise function.
