Piecewise Functions. Essential Question How can you describe a function that is represented by more than one equation?

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1 COMMON CORE Learning Standards HSA-CED.A. HSA-REI.D. HSF-IF.C.7b CONSTRUCTING VIABLE ARGUMENTS.7 To be proficient in math, ou need to justif our conclusions and communicate them to others. Piecewise Functions Essential Question How can ou describe a function that is represented b more than one equation? Work with a partner. a. Does the graph represent as a function of? Justif our conclusion. b. What is the value of the function when =? How can ou tell? c. Write an equation that represents the values of the function when. f() =, if d. Write an equation that represents the values of the function when >. f() =, if > Writing Equations for a Function e. Combine the results of parts (c) and (d) to write a single description of the function. f() =, if, if > Writing Equations for a Function Work with a partner. a. Does the graph represent as a function of? Justif our conclusion. b. Describe the values of the function for the following intervals. f() =, if <, if <, if <, if < Communicate Your Answer. How can ou describe a function that is represented b more than one equation?. Use two equations to describe the function represented b the graph. Section.7 Piecewise Functions 7

2 .7 Lesson What You Will Learn Core Vocabular piecewise function, p. 8 step function, p. Previous absolute value function verte form verte Evaluate piecewise functions. Graph and write piecewise functions. Graph and write step functions. Write absolute value functions. Evaluating Piecewise Functions Core Concept Piecewise Function A piecewise function is a function defined b two or more equations. Each piece of the function applies to a different part of its domain. An eample is shown below. f() =, if +, if > The epression represents the value of f when is less than f() = +, > or equal to. The epression + represents the value of f when is greater than. f() =, Evaluating a Piecewise Function Evaluate the function f above when (a) = and (b) =. a. f() = Because, use the first equation. f() = Substitute for. f() = Simplif. The value of f is when =. b. f() = + Because >, use the second equation. f() = () + Substitute for. f() = 9 Simplif. The value of f is 9 when =. Evaluate the function. f() =, +,, if < if 5 if > 5. f( 8). f( ). f(). f() 5. f(5). f() 8 Chapter Writing Linear Functions

3 Graphing and Writing Piecewise Functions, Graph =, Graphing a Piecewise Function if <. Describe the domain and range. if Step Graph = for <. Because is not equal to, use an open circle at (, ). Step Graph = for. Because is greater than or equal to, use a closed circle at (, ). The domain is all real numbers. The range is >. =, < =, Graph the function. Describe the domain and range. 7. = +, if 8. =, if >,, if < if Writing a Piecewise Function Write a piecewise function for the graph. Each piece of the function is linear. Left Piece When <, the graph is the line given b = +. Right Piece When, the graph is the line given b =. So, a piecewise function for the graph is f() = +,, if < if. Write a piecewise function for the graph. 9.. Section.7 Piecewise Functions 9

4 STUDY TIP The graph of a step function looks like a staircase. Graphing and Writing Step Functions A step function is a piecewise function defined b a constant value over each part of its domain. The graph of a step function consists of a series of line segments. 8 f() =, if <, if <, if < 5, if < 8, if 8 < 7, if < Graphing and Writing a Step Function You rent a karaoke machine for 5 das. The rental compan charges $5 for the first da and $5 for each additional da. Write and graph a step function that represents the relationship between the number of das and the total cost (in dollars) of renting the karaoke machine. Step Use a table to organize the information. Number of das Total cost (dollars) < 5 < 75 < < 5 < 5 5 Step Write the step function. 5, if < 75, if < f() =, if < 5, if < 5, if < 5 Step Graph the step function. Karaoke Machine Rental Total cost (dollars) Number of das. A landscaper rents a wood chipper for das. The rental compan charges $ for the first da and $5 for each additional da. Write and graph a step function that represents the relationship between the number of das and the total cost (in dollars) of renting the chipper. Chapter Writing Linear Functions

5 REMEMBER The verte form of an absolute value function is g() = a h + k, where a. The verte of the graph of g is (h, k). STUDY TIP Recall that the graph of an absolute value function is smmetric about the line = h. So, it makes sense that the piecewise definition splits the function at = 5. Writing Absolute Value Functions The absolute value function f() = can be written as a piecewise function. f() =,, if < if Similarl, the verte form of an absolute value function g() = a h + k can be written as a piecewise function. a[ ( h)] + k, g() = a( h) + k, if h < if h Writing an Absolute Value Function In holograph, light from a laser beam is split into two beams, a reference beam and an object beam. Light from the object beam reflects off an object and is recombined with the reference beam to form images on film that can be used to create three-dimensional images. a. Write an absolute value function that represents the path of the reference beam. b. Write the function in part (a) as a piecewise function. a. The verte of the path of the reference beam is (5, 8). So, the function has the form g() = a Substitute the coordinates of the point (, ) into the equation and solve for a. g() = a Verte form of the function = a Substitute for and for g().. = a Solve for a. So, the function g() = represents the path of the reference beam. b. Write g() = as a piecewise function..[ ( 5)] + 8, if 5 < g() =.( 5) + 8, if 5 Simplif each epression and solve the inequalities. So, a piecewise function for g() = is g() =.,. +, if < 5 if 5. 8 (5, 8) mirror reference reference beam object beam beam splitter (, ) laser object beam mirror 8 film plate. WHAT IF? The reference beam originates at (, ) and reflects off a mirror at (5, ). a. Write an absolute value function that represents the path of the reference beam. b. Write the function in part (a) as a piecewise function. Section.7 Piecewise Functions

6 .7 Eercises Dnamic Solutions available at BigIdeasMath.com Vocabular and Core Concept Check. VOCABULARY Compare piecewise functions and step functions.. WRITING Use a graph to eplain wh ou can write the absolute value function = as a piecewise function. and Modeling with Mathematics In Eercises, evaluate the function. (See Eample.) 5, if < f() = +, if +, g() =, 5, if if < < if. f( ). f( ) 5. f(). f(5) 7. g( ) 8. g( ) 9. g(). g(). g(). g(5). MODELING WITH MATHEMATICS On a trip, the total distance (in miles) ou travel in hours is represented b the piecewise function d() = 55, 5, if if < 5. How far do ou travel in hours?. MODELING WITH MATHEMATICS The total cost (in dollars) of ordering custom shirts is represented b the piecewise function 7 +, c() = 5.8 +, +, if < 5 if 5 < 5. if 5 Determine the total cost of ordering shirts. In Eercises 5, graph the function. Describe the domain and range. (See Eample.) 5. =, if <, if. =,,, 7. = +, 8. = + 8,, 9. =,, +, +,. = +,, if if > if if > if < if if < if if > if if < < if. ERROR ANALYSIS Describe and correct the error in, if < 5 finding f(5) when f() = + 8, if 5. f(5) = (5) = 7. ERROR ANALYSIS Describe and correct the error in graphing = +, if, if >. 5 Chapter Writing Linear Functions

7 In Eercises, write a piecewise function for the graph. (See Eample.) MODELING WITH MATHEMATICS The cost to join an intramural sports league is $8 per team and includes the first five team members. For each additional team member, there is a $ fee. You plan to have nine people on our team. Write and graph a step function that represents the relationship between the number p of people on our team and the total cost of joining the league. (See Eample.). MODELING WITH MATHEMATICS The rates for a parking garage are shown. Write and graph a step function that represents the relationship between the number of hours a car is parked in the garage and the total cost of parking in the garage for da In Eercises 7, write the absolute value function as a piecewise function = + 8. = 9. =. = + 5. = +. =. = 5 8. = + In Eercises, graph the step function. Describe the domain and range.. f() =. f() =. f() =,, 5,,,, 8,, 9,, 5,, if < if < if < if < 8 if < if < if < if < 5 if < if < if < 9 if 9 < 5. = +. = MODELING WITH MATHEMATICS You are sitting on a boat on a lake. You can get a sunburn from the sunlight that hits ou directl and also from the sunlight that reflects off the water. (See Eample 5.) 5. f() =,,,, if < 5 if 5 < if < if < a. Write an absolute value function that represents the path of the sunlight that reflects off the water. b. Write the function in part (a) as a piecewise function. Section.7 Piecewise Functions

8 8. MODELING WITH MATHEMATICS You are tring to 5. HOW DO YOU SEE IT? The graph shows the total cost C of making photocopies at a cop shop. make a hole in one on the miniature golf green. Making Photocopies 5 Total cost (dollars) 5 7 C a. Write an absolute value function that represents the path of the golf ball. 5 Number of copies b. Write the function in part (a) as a piecewise function. a. Does it cost more mone to make photocopies or photocopies? Eplain. 9. REASONING The piecewise function f consists of two linear pieces. The graph of f is shown. b. You have $ to make photocopies. Can ou bu more than 5 photocopies? Eplain. 5. USING STRUCTURE The output of the greatest integer function is the greatest integer less than or equal to the input value. This function is written as f() =. Graph the function for <. Is it a piecewise function? a step function? Eplain. a. What is the value of f( )? 5. THOUGHT PROVOKING Eplain wh b. What is the value of f(8)? = 5. CRITICAL THINKING Describe how the graph of each piecewise function changes when < is replaced with and is replaced with >. Do the domain and range change? Eplain. a. f() = b. f() = +,, if < +, if +,, if if 55. MAKING AN ARGUMENT During a 9-hour snowstorm, it snows at a rate of inch per hour for the first hours, inches per hour for the net hours, and inch per hour for the final hour. a. Write and graph a piecewise function that represents the depth of the snow during the snowstorm. 5. USING STRUCTURE Graph =,, does not represent a function. How can ou redefine so that it does represent a function? if < if +, if. if > b. Your friend sas inches of snow accumulated during the storm. Is our friend correct? Eplain. Describe the domain and range. Maintaining Mathematical Proficienc Reviewing what ou learned in previous grades and lessons Write the sentence as an inequalit. Graph the inequalit. (Section.5) 5. A number r is greater than and no more than. 57. A number t is less than or equal to or no less than 8. Graph f and h. Describe the transformations from the graph of f to the graph of h. 58. f() = ; h() = + Chapter 59. f() = ; h() = 8 Writing Linear Functions (Section.). f() = ; h() = + 5