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Lesson 15 NYS COMMON CORE MATHEMATICS CURRICULUM M3 ALGEBRA I Lesson 15: Piecewise Functions Classwork Opening Exercise For each real number π, the absolute value of π is the distance between 0 and π on the number line and is denoted |π|. 1. Solve each one variable equation. a. 2. |π₯| = 6 b. |π₯ β 5| = 4 c. 2|π₯ + 3| = β10 Determine at least five solutions for each two-variable equation. Make sure some of the solutions include negative values for either π₯ or π¦. a. π¦ = |π₯| b. π¦ = |π₯ β 5| c. π₯ = |π¦| Exploratory Challenge 1 For parts (a)β(c) create graphs of the solution set of each two-variable equation from Opening Exercise 2. a. b. Lesson 15: Date: Piecewise Functions 5/3/17 © 2014 Common Core, Inc. Some rights reserved. commoncore.org S.94 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 15 NYS COMMON CORE MATHEMATICS CURRICULUM M3 ALGEBRA I c. d. Write a brief summary comparing and contrasting the three solution sets and their graphs. For parts (e)β(j) consider the function π(π₯) = |π₯| where π₯ can be any real number. e. Explain the meaning of the function π in your own words. f. State the domain and range of this function. g. Create a graph of the function π. You might start by listing several ordered pairs that represent the corresponding domain and range elements. Lesson 15: Date: Piecewise Functions 5/3/17 © 2014 Common Core, Inc. Some rights reserved. commoncore.org S.95 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 15 NYS COMMON CORE MATHEMATICS CURRICULUM M3 ALGEBRA I h. How does the graph of the absolute value function compare to the graph of π¦ = |π₯|? i. Define a function whose graph would be identical to the graph of π¦ = |π₯ β 5|? j. Could you define a function whose graph would be identical to the graph of π₯ = |π¦|? Explain your reasoning. k. Let π1 (π₯) = βπ₯ for π₯ < 0 and let π2 (π₯) = π₯ for β₯ 0 . Graph the functions π1 and π2 on the same Cartesian plane. How does the graph of these two functions compare to the graph in part (g)? Definition: The absolute value function π is defined by setting π(π₯) = |π₯| for all real numbers. Another way to write π is as a piecewise linear function: βπ₯ π(π₯) = { π₯ Lesson 15: Date: π₯<0 π₯β₯0 Piecewise Functions 5/3/17 © 2014 Common Core, Inc. Some rights reserved. commoncore.org S.96 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 15 NYS COMMON CORE MATHEMATICS CURRICULUM M3 ALGEBRA I Example 1 Let π(π₯) = |π₯ β 5|. The graph of π is the same as the graph of the equation π¦ = |π₯ β 5| you drew in Exploratory Challenge 1, part (b). Use the redrawn graph below to re-write the function π as a piecewise function. Label the graph of the linear function with negative slope by π1 and the graph of the linear function with positive slope by π2 as in the picture above. Function π1 : Slope of π1 is β1 (why?), and the π¦-intercept is 5; therefore, π1 (π₯) = βπ₯ + 5. Function π2 : Slope of π2 is 1 (why?), and the π¦-intercept is β5 (why?); therefore, π2 (π₯) = π₯ β 5. Writing π as a piecewise function is just a matter of collecting all of the different βpiecesβ and the intervals upon which they are defined: βπ₯ + 5 π(π₯) = { π₯β5 π₯<5 π₯β₯5 Exploratory Challenge 2 The floor of a real number π₯, denoted by βπ₯β, is the largest integer not greater than π₯. The ceiling of a real number π₯, denoted by βπ₯β, is the smallest integer not less than π₯. The sawtooth number of a positive number is the βfractional partβ of the number that is to the right of its floor on the number line. In general, for a real number π₯, the sawtooth number of π₯ is the value of the expression π₯ β βπ₯β. Each of these expressions can be thought of as functions with domain the set of real numbers. Lesson 15: Date: Piecewise Functions 5/3/17 © 2014 Common Core, Inc. Some rights reserved. commoncore.org S.97 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 15 NYS COMMON CORE MATHEMATICS CURRICULUM M3 ALGEBRA I a. Complete the following table to help you understand how these functions assign elements of the domain to elements of the range. The first and second rows have been done for you. π₯ πππππ(π₯) = βπ₯β πππππππ(π₯) = βπ₯β π ππ€π‘πππ‘β(π₯) = π₯ β βπ₯β 4.8 4 5 0.8 β1.3 β2 β1 0.7 2.2 6 β3 β 2 3 π b. Create a graph of each function. πππππ(π) = βπβ c. πππππππ(π) = βπβ ππππππππ(π) = π β βπβ For the floor, ceiling, and sawtooth functions, what would be the range values for all real numbers π₯ on the interval [0,1)? The interval (1,2]? The interval [β 2, β 1)? The interval [1.5,2.5]? Lesson 15: Date: Piecewise Functions 5/3/17 © 2014 Common Core, Inc. Some rights reserved. commoncore.org S.98 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 15 NYS COMMON CORE MATHEMATICS CURRICULUM M3 ALGEBRA I Relevant Vocabulary Piecewise-Linear Function: Given a number of non-overlapping intervals on the real number line, a (real) piecewiselinear function is a function from the union of the intervals to the set of real numbers such that the function is defined by (possibly different) linear functions on each interval. Absolute Value Function: The absolute value of a number π₯, denoted by |π₯|, is the distance between 0 and π₯ on the number line. The absolute value function is the piecewise-linear function such that for each real number π₯, the value of the function is |π₯|. We often name the absolute value function by saying, βLet π(π₯) = |π₯| for all real numbers π₯.β Floor Function: The floor of a real number π₯, denoted by βπ₯β, is the largest integer not greater than π₯. The floor function is the piecewise-linear function such that for each real number π₯, the value of the function is βπ₯β. We often name the floor function by saying, βLet π(π₯) = βπ₯β for all real numbers π₯.β Ceiling Function: The ceiling of a real number π₯, denoted by βπ₯β, is the smallest integer not less than π₯. The ceiling function is the piecewise-linear function such that for each real number π₯, the value of the function is βπ₯β. We often name the ceiling function by saying, βLet π(π₯) = βπ₯β for all real numbers π₯.β Sawtooth Function: The sawtooth function is the piecewise-linear function such that for each real number π₯, the value of the function is given by the expression π₯ β βπ₯β. The sawtooth function assigns to each positive number the part of the number (the non-integer part) that is to the right of the floor of the number on the number line. That is, if we let π(π₯) = π₯ β βπ₯β for all real numbers π₯, then 1 3 1 3 1 3 1 3 π ( ) = , π (1 ) = , π(1000.02) = 0.02, π(β0.3) = 0.7, etc. Lesson 15: Date: Piecewise Functions 5/3/17 © 2014 Common Core, Inc. Some rights reserved. commoncore.org S.99 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 15 NYS COMMON CORE MATHEMATICS CURRICULUM M3 ALGEBRA I Problem Set 1. Explain why the sawtooth function, π ππ€π‘πππ‘β(π₯) = π₯ β βπ₯β for all real numbers π₯, takes only the βfractional partβ of a number when the number is positive. 2. Let π(π₯) = βπ₯β β βπ₯β where π₯ can be any real number. In otherwords, π is the difference between the ceiling and floor functions. Express π as a piecewise function. 3. The Heaviside function is defined using the formula below. β1, π₯ < 0 π»(π₯) = { 0, π₯ = 0 1, π₯ > 0 Graph this function and state its domain and range. 4. The following piecewise function is an example of a step function. 3 π(π₯) = {1 2 5. 6. β5 β€ π₯ < β2 β2 β€ π₯ < 3 3β€π₯β€5 a. Graph this function and state the domain and range. b. Why is this type of function called a step function? Let π(π₯) = |π₯| π₯ where π₯ can be any real number except 0. a. Why is the number 0 excluded from the domain of π? b. What is the range of f? c. Create a graph of π. d. Express π as a piecewise function. e. What is the difference between this function and the Heaviside function? Graph the following piecewise functions for the specified domain. a. π(π₯) = |π₯ + 3| for β5 β€ π₯ β€ 3 b. π(π₯) = |2π₯| for β3 β€ π₯ β€ 3 c. π(π₯) = |2π₯ β 5| for 0 β€ π₯ β€ 5 d. π(π₯) = |3π₯ + 1| for β2 β€ π₯ β€ 2 e. π(π₯) = |π₯| + π₯ for β4 β€ π₯ β€ 3 π₯ if π₯ β€ 0 π(π₯) = { π₯ + 1 if π₯ > 0 2π₯ + 3 if π₯ < β1 π(π₯) = { 3 β π₯ if π₯ β₯ β1 f. g. Lesson 15: Date: Piecewise Functions 5/3/17 © 2014 Common Core, Inc. Some rights reserved. commoncore.org S.100 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 15 NYS COMMON CORE MATHEMATICS CURRICULUM M3 ALGEBRA I 7. Write a piecewise function for each graph below. a. b. y 1 1 x Graph of π c. d. Lesson 15: Date: Piecewise Functions 5/3/17 © 2014 Common Core, Inc. Some rights reserved. commoncore.org S.101 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.