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A lesson about piecewise function with activity and answer key.

NICO MONTEROSeguir

- 1 FUNCTIONS Lesson 1.1 Functions in Real-life In mathematics, a function is a relationship between two sets of elements in which no element in the first set relates to more than one element in the second. We can think of a function as a rule that takes inputs from the first set and relates them to an element in the second set, which is the output (Function Application for the Real World, 2017).For example, if you take tests, there is a function that tells you if you get this score on your test ,you will get this grade. If you go grocery shopping, there is a function that tells you if you buy this many bottles of water ,you need to pay this much money. If you pay for cable TV, there is a function that tells you if you want this many channels, you need to pay this much money (Functions: Identification, Notation & Practice Problems,2013). Guide Questions: 1. In the example: If you pay for cable TV, there is a function that tells you if you want this many channels, you need to pay this much money. What is the input variable?the output variable? _____________________________________________________________________ 2. How did you identify the input variable from the output variable? _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ In addition, functions are used as representation of real-life situations that shows relationship between the sets of values or quantities such as test score and grade, and number of bottles of water and amount paid. The test score does changes in the grade shows relationship, that is grade (output/dependent variable) is dependent to test score (input/independent variable). We can use function notation for functions, such as f(x) = y, where: x is the input/independent variable, y is the output/dependent variable f is the name of the function (we can also use letters other than f). f(x) is the value of the function at x.
- 2 In this lesson, you are expected to construct mathematical models to represent real-life situations using functions. Example 1: Give a function P that can represent the cost of buying x large size t-shirt if each t-shirt costs P160. Solution: Step 1: Identify the input and output in the situation and assign corresponding variable for each. Remember that, output always depends on input. ➢ The input is number of large size t-shirt (x) and the output is the cost of t-shirts (C). Step 2: Write the function notation to be used to represent the situation. ➢ Since the cost of buying (C) depends on the number of large size t-shirt (x), then the function notation is C(x). Step 3: Construct a mathematical expression based on the mathematical phrase present in the situation. This will be the subfunctions. ➢ Each t-shirt costs P160 is translated as mathematical expression as: 160x Step 4: Combine the function notation and mathematical expression using (=) sign. The result will be the linear function that represent the cost (C) of buying large size t-shirts (x). ➢ C(x) = 160x Note: Remember, functions can also be written in different ways using other variables such as C(x),h(x), and g(x). Similarly, functions may take other input values other than x such as P(a), h (r), and g(m). PIECEWISE FUNCTIONS. Some situations can only be described by more than one formula, depending on the value of the independent variable. Definition: A piecewise function is defined by multiple subfunctions, where each subfunction applies to a certain interval of the main function’s domain. Example 2: A videoke machine can be rented for P1,000 for three days, but for the fourth day onwards, an additional cost of P400 per day is added. Represent the cost of renting a videoke machine as a piecewise function of the number of days it is rented.
- 3 Solution: Step 1: Identify the input and output in the situation and assign corresponding variable for each. Remember that, output always depends on input. ➢ The input is the number of days (x) rented and the output is the cost of renting (r). Step 2: Write the function notation to be used to represent the situation. ➢ Since the cost of renting (r) depends on number of days (x) rented, then the function notation is r(x). Step 3: Construct a mathematical expression based on the mathematical phrase present in the situation. This will be the subfunctions. ➢ rented for P1,000 for three days is translated as mathematical expression as:1000,𝑖𝑓 0 ≤ 𝑥 ≤ 3 ➢ an additional cost of P400 per day is added to 1000 is translated as mathematical expression as: 1000 + 400(𝑥 − 3) , 𝑖𝑓 𝑥 > 3 Step 4: Combine the function notation and subfunctions using equal (=) sign. The result will be the piecewise function that represent the cost of renting (r) in terms of number of days (x). ➢ 𝑟(𝑥) = { 1000 , 𝑖𝑓 0 ≤ 𝑥 ≤ 3 1000 + 400(𝑥 − 3) , 𝑖𝑓 𝑥 > 3 Function Notation Subfunctions Intervals r(x) 1000 1000 + 400(x – 3) 0 ≤ 𝑥 ≤ 3 𝑥 > 3 Notes: • 0 ≤ 𝑥 ≤ 3 and 𝑥 > 3 are intervals of the piecewise function based on the given information in the problem. • fourth day onwards means four days and up • additional cost means increased in cost (addition). Example 3: The cost of hiring a catering service to serve food for a party is P150 per head for 20 persons or less, P130 per head for 21 to 50 persons, and P110 per head for 51 to 100 persons. For more than 100 persons, the cost is at P100 per head. Represent the total cost as a piecewise function of the number of attendees of the party. Solution: Step 1: Identify the input and output in the situation and assign corresponding variable for each. Remember that, output always depends on input.
- 4 ➢ The input is the number of attendees/persons (x) and the output is the cost of hiring a catering service (f). Step 2: Write the function notation to be used to represent the situation. ➢ Since the cost of hiring a catering services (f) depends on the number of attendees/persons (x), then the function notation is f(x). Step 3: Construct a mathematical expression based on the mathematical phrase present in the situation. This will be the subfunctions. ➢ P150 per head for 20 persons or less is translated as mathematical expression as:150𝑥 , 𝑖𝑓 0 ≤ 𝑥 ≤ 20 ➢ P130 per head for 21 to 50 persons is translated as mathematical expression as: 130𝑥 , 𝑖𝑓 21 ≤ 𝑥 ≤ 50 ➢ P110 per head for 51 to 100 persons is translated as mathematical expression as: 110𝑥 , 𝑖𝑓 51 ≤ 𝑥 ≤ 100 ➢ For 100 or more persons, the cost is at P100 per head is translated as mathematical expression as: 100𝑥 , 𝑖𝑓 x > 100 Step 4: Combine the function notation and subfunctions using equal (=) sign. The result will be the piecewise function that represent the cost of renting (r) in terms of number of days (x). 150x if 0 ≤ 𝑥 ≤ 20 , x ∈ ℕ 130x if 21 ≤ 𝑥 ≤ 50, x ∈ ℕ 110x if 51 ≤ 𝑥 ≤ 100, x ∈ ℕ 100x if x > 100, x ∈ ℕ Note: • 20 persons or less means 20 persons and below (less than) • 100 or more persons means 100 persons and above (greater than) • 51 to 100 persons means less than or equal to 100 but greater than or equal to 51. • More than 100 means greater than 100. • x ∈ ℕ, 𝑟𝑒𝑎𝑑𝑠 𝑎𝑠 𝑥 𝑖𝑠 𝑎𝑛 𝑒𝑙𝑒𝑚𝑒𝑛𝑡 𝑜𝑓 𝑛𝑎𝑡𝑢𝑟𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟𝑠. X should be an element of natural number because the number of attendees should be in whole number. There will be never 1.5 persons/attendees. f(x) = Tips: The number of input values or independent variable gives the number of subfunctions of the piecewise function.
- 5 I. Activity A. Model it with Functions After knowing more on functions, let us now apply these in the real-life situations. You are task to model the situation with functions, by constructing mathematical model to represents the real-life scenario. Put your answer on a sheet of paper. Scenario # 1: A computer shop operates for a cause at this time of pandemic and charges 25 pesos for the first two hours, but for the succeeding hours, an additional charge of 15 pesos per hour is added. 1-2. The number of hours (h) is the _________and the charges in using the computer (c) is _________. 1. The function notation to be used to represent the situation is ________. 2. Using the information given in the scenario, the subfunctions with corresponding intervals are _________ and _________. 3. The piecewise function that represents the charges in using the computer in terms of the number of hours is _________. Scenario # 2: You plan to bake and sell pandesal varieties each day to raise funds for buying face masks for the frontliners facing the pandemic. You sell it for P8.00 each for the first 80 pandesals brought by a customer. After the first 80 pandesals up to 160 pandesals sold by the same customer you lower the price to P5.50 each. After 160 pandesals sold by the same customer the price will decrease to P3.00 each. 6-7. The input used in the situation is _________ and the output is _________. 8. The function notation to be used to represent the situation is ________. 9. Using the information given in the scenario, the subfunctions with corresponding intervals are _________ , _________ , and __________. 10. The piecewise function that describes the of pandesal is ________
- 6 Key to Corrections A. Model it with Functions 1. input 6. number of pandesal (n) 2. output 7. Sales (s) or price (p) 3. h(c) 8. s(n), may vary according to 4. 25 if 0 ≤ c ≤ 2 variables used 25 + 15(c – 2) if c > 2 9. 8n if 0 ≤ n ≤ 80 5. ℎ(𝑐) = { 25 0 ≤ 𝑐 ≤ 2 25 + 15(𝑐 − 2) 𝑐 > 2 5.50n if 81≤ n ≤ 160 3n if n > 160 10. s(n)= { 8n if 0 ≤ n ≤ 80 5.50n if 81 ≤ n ≤ 160 3n if n > 160 References Smith, Jessica. 2019. “How Are Linear Equations Used in Everyday Life?” Sciencing, N.p. Last modified August 7, 2020. https://sciencing.com/linear-equations-used-everyday-life- 6022370.html. Abramson, Carmichael, and David Lippman.n.d. “College Algebra.” Lumen Learning, N.p. Last accessed August 10,2020. https://courses.lumenlearning.com/waymakercollegealgebra/chapter/ evaluate-and-solve-functions/. N.p. n.d. “Function Notation (Examples, Solutions, Videos).” Onlinemathlearning. Last accessed August 11,2020. https://www.onlinemathlearning.com/function-notation.html. "Function Application for the Real World." Study.com. September 15, 2017. https://study.com/academy/lesson/function-application-for-the-real- world.html. "Functions: Identification, Notation & Practice Problems." Study.com. January 15, 2013. https://study.com/academy/lesson/how-to-use- function-notation.html. Barcelona, Belecina, and John Lleonard C. Villanueva. 2016. Senior High Conceptual Math and Beyond: General Mathematics, 1st Edition. Edited by Rene R. Belecina.Quezon City,Philippines. Brilliant Creations Publishing,Inc.
- 7 Crisologo, Ocampo, Miro, Tresvalles, Hao, and Emellie G. Palomo. 2016.General Mathematics: Teacher’s Guide, 1st Edition. Edited by Christian Paul O. Chan Shio and Mark L. Loyola. Pasig City, Philippines. Department of Education. Oronce, Orlando A. 2016. General Mathematics,1st Edition.Manila, Philippines. Rex Book Store, Inc. (RBSI).