Functions as Models Learner's Module in General Mathematics Quarter 1 ● Module 1 ● Week 1 JOPIE B. FERRER Developer Department of Education • Cordillera Administrative Region NAME:________________________ GRADE AND SECTION ________________ TEACHER: ____________________ SCORE _____________________________ Republic of the Philippines DEPARTMENT OF EDUCATION Cordillera Administrative Region SCHOOLS DIVISION OF BAGUIO CITY No. 82 Military Cut-off, Baguio City Published by: DepEd Schools Division of Baguio City Curriculum Implementation Division Learning Resource Management and Development System COPYRIGHT NOTICE 2020 Section 9 of Presidential Decree No. 49 provides: “No copyright shall subsist in any work of the Government of the Philippines. However, prior approval of the government agency of office wherein the work is created shall be necessary for exploitation of such work for profit.” This material has been developed for the implementation of K-12 Curriculum through the DepEd Schools Division of Baguio City – Curriculum Implementation Division (CID). It can be reproduced for educational purposes and the source must be acknowledged. Derivatives of the work including creating an edited version, an enhancement or a supplementary work are permitted provided all original work is acknowledged and the copyright is attributed. No work may be derived from this material for commercial purposes and profit. ii What I Need to Know Hello learner! This module was designed and written with you in mind. Primarily, its scope is to represent real-life situations using functions, including piecewise functions and to evaluate a function. While going through this module, you are expected to: 1. 2. 3. 4. 5. define functions and relations; distinguish between functions and relations; identify the different ways of determining a function; describe real-life situations using piecewise functions by choice; and determine the value of something in a thoughtful way Now, here is an outline of the different parts of your learning module. The descriptions will guide you on what to expect on each part of the module. Icon Label Description What I need to know This states the learning objectives that you need to achieve as you study this module. What I know This is to check what you already know about the lesson on this module. If you answered all the questions here correctly, then you may skip studying this module. What’s In This connects the current lesson with a topic or concept necessary to your understanding. What’s New This introduces the lesson to be tackled through an activity. What’s In it This contains a brief discussion of the learning module lesson. Think of it as the lecture section of the lesson. What’s More These are activities to check your understanding and to apply what you have learned from the lesson. What I have Learned This generalizes the essential ideas tackled from this module. What I Can Do This is a real-life application of what you have learned. Post-Assessment This is an evaluation of what you have learned from this learning material. Additional Activity This is an activity that will strengthen and fortify your knowledge about the lesson. 2 What I Know If you answer all the test items correctly in this pretest, then you may skip studying this learning material and proceed to the next learning module. I. TRUE OR FALSE. Write the word TRUE if the given statement is correct and FALSE if otherwise. (5 points) __________ 1. A function from to is a set of ordered pair such that to each , there corresponds a unique . __________ 2. All functions are relations. __________ 3. Evaluating a function means to find the value of the output f (x) . __________ 4. Horizontal line test is used to determine if a graph represents a function. __________ 5. A relation is a function defined by two or more formulas on different parts of its domain. II. IDENTIFICATION. Determine whether each of the following items is a function or simply a relation. Write FUNCTION if can be a function and RELATION if only a relation on the space provided. (5 points) 2 1. _____________________________ 4. _____________________________ 2. _____________________________ A 1 B 2 C 3 5. _____________________________ 3. _____________________________ 3 III. MULTIPLE CHOICE. Choose the best answer. Write the letter of your choice in CAPITAL LETTERS on the space provided for. (5 points) _____ 1. Which of the following is a function? A. B. C. D. _____ 2. Which set considers all possible values of ? A. Domain B. Range C. Function D. Relation _____ 3. Evaluate A. 8 B. 1 D. 6 _____ 4. Given A. B. at C. 25 C. , what is the range? D. _____ 5. Which of the following graphs is an example of a piecewise function? A. C. B. D. 4 What’s In There are many situations wherein a variable depends on another variable. For example, the distance travelled by a car depends on its rate, the savings of the government depends on its expenditures, the area of a circle depends on the length of its diameter, and so on. These relationships are mathematically described by functions. Many real-life situations and problems can be represented and solved by mathematical models such as functions. The concept of sets is also considered in this topic. Remember that “set” in mathematical concept refers to the collection of things or objects with a common characteristic. An example of set is the group chat in the messenger that you belong to. In this group chat, there are members with a common goal or purpose. What’s New Activity : I Got This! Write two (2) things or objects that belong to the given group. Write your answer in column B. A B Even numbers Indigenous groups in the Philippines Subjects when you were in Grade 10 or in the ALS Program Primary colors Provinces in Cordillera Administrative Region What’s In It FUNCTIONS AND RELATIONS Definition of Terms Relation – a relation from t is a set of ordered pairs that to each , there corresponds at least one . 5 such Function – a function from to is a set of ordered pairs that to each , there corresponds a unique or exactly one such . Note: The symbol is read as “element”. This also refers to the member of the given set. In addition to, the small letter is the element or member of set written in capital letter. In naming of a set, capital letters are used. This goes to and from the definition of relation and function. The first component in the ordered pair is and the second component is . Domain – is the set of first coordinates in the ordered pairs. It is the set of all possible values of . In this module, “D” is used to represent the domain of the given set. Range – is the set of second coordinates in the ordered pairs. It is the set of all possible values of . In this module, “R” is used to represent the range of the given set. Three Things Necessary to Form a Relation A. Non – empty set X this set also refers to the domain of the given relation B. Non – empty set Y this set is not necessarily the range of the given relation. There are instances that not all in the second set are mapped from the first set. In this case, the range is just a subset of the second set. C. The pairing or correspondence or mapping of the two sets this pertains to the relationship existing between of and Ways to Describe a Relation A. Listing of ordered pairs Example: D. Graph Example: B. Arrow diagram Example: X Y 1 a 2 b 3 c E. Equation Examples: C. Table Example: NOTE: All functions are relations but not all relations are functions. 6 Ways of Determining Whether a Relation is a Function or Not A. Listing of ordered pairs If there are two or more ordered pairs with the same first component, then the relation is not a function. Examples: 1. The first components of set P are 1, 2, 3, and 16. Thus, there is no same first component in the ordered pairs. This is a function. The domain is and the range is . Since 1 is repeated as second component, just write it once in the set of range. 2. The first components of set T are 1, 1, 2, and 3. Thus, there is a repetition of 1 in the ordered pairs. This is not a function. It is just merely a relation. The domain is and the range is . Since 1 is repeated as first component, just write it once in the set of domain. B. Arrow diagram If the first set is mapped or paired to two or more in the second set, then it is not a function. Examples: X Y X 1. Y 2. 1 a 1 a 2 b 2 b 3 c 3 c Every element in set X is paired to at least one and exactly one element in set Y. Therefore, it is both a relation and function. The domain is and the range is . A Every element in set X is paired to at least one and exactly one element in the second set. Therefore, it is both a relation and function. The domain is and the range is . B 3. A B 4. 1 a 1 a 2 b 2 b 3 c 3 c 7 Every element in set A is paired to at least one element in set B. Therefore, it is only a relation. The domain is and the range is . C. Table If the two or more function. Not all in set A is paired to at least one or exactly one element in set B. Therefore, it is not a relation nor a function. (Always refer to the definition). -values are the same, then it is not a Examples: 1. 2. There is no repetition of values, therefore, it is both a relation and function. The domain is and the range is R . There is a repetition of values, therefore, it is just simply a relation. The domain is and the range is R D. Equation If the exponent of the dependent variable is an odd integer, then it is a function. If the first component is not constant, then it is a function. Examples: The exponent of y is 1. It is both relation and function. The exponent of y is 2. It is just a relation. The first component x is constant. It is just a relation. The first component x is constant. It is just a relation. E. Graph Vertical line test is used to test if the graph is a function or not. If it intersects the graph at more than one point, then it is not a function. (In the given examples, the broken lines are the vertical lines for testing). 8 Examples: Every vertical line in the graph, there is only one point of intersection in the graph. This concludes that both graphs above are functions. The vertical line intersects the graph at two points. This concludes that both graphs above are just simply relations. PIECEWISE FUNCTIONS Sometimes a function is defined by different formulas on different parts of its domain. This function is called a piecewise function. It is also known as piecewisedefined function. In this discussion, is being replaced with or can be denoted by other symbols or letters Well-known Examples of Piecewise Functions 1. Absolute Value Function | | is defined as { Examples: 𝑦 𝑥 𝑦 𝑥 9 | | | | | | | | | | | | | | | | 2. Greatest Integer Function ⌊ ⌋ is defined as ⌊ ⌋ the largest integer that is less than or equal to . It is also known as Floor Function. The deleted point (○) denotes discontinuity (not included) in the graph while the solid point (●) signifies continuity (included) in the graph. Examples: The number line will help you on this. Locate the number given in the number line and look at the integer on its left. That is the answer. If the number given is already an integer, then that is the answer. ⌊ ⌊ ⌋ ⌋ ⌊ ⌋ ⌊ ⌋ ⌊ ⌋ ⌊ ⌋ ⌊ ⌋ 3. Unit Step Function This function is also known as Heaviside Step Function named after Oliver Heaviside (1850–1925). The function is defined as: { Examples: 1. 2. 3. 4. If If If If , then , then , then , then Note: This will be discussed further on the next topic on Evaluation of Functions. What’s More Activity 1: Find Me! How large is the value of the output in the function if we input ? The amount of space that a spherical container can hold, if the radius of the 10 container can hold, if the radius of the container is 12 cm, can be calculated by using the formula . What is the volume of the container? In this activity, we shall learn how to evaluate functions. Evaluate the following algebraic expressions, given that . Show your solution. and Example: Copy the expression. Substitute the values for each variable. Multiply the first term. Add the two terms. 1. 2. 3. 4. Activity 2: Do Not Guesstimate! To evaluate a function means to find the value of the output input . Study the given examples. Example 1: If , evaluate at . Copy the given function. Replace with . Apply PEMDAS rule. Multiply and . Combine the terms. Example 2: Given , find Copy the given function. Replace Divide. 11 with . , given the Example 3: Given , find Copy the function. Replace x with x+1. Apply the FOIL method. Combine similar terms. Now, it’s your turn to evaluate! Always remember that to evaluate is just to substitute the value of in the given function. Algebraic process will just follow. Show your solution if necessary. 1. Given h . √ 2. What is | , evaluate at 3. If in |? , evaluate What I Have Learned Activity: Now, Decide! Determine whether each of the following items is a function or simply a relation. Write F for function and R for relation on the blank provided for. _____ 1. _____ 2. _____ 3. _____ 4. _____ 5. _____ 6. _____7 . . (Grade) (Subject) 92 92 85 Math Science English 12 87 90 Filipino EsP 89 93 93 TLE MAPEH AP . X Y Tuba Lagawe Benguet _____ 8. Mt. Province Ifugao Baguio City Sagada _____ 9. _____ 10. What I Can Do Activity: Into Pieces The process of evaluating piecewise functions is similar to evaluating ordinary functions, except for the fact that we need to choose the piece or chunk of the function to be used for the given value of . We can do this by examining the appropriate interval where the input or the value of belongs. The process of hiring a catering service to serve food for a party is ₱150 per head for 20 persons or less, ₱130 per head for 21 to 50 persons, and ₱110 per head for 51 to 100 persons. For 100 or more persons, the cost is at ₱100 per head. The piecewise defined function of the number of attendees of the party is given by: { Example: How much is to be paid if there are 10 attendees in the party? Since and can be found in the interval we use . Copy the expression from the function and equate to Replace x with 10. Multiply. . Now, it’s your turn to show what you can do. What is the cost of the party if there are 25 attendees? 110 attendees? 1. 2. 13 Assessment I. TRUE OR FALSE. Write the word TRUE if the given statement is correct and FALSE if otherwise. (5 points) __________ 1. A function from to is a set of ordered pair such that to each , there corresponds a unique . __________ 2. All functions are relations. __________ 3. Evaluating a function means to find the value of the output f (x) . __________ 4. Horizontal line test is used to determine if a graph represents a function. __________ 5. A relation is a function defined by two or more formulas on different parts of its domain. II. IDENTIFICATION. Determine whether each of the following items is a function or simply a relation. Write FUNCTION if can be a function and RELATION if only a relation on the space provided. 2 1. _____________________________ 4. _____________________________ 2. _____________________________ A 1 B 2 C 3 5. _____________________________ 3. _____________________________ 14 III. MULTIPLE CHOICE. Choose the best answer. Write the letter of your choice in CAPITAL LETTERS on the space provided. (5 points) _____ 1. Which of the following is a function? A. B. C. D. _____ 2. Which set considers all possible values of ? A. Domain B. Range C. Function D. Relation _____ 3. Evaluate A. 8 B. 1 D. 6 _____ 4. Given A. B. at C. 25 , what is the range? C. D. _____ 5. Which of the following graphs is an example of a piecewise function? A. C. B. D. 15 Additional Activity Activity: Tax Rates in the Philippines Based on the existing tax structure, you are to define a piecewise function that models this situation. You may give the function through (a) equation, (b) table showing some values, or (c) graph. Tax Rate Table: Source: https://www.pinterest.ph/pin/722898177661051772/ 16 I. II. WHAT I HAVE LEARNED 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. F R F R R F R R F R WHAT I CAN DO 1. 2. ₱ 3, 250 ₱ 11, 000 I. II. III. ASSESSMENT 1. TRUE 2. TRUE 3. TRUE 4. FALSE 5. FALSE 1. FUNCTION 2. FUNCTION 3. FUNCTION 4. RELATION 5. FUNCTION 1. B 2. A 3. C 4. D 5. B WHAT’S NEW Answers may vary WHAT I KNOW III. 17 1. TRUE 2. TRUE 3. TRUE 4. FALSE 5. FALSE 1. FUNCTION 2. FUNCTION 3. FUNCTION 4. RELATION 5. FUNCTION 1. B 2. A 3. C 4. D 5. B Examples: 1. 0, 2, 4, 6, 8, 10 2. Isneg, Kankanaey, Ibaloi, Kalanguya, Ifugao, Bontoc, Isneg, Tinggian, Iyaplay Math, Science, Filipino, English Red, Blue, Yellow Benguet, Kalinga, Abra, Mt. Province, Ifugao, Apayao, 3. 4. 5. ADDITIONAL ACTIVITY Answers may vary WHAT’S MORE Activity 1: 1. 27 2. 24 3. 16 4. -116 Activity 2: 1. ±3 2. 25 2 3. 4x + 3x + 2 ANSWER KEY REFERENCES Verzosa, Debbie Marie B., et.al. Teaching Guide for Senior High School – General Mathematics. Quezon City: Commission on Higher Education, 2016. Belecina, Rene R., et.al. General Mathematics. Quezon City: Brilliant Creations Publishing, Inc., 2016. “Train Law (2020) Income Tax Tables in the Philippines – Pinoy Money Talk.” Pinterest. https://www.pinterest.ph/pin/722898177661051772/ 18 For inquiries or feedback, please write or call: Department of Education – Schools Division of Baguio City No. 82 Military Cut-off Road, Baguio City Telefax: 422-4326 / 422-7819 Email Address: depedbaguiocity@gmail.com Social Media: facebook.com/DepEdTayoBaguioCity