1_Q1-General-Mathematics(1).pdf

SENIOR HIGH SCHOOL General Mathematics Quarter 1 – Module 1 (Week 1) Real Life Functions i About the Module This module was designed and written with you, students, in mind. It is here to help you master the nature of functions. The scope of this module permits...

SENIOR HIGH SCHOOL General Mathematics Quarter 1 – Module 1 (Week 1) Real Life Functions i About the Module This module was designed and written with you, students, in mind. It is here to help you master the nature of functions. The scope of this module permits it to be used in many different learning situations. The language used recognizes the diverse vocabulary level of students. The lessons are arranged based on the Most Essential Learning Competencies (MELCs) released by the Department of Education (DepEd) for this school year 2020 – 2021. This module is divided into four parts, namely: Lesson 1 – Real Life Functions Lesson 2 – Evaluation of Functions Lesson 3 – Operations of Functions o Lesson 3.1 – Addition of Functions o Lesson 3.2 – Subtraction of Functions o Lesson 3.3 – Multiplication of Functions o Lesson 3.4 – Division of Functions o Lesson 3.5 – Composition of Functions Lesson 4 – Problem Solving involving Functions After going through this module, you are expected to: 1. represent real-life situations using functions, including piece-wise functions. 2. evaluate a function. 3. perform addition, subtraction, multiplication, division and composition of functions. solve problems involving functions ii  What I Know (Pre-test) Test I: Choose the letter of the correct answer and write them on a separate sheet of paper. 1. What is a function? A. Set of inputs and outputs C. Every input has only ONE output B. Set of x-values and y-values D. Inputs have different outputs every time 2. Which of the following set of values is a function? A. {(3,4),(4,-3),(7,4),(3, 8)} C. {(6,-5),(7,-3),(8,-1),(9,1)} B. {(2,-2),(5,9),(5,-7),(1,4)} D. {(9,5),(10,5),(9,-5),(10,-5)} 3. According to the table, does the relationship represent a function? Time (Minutes) 0 40 87 160 240 Temperature (oC) 15 20 25 35 49 A. Yes, the relationship is a function C. No, the relationship is not a because every minute has a unique function because the increase in temperature. temperature is not the same. B. Yes, the relationship is a function D. No, the relationship is not a because the temperature is increasing function because the increase in the as time passes. minutes is higher than the temperature. 4. Students were asked to state the different ways of determining if a relation is a function. Which student is NOT correct? A. Maria said, “In a table, if the x C. Sam said, “In an ordered pair, if the values have more than one x values have more than one corresponding y value, it is not a corresponding y value, it is not a function.” function.” B. Jennifer said, “In a graph you can D. Matthew said, “In a mapping use the vertical line test and if the line diagram you can use the vertical line intercepts only once, it is a function.” test and if the line intercepts only once, it is a function” 5. Which of the following relations is NOT a function? A. {(3,3),(4,5),(5,5),(5,4)} C. {(1,1),(2,2),(3,3),(4,4)} B. {(2,3),(4,5),(5,6),(6,7)} D. {(-1,2),(0,2),(1,2),(2,3)} 6. Evaluate f(x) = x2 for f(-8). A. -64 B. -16 C. 16 D. 64 7. Evaluate f(x) = –2x2 + 1 for f(-3). A. -11 B. -17 C. 19 D. 37 1 8. Using the graph at the right, evaluate the value of f(1). A. -3 C. 2 B. 0 D. 3 9. Using the diagram at the right, evaluate the value of f(2). A. -3 C. 4 B. 0 D. 7 10. Refer to the graph at the right, evaluate the value of f(0). A. -2 C. 2 B. 0 D. 6 Test II. Match Column A to the answers in column B given the following. 𝑓 (𝑥 ) = 3𝑥 + 2 𝑔(𝑥 ) = 4 − 5𝑥 Column A Column B _____11. (𝑓 + 𝑔)(𝑥) A. 2𝑥 − 8 _____12. (𝑓 − 𝑔)(𝑥) B. 8𝑥 − 2 _____13. (𝑓 ∙ 𝑔)(𝑥) C. −2𝑥 + 6 𝑓 3𝑥+2 _____14. (𝑔) (𝑥) D. −5𝑥+4 3𝑥+2 E. 5𝑥−4 _____15. (𝑓 ⃘ 𝑔)(𝑥) F. −15𝑥 + 14 G. −15𝑥 2 + 2𝑥 + 8 2  Lesson Real Life Functions 1 What I Need to Know At the end of this lesson, you are expected to: o determine functions and relations; o illustrate functions through mapping diagrams, sets and graphs; and o represent real-life situations using functions. What’s In Hi I’m teacher Mathew. I’m a Math enthusiast. I will help towe know and Let’s begin? remember involving sets. start first withsoathat you will review understand on some ” Before we will start with the lesson proper, there are terms that you need and assist you to understand this module better. Shall functions definitions ☑Sets – collection of well-defined and distinct objects, called elements that share a common characteristic (Grade 7 Mathematics LM, 2013). ☑Ordered Pair – is a pair of objects taken in a specific order. An ordered pair is written by listing two members in a specific order, separating them by a comma and enclosing the pair in parentheses (Aggarwal, 2014). ☑Relation – is a rule that relates values from a set of values (called the domain) to a second set of values (called the range). A relation is also a set of ordered pairs (x,y) (General Mathematics LM, 2016). ☑Domain – domain is the set of all x or input values (General Mathematics LM, 2016). ☑Range – range is the set of all y or output values (General Mathematics LM, 2016). What’s New What makes RELATION a FUNCTION? A function is a set of ordered pairs for which every x-value must be associated to only one y-value (Estela, n.d.). 3 Here are some examples of functions through SET (ROSTER NOTATION), MAPPING AND GRAPHING IN COORDINATE SYSTEM. Mapping - A mapping shows how the elements are paired. It is like a flow chart for a function, showing the input and output values. Any of the x-values (domain) should not have more than one pair to consider them function (Varsity Tutors, n.d.). Domain Range Function Function Not Function SETS – functions can appear in sets too. In this example, we will only discuss roster notation or the list of elements, separated by commas and enclosed in curly braces. It is important to note that the x-values should not occur more than once. Example: {(2,3),(4,5),(5,6),(6,7)} 2 4 5 6 Note that no x-value is repeated. Hence, the set is a function. {(3,3),(4,5),(5,5),(5,4)} 3 4 5 5 Note that x-value 5 is repeated. Hence, the set is NOT a function. GRAPHING – functions can be determined in graphing also. We can use a special kind of test which is called the Vertical Line Test (VLT). The method is to create imaginary vertical lines across the graph, if the line hits exactly one point on the graph, then it is a function. Figure 1.Vertical Line Test 4 Functions in Real Life 1. Circumference of a Circle: A circle’s circumference (C) is a function of its diameter (d) or C(d). 2. Shadow: The length (L) of person’s shadow along the floor is a function of their height (h) or L(h). 3. Driving a Car: When driving a car, your location (D) is a function of time (t) or D(t). 4. Temperature: Based on a variety of inputs, or factors, (F) we get a certain temperature (t) or F(t). 5. Money: The amount (A) of money you have is a function of the time (t) spent earning it or A(t). What’s More NOW IT’S YOUR TURN! Directions: Choose the letter of the correct answer. 1. Which of the following set of values is a function? A. {(2,3),(4,-3),(6,4),(8, 8)} C. {(6,-5),(7,-3),(8,-1),(8,1)} B. {(2,-2),(5,9),(5,-7),(1,4)} D. {(9,5),(10,5),(11,-5),(11,-5) 2. Which of the following relations is NOT a function? A. {(1,2),(3,4),(5,6),(7,8)} C. {(1,1),(2,2),(3,3),(4,4)} B. {(2,3),(3,4),(4,5),(5,6)} D. {(-1,2),(0,2),(-1,2),(2,3) 3. According to the table, does the relationship represent a function? TIME (Minutes) Temperature (oC) 0 15 40 20 87 25 160 35 240 49 A. Yes, the relationship is a function C. No, the relationship is not a because every minute has a unique function because the increase in temperature. temperature is not the same. B. Yes, the relationship is a function D. No, the relationship is not a because the temperature is function because the increase in the increasing as time passes. minutes is higher than the temperature. 5 4. Refer to the image below, is the relation a function? Why? A. Yes, because the x-value 11 has C. No, because the x-value 11 has two y-values pair with it. two y-values pair with it. B. Yes, because each x-value has D. No, because each x-value has only one y-value paired with it. only one y-value paired with it. 5. Which of the graphs in the image below is a function? A. Graph 1 C. Graph 3 B. Graph 2 D. Graph 4 What I Need to Remember A relation is a function when every x-value is associated to only one y-value. You can illustrate functions through graphing, mapping or sets. Functions can be seen in our daily lives like driving a car, length of shadow, temperature, money and many more. Lesson Evaluation of Functions 2 What I Need to Know At the end of this lesson, you are expected to: substitute values in a function, and evaluate a function. 6  What’s In What is a function? A function relates an input to an output. It is like a machine that has an input and an output where the input is somehow related to the output. How to write a function? “f(x) = …” is the classic way of writing a function. You can also replace the value of the variable and the value of the function like using g(x) or h(x). An example of a function is this, f(x) = 2x + 1 (Pierce, n.d.). What’s New Evaluation of Functions To evaluate a function is to replace/substitute its variables with a given number of expressions. Example 1: Example 3: Evaluate the function f(x) = 3x + 4 for x = 2 Evaluate g(x) = 1 – x + x2 for x = 3 Just replace the variable “x” with “2”: Again, just replace the variable “x” with “3” f(2) = 3(2) + 4 g(3) = 1 – (3) + (3)2 =6+4 =1–3+9 = 10 = -2 + 9 = 7 Example 2: Evaluate the function h(x) = x2 + 2 for x = -3 Replace the variable “x” with “-3” h(-3) = (-3)2 + 2 =9+2 = 11 What’s More NOW IT’S YOUR TURN! Directions: Evaluate the following functions. 1. f(x) = 2 + x – x2 Find f (2), f(-1) 4. f (x) = x2 – 3x Find f (−8), f(0) 2. g(a) = 3a – 2 Find g (1), g(-2) 5. h(n) = −2n2 + 4 Find h (4), h(-2) 3. p(a) = −43a Find p (1), p(0) 7  What I Need to Remember The classic way of writing a function is “f(x) = …” To evaluate a function is to replace/substitute its variables with a given number of expressions. Lesson Operations of Functions 3.1 Addition What I Need to Know At the end of this lesson, you are expected to: follow the steps in adding functions. add functions. What’s In ” “Yippee! You made it to our third lesson of this module. Before we will start with theI lesson am Teacher proper, Mathew there are and termsI that will you guide need to you until the end of the lesson. Let’s recall know and remember first so that you will understand functions how to solve add integers before proceeding to the addition of functions. Addition of Integers Rule 1: The sum of two positive Rule 2: The sum of two negative integers is a positive integer. integers is a negative integer. Adding Positive Integers Adding Negative Integers Integers Sum Integers Sum + 2 + +9 = + 11 -2 + -9 = -11 + 17 + +5 = + 22 -5 + -8 = -13 Rule 3: If the signs are different, subtract the numbers and use the sign of the larger number. Adding Unlike Signs Integers Integers Sum -2 + 9 = +7 5 + -8 = -3 8  What’s New How do we add functions? The sum of functions can be written as 𝑓(𝑥) + 𝑔(𝑥) or (𝑓 + 𝑔)(𝑥). Find the sum of the following functions. Example 1: 𝒇(𝒙) = 𝟑𝒙 + 𝟐 𝒈(𝒙) = 𝟒 − 𝟓𝒙 Write the functions into (𝒇 + 𝒈)(𝒙) = 𝒇(𝒙) + 𝒈(𝒙) Solution: (𝑓 + 𝑔)(𝑥) = (𝟑𝒙 + 𝟐) + (𝟒 − 𝟓𝒙) Note: Put the relationship inside parentheses to distinguish them (𝑓 + 𝑔)(𝑥) = (3𝑥 + 2) + (4 − 5𝑥) → remove ( ) using distributive property (𝑓 + 𝑔)(𝑥) = 3𝑥 + 2 + 4 − 5𝑥 → combine like terms (𝑓 + 𝑔)(𝑥) = −2𝑥 + 6 → find the sum So, the answer for (𝑓 + 𝑔)(𝑥) is −𝟐𝒙 + 𝟔 Example 2: 𝒇(𝒙) = 𝟒𝒙 + 𝟐𝒙𝟐 − 𝟐 𝒈(𝒙) = 𝟒 − 𝟑𝒙 Write the functions into (𝒇 + 𝒈)(𝒙) = 𝒇(𝒙) + 𝒈(𝒙) Solution: (𝑓 + 𝑔)(𝑥) = (4𝑥 + 2𝑥 2 − 2) + (4 − 3𝑥) (𝑓 + 𝑔)(𝑥) = 4𝑥 + 2𝑥 2 − 2 + 4 − 3𝑥 → combine like terms (𝑓 + 𝑔)(𝑥) = 2𝑥 2 + 𝑥 + 2 → find the sum Therefore, (𝑓 + 𝑔)(𝑥) is 𝟐𝒙𝟐 + 𝒙 + 𝟐 What’s More NOW IT’S YOUR TURN! Directions: Find (𝑓 + 𝑔)(𝑥) using the functions given. 1. 𝑓(𝑥) = 3𝑥 + 3 2. 𝑓(𝑥) = 2𝑥 + 5 3. 𝑓(𝑥) = −15𝑥 3 − 2𝑥 + 5 𝑔(𝑥) = −4𝑥 + 1 𝑔(𝑥) = 4𝑥 2 + 2𝑥 − 2 𝑔(𝑥) = 3𝑥 2 + 𝑥 − 7 What I Need to Remember The sum of functions can be written as 𝑓(𝑥) + 𝑔(𝑥) or (𝑓 + 𝑔)(𝑥). 9  Lesson Operations of Functions 3.2 Subtraction What I Need to Know At the end of this lesson, you are expected to: follow the steps in subtracting functions. subtract functions. What’s In REVIEW Subtraction of Integers Rule: In subtracting integers, change the sign of subtrahend and proceed to addition. Example 1: Example 2: Integers Steps Integers Steps 5 – (+17) + Given -2 – (9) Given Change the sign of Change the sign of 5 + (-17) subtrahend and -2 + (-9) subtrahend and proceed to proceed to addition addition -12 Adding unlike signs -11 Adding Like Signs Example 3: Example 4: Integers Steps Integers Steps -2 – (-9) Given 2 – (-9) Given Change the sign of Change the sign of -2 + (+9) subtrahend and 2 + (+9) subtrahend and proceed to proceed to addition addition 7 Adding Unlike Signs 11 Adding Like Signs What’s New How do we subtract functions? The difference of functions can be written as 𝒇(𝒙) − 𝒈(𝒙) or (𝒇 − 𝒈)(𝒙). 10 Find the difference of the following functions. Example 1 𝒇(𝒙) = 𝟑𝒙 + 𝟐 𝒈(𝒙) = 𝟒 − 𝟓𝒙 Solution: Write the functions into (𝒇 − 𝒈)(𝒙) = 𝒇(𝒙) − 𝒈(𝒙) (𝑓 − 𝑔)(𝑥) = (3𝑥 + 2) − (4 − 5𝑥) → remove parentheses using distributive property (𝑓 − 𝑔)(𝑥) = 3𝑥 + 2 − 4 + 5𝑥 → combine like terms (𝑓 − 𝑔)(𝑥) = 8𝑥 − 2 So, (𝑓 − 𝑔)(𝑥) is 𝟖𝒙 − 𝟐 Example 2 𝒃(𝒙) = 𝟒𝒙 + 𝟐𝒙𝟐 − 𝟐 𝒉(𝒙) = 𝟒 − 𝟑𝒙 Solution: Write the functions into (𝒃 − 𝒉)(𝒙) = 𝒃(𝒙) − 𝒉(𝒙) (𝑏 − ℎ)(𝑥) = (4𝑥 + 2𝑥 2 − 2) − (4 − 3𝑥) → remove parentheses using distributive property (𝑏 − ℎ)(𝑥) = 4𝑥 + 2𝑥 2 − 2 − 4 + 3𝑥 → combine like terms (𝑏 − ℎ)(𝑥) = 2𝑥 2 + 7𝑥 − 6 So, (𝑏 − ℎ)(𝑥) is 𝟐𝒙𝟐 + 𝟕𝒙 − 𝟔 What’s More NOW IT’S YOUR TURN! Directions: Find (𝑓 − 𝑔)(𝑥) using the given functions. 1. 𝑓(𝑥) = 3𝑥 + 3 2. 𝑓(𝑥) = 2𝑥 + 5 3. 𝑓(𝑥) = −15𝑥 3 − 2𝑥 + 5 𝑔(𝑥) = −4𝑥 + 1 𝑔(𝑥) = 4𝑥 2 + 2𝑥 − 2 𝑔(𝑥) = 3𝑥 2 + 𝑥 − 7 What I Need to Remember o The difference of functions can be written as 𝒇(𝒙) − 𝒈(𝒙) or (𝒇 − 𝒈)(𝒙). Lesson Operations of Functions 3.3 Multiplication What I Need to Know At the end of this lesson, you are expected to: follow the steps in multiplying functions. multiply functions. 11  What’s In REVIEW Multiplication of Integers Rule: The product of two like signs integers is always positive and the product of two unlike signs is always negative. Multiplying Like Signs Multiplying Unlike Signs Integers Product Integers Product +5 (+17) = 85 -2 (9) = -18 -5 (-8) = 40 5 (-8) = -40 What’s New How do we multiply functions? The product of functions can be written as 𝒇(𝒙) ∙ 𝒈(𝒙) or (𝒇 ∙ 𝒈)(𝒙). Find the product of the following functions. Example 1 𝑓(𝒙) = 𝟑𝒙 𝒈(𝒙) = 𝟒 − 𝟓𝒙 Step 1: Write the functions into (𝒇 ∙ 𝒈)(𝒙) = 𝒇(𝒙) ∙ 𝒈(𝒙) Solution: (𝑓 ∙ 𝑔)(𝑥) = (3𝑥)(4 − 5𝑥) → remove parentheses using distributive property (𝑓 ∙ 𝑔)(𝑥) = (3𝑥)(4) − (3𝑥)(5𝑥) → perform the operations (𝑓 ∙ 𝑔)(𝑥) = 12𝑥 − 15𝑥 2 → if there are like terms, combine them and arrange the variables in descending power. So, (𝑓 ∙ 𝑔)(𝑥) is −𝟏𝟓𝒙𝟐 + 𝟏𝟐𝒙 Example 2, 𝑓(𝑥) = 3𝑥 − 2 𝑔(𝑥) = 3 − 5𝑥 Solution: Write the functions into (𝒇 ∙ 𝒈)(𝒙) = 𝒇(𝒙) ∙ 𝒈(𝒙) Solution: (𝑓 ∙ 𝑔)(𝑥) = (3𝑥 − 2)(3 − 5𝑥) → remove parentheses using distributive property (𝑓 ∙ 𝑔)(𝑥) = (3𝑥 − 2)(3 − 5𝑥) → perform the operations (𝑓 ∙ 𝑔)(𝑥) = 9𝑥 − 15𝑥 2 − 6 + 10𝑥 → combine like terms and arrange variables in descending power (𝑓 ∙ 𝑔)(𝑥) = −15𝑥 2 + 19𝑥 + 6 So, (𝑓 ∙ 𝑔)(𝑥) is −𝟏𝟓𝒙𝟐 + 𝟏𝟗𝒙 − 𝟔 12  What’s More NOW IT’S YOUR TURN! Directions: Find (𝑓 ∙ 𝑔)(𝑥) using the two functions given in each number. 1. 𝑓(𝑥) = 3𝑥 2. 𝑓(𝑥) = 2𝑥 3. 𝑓(𝑥) = 5𝑥 + 2 𝑔(𝑥) = −4𝑥 + 1 𝑔(𝑥) = 4𝑥 2 + 2𝑥 − 2 𝑔(𝑥) = −3𝑥 2 − 5 What I Need to Remember The multiplication of functions can be written as 𝒇(𝒙) ∙ 𝒈(𝒙) or (𝒇 ∙ 𝒈)(𝒙) or (𝒇𝒈)(𝒙). Lesson Operations of Functions; 3.4 Division What I Need to Know At the end of this lesson, you are expected to: follow the steps in dividing functions, and divide functions. What’s In REVIEW Division of Integers Rule: The quotient of two like signs integers is always positive and the quotient of two unlike signs integers is always negative. Dividing Like Signs Integers Dividing Unlike Signs Integers Integers Quotient Integers Quotient 15 ÷ (+3) = + 5 -20 ÷ 4 = -5 -40 ÷ (-8) = 5 27 ÷ (-3) = -9 13  What’s New How to divide functions? 𝒇 𝒇(𝒙) The quotient of functions can be written 𝒇(𝒙) ÷ 𝒈(𝒙) or ( ) (𝒙) =. 𝒈 𝒈(𝒙) Find the quotient of the following functions. Example 1 𝑓(𝑥) = 12𝑥 − 4 𝑔(𝑥) = −2 𝒇 𝒇 ( 𝒙) Solution: Write the functions into (𝒈) (𝒙) = 𝒈(𝒙) 𝑓 (12𝑥−4) Solution: (𝑔) (𝑥) = −2 → divide using distributive property 𝑓 12𝑥 4 (𝑔) (𝑥) = − −2 −2 → perform the operation 𝑓 (𝑔) (𝑥 ) = −6𝑥 + 2 𝑓 So, (𝑔) (𝑥) is -6x + 2 Example 2 𝑓(𝑥) = 3𝑥 2 − 12𝑥 𝑔(𝑥) = 6𝑥 − 18𝑥 3 𝑓 (3𝑥 2 −12𝑥) Solution: (𝑔) (𝑥) = (6𝑥−18𝑥 3 ) → check for common factors in the numerator and denominator 𝑓 3𝑥(𝑥−4) (𝑔) (𝑥) = 6𝑥(1−3𝑥 2 ) → check if there is a common factor in the numerator and denominator 𝑓 3𝑥(𝑥−4) (𝑔) (𝑥) = 6𝑥(1−3𝑥 2 ) → common factor is 3x, simplify 𝑓 𝑥−4 (𝑔) (𝑥) = 2(1−3𝑥 2 ) 𝑓 𝒙−𝟒 So, (𝑔) (𝑥) is 𝟐(−𝟑𝒙𝟐+𝟏) Example 3 𝑓(𝑥) = 𝑥 2 + 2𝑥 − 8 𝑔(𝑥) = 𝑥 2 − 4 𝑓 (𝑥 2 +2𝑥−8) Solution: (𝑔) (𝑥) = (𝑥 2 −4) → factor the numerator and the denominator 𝑓 (𝑥+4)(𝑥−2) (𝑔) (𝑥) = (𝑥+2)(𝑥−2) → check for common factors then simplify, common factor is x-2 𝑓 (𝑥+4)(𝑥−2) (𝑔) (𝑥) = (𝑥+2)(𝑥−2) 𝑓 𝑥+4 (𝑔) (𝑥) = 𝑥+2 𝑓 𝒙+𝟒 So, (𝑔) (𝑥) is 𝒙+𝟐 14  What’s More NOW IT’S YOUR TURN! 𝑓 Directions: Find ( ) (𝑥) using the two functions given in each number 𝑔 1. 𝑓(𝑥) = 3𝑥 − 15 2. 𝑓(𝑥) = 25𝑥 − 10𝑥 2 3. 𝑓(𝑥) = 𝑥 2 − 2𝑥 − 8 𝑔(𝑥) = −6𝑥 + 12 𝑔(𝑥) = −15𝑥 3 − 5𝑥 𝑔(𝑥) = 𝑥 2 − 16 What I Need to Remember o The division of functions can be written as 𝒇(𝒙) ÷ 𝒈(𝒙) or 𝒇 𝒇(𝒙) ( ) (𝒙) = 𝒈 𝒈(𝒙) Lesson Operations of Functions; 3.5 Composition of Functions What I Need to Know At the end of this lesson, you are expected to: follow the steps in solving composition of functions. perform the operation of composition of functions given two functions. What’s New What is composition of functions? The term "composition of functions" (or "composite function") refers to the combining of two or more functions in a manner where the output from one function becomes the input for the next function. The composition of functions can be written as (𝒇 𝒐 𝒈)(𝒙) = 𝒇(𝒈(𝒙)) 15 Solve the composition of functions. Example 1 𝑓(𝑥) = 3𝑥 𝑔(𝑥) = 4 − 5𝑥 Solution: Write the functions into (𝒇 𝒐 𝒈)(𝒙) = 𝒇(𝒈(𝒙)) (𝑓 𝑜 𝑔)(𝑥) = 3𝑥 →substitute x in f(x) by g(x) (𝑓 𝑜 𝑔)(𝑥) = 3(4 − 5𝑥) → use distributive property (𝑓 𝑜 𝑔)(𝑥) = 12 − 15𝑥 → combine like terms if there are and simplify (𝑓 𝑜 𝑔)(𝑥) = −15𝑥 + 12 So, (𝑓 𝑜 𝑔)(𝑥) is −𝟏𝟓𝒙 + 𝟏𝟐 Example 2 𝑓(𝑥) = 2𝑥 + 12 𝑔(𝑥) = 12 − 𝑥 Solution: Write the functions into (𝒇 𝒐 𝒈)(𝒙) = 𝒇(𝒈(𝒙)) (𝑓 𝑜 𝑔)(𝑥) = 2𝑥 + 12 →substitute x in f(x) by g(x) (𝑓 𝑜 𝑔)(𝑥) = 2(12 − 𝑥) + 12 →use distributive property (𝑓 𝑜 𝑔)(𝑥) = 24 − 2𝑥 + 12 →combine like terms (𝑓 𝑜 𝑔)(𝑥) = −2𝑥 + 36 So, (𝑓 𝑜 𝑔)(𝑥) is −𝟐𝒙 + 𝟑𝟔 What’s More NOW IT’S YOUR TURN! Directions: Perform compositions of functions using the following. 𝒇(𝒙) = 𝟑𝒙 + 𝟐 𝒈(𝒙) = 𝒙 − 𝟒 1. Find (𝑓 𝑜 𝑔)(𝑥) 2. Find (𝑔 𝑜 𝑓)(𝑥) What I Need to Remember The term "composition of functions" (or "composite function") refers to the combining of two or more functions in a manner where the output from one function becomes the input for the next function. The composition of functions can be written as (𝑓 𝑜 𝑔)(𝑥) = 𝑓(𝑔(𝑥)) 16  Lesson Problem Solving involving 4 Functions What I Need to Know At the end of this lesson, you are expected to: solve word problems using operations and compositions of functions. What’s New Problems Involving Functions Problem 1: Kamao Network charges P500 monthly cable connection fee plus P125 for each hour of pay-per-view (PPV) event regardless of a full hour or a function of an hour. What is the monthly bill of a customer who watched 25 hours of PPV events? Solution: Given: P500 - connection fee P125 - additional fee for each hour of PPV 25 hrs. - number of hours the customer watched PPV Find: Monthly bill of a customer who watched 25 hours of PPV events Write the Function Rule of the problem using x as the variable where x is the number of hours spent watching PPV Function Rule: 𝑓(𝑥) = 500 + 125𝑥 where, 500 is the initial connection fee 125 is the additional fee per hour x is the number of hours spent watching PPV Solution: 𝑓(𝑥) = 500 + 125𝑥 𝑤ℎ𝑒𝑟𝑒 𝑥 = 25 𝑓(25) = 500 + 125(25) 𝑓(25) = 500 + 3,125 𝑓(25) = 3,625 Therefore, the monthly bill of a customer who watched 25 hours of PPV events is P 3,625. Problem 2: Aryan wants to borrow a car from ACE Car Rentals. The car rental charge is P600 upon application plus P25 per kilometer travelled. Determine the function that represents the daily cost by the number of kilometers travelled. If a total amount of P3,500 was paid after returning the car, how many kilometers did Aryan travel? Solution: Given: P600 - initial charge of the car rental P25 - additional fee for each kilometer travelled P3,500 - total amount paid after returning the car Find: Distance travelled by Aryan 17  Write the Function Rule of the problem with x as the variable where x is the number of distance travelled by Aryan. Function Rule: 𝑓(𝑥) = 600 + 25𝑥 where, 600 is the initial charge 25 is the additional fee per kilometer x is the number of kilometers 𝑓(𝑥) = 600 + 25𝑥 → we are looking for the value of x, therefore substitute f(x) by 3500 3500 = 600 + 25𝑥 → perform the operation using the addition property of equality 3500 − 600 = 600 + 25𝑥 − 600 → simplify 2900 = 25𝑥 → use division property of equality to find x 2900 25𝑥 = 25 25 116 = 𝑥 So, the total amount of kilometers covered by Aryan is 116 kilometers. What’s More NOW IT’S YOUR TURN! Solve the following problems and show your solutions. 1. The jeepney drivers charge a fare P7 upon riding on the vehicle and charge P2 per kilometer of distance travelled. If you are travelling from your home to your school which is 5 kilometers away, how much will you pay the jeepney driver? 2. At Joe's pizzeria a pizza costs P40 with the first topping, and then an additional P15 for each additional topping. If you ordered and paid a total of P130 for toppings, how many toppings did you add? What I Need to Remember You can solve problems by identifying the given and what is being asked. Afterwards, translate word phrases to mathematical phrases and perform the operations. 18  What I Can Do Here is a table of the temperature and its equivalent. Fill out the missing values with respect to the conversion of oC to oF. 9 ℉(℃) = ℃ + 32 5 Temperature (oC) Temperature (oF) 25 Example: 9 ℉(℃) = ℃ + 32 5 9 ℉(25) = (25) + 32 5 ℉(25) = 9(5) + 32 ℉(25) = 45 + 32 ℉(25) = 77 30 45 Temperature (oC) Temperature (oF) Example: 122 9 ℉(℃) = ℃ + 32 5 9 122 = ℃ + 32 5 9 122 − 32 = ℃ 5 9 90 = ℃ 5 450 = 9℃ 50 = ℃ 131 149 19  Assessment (Post-test) Test I. Choose the letter of the correct answer and write them on a separate sheet of paper. 1. What is a function? A. Set of inputs and outputs C. Every input has only ONE output B. Set of x-values and y-values D. Inputs have different outputs every time 2. Which of the following set of values is a function? A. {(2,4),(2,-3),(7,4),(3, 8)} C. {(6,-5),(7,-3),(8,-1),(8,1)} B. {(2,-2),(5,9),(5,-7),(1,4)} D. {(7,5),(-8,5),(-9,-5),(10,-5)} 3. According to the table below, does the relationship represent a function? TIME (Minutes) Temperature (oC) 0 25 10 30 20 32 30 36 50 36 A. Yes, the relationship is a function C. No, the relationship is not a because every minute has a unique function because the increase in temperature. temperature is not the same. B. Yes, the relationship is a function D. No, the relationship is not a because the temperature is function because the increase in the increasing as time passes. minutes is higher than the temperature. 4. Students were asked to state the different ways of determining if a relation is a function. Which student is NOT correct? A. Maria said, “In a table, if the x C. Sam said, “In an ordered pair, if values have more than one the x values have more than one corresponding y value, it is not a corresponding y value, it is not a function.” function.” B. Jennifer said, “In a graph you can D. Matthew said, “In a mapping use the vertical line test and if the diagram you can use the vertical line line intercepts only once, it is a test and if the line intercepts only function.” once, it is a function” 5. Which of the following relations is NOT a function? A. {(3,3),(4,5),(5,5),(6,4)} C. {(1,1),(2,2),(3,3),(4,4)} B. {(2,3),(4,5),(5,6),(5,7)} D. {(-1,2),(0,2),(1,2),(2,3)} 6. Evaluate f(x) = x2 for f(-3) A. -9 B. -6 C. 6 D. 9 7. Evaluate f(x) = –2x2 + 1 for f(-2) A. -7 B. -5 C. 7 D. 9 20 8. Using the graph on the right, evaluate the function with value f(1). A. 3 C. 0 B. 2 D. -3 9. Using the diagram on the right, evaluate the function with the value f(2). A. 7 C. 0 B. 4 D. -3 10. Refer to the graph on the right and evaluate for the function with the value f(0). A. 6 C. 0 B. 2 D. -2 Test II. Match Column A to the answers in column B given the following. 𝑓 (𝑥 ) = 2𝑥 + 6 𝑔(𝑥 ) = 4 − 4𝑥 Column A Column B _____11. (𝑓 + 𝑔)(𝑥) A. 6𝑥 + 2 _____12. (𝑓 − 𝑔)(𝑥) B. 2𝑥 − 10 _____13. (𝑓 ∙ 𝑔)(𝑥) C. −2𝑥 + 10 𝑓 𝑥+3 _____14. (𝑔) (𝑥) D. 2(−𝑥+1) 𝑥+3 E. − 2(−𝑥+2) _____15. (𝑓 ⃘ 𝑔)(𝑥) F. −8𝑥 + 14 G. −8𝑥 2 − 16𝑥 + 24 21 References Textbook Aggarwal, P.M.L (2014). APC Mastering Mathematics. APC Mastering Mathematics - Class 7 (ICSE) - Avichal Publishing Company. Avichal Publishing Company ISBN 9788178555119 Gefland, I.M. et al. (2003). Algebra. Springer Science & Business Media, ISBN 0817636773, 9780817636777 Oronce, O. A. (2016). General Mathematics. Manila, Philippines: Rex Book Store. pp 1-21. Websites Pierce, R. (2020). Math is Fun. Accessed July 7, 2020. http://www.mathsisfun.com/ wikiHow. n.d. Accessed July 7, 2020. https://www.wikihow.com/ PDF Verzosa, Debbie Marie B. et al. (2016). General Mathematics Teaching Guide [PDF File]. Quezon City, Philippines: Commission on Higher Education. pp 1-21. Retrieved from https://www.lrmds.gov.ph Images All images used in this module are creative commons results taken from the Insert Ribbon – Online Pictures from MS Office 365. Figure 1: retrieved from https://sites.google.com/site/thefunctionsong/lyrics/vertical-line-test Avatars All avatars used in this module are created originally using the Bitmoji mobile application. Created on July 15, 2020. Icons All icons used in this module is taken from MS Office 365. Graphs All graphical representation in this module is a snip/screenshot using the graphing calculator site of www.desmos.com. Congratulations! You are now ready for the next module. Always remember the following: 1. Make sure every answer sheet has your ▪ Name ▪ Grade and Section ▪ Title of the Activity or Activity No. 2. Follow the date of submission of answer sheets as agreed with your teacher. 3. Keep the modules with you AND return them at the end of the school year or whenever face-to-face interaction is permitted. 23