Math 11-CORE Gen Math-Q1-Week-2 (1).pdf

4 RATIONAL FUNCTIONS, RATIONAL EQUATIONS AND INEQUALITIES for General Mathematics Senior High School (CORE) Quarter 1 / Week 2 1  FOREWORD This Self-Learning Kit for General Mathematics is designed specifically for Grade 1...

4 RATIONAL FUNCTIONS, RATIONAL EQUATIONS AND INEQUALITIES for General Mathematics Senior High School (CORE) Quarter 1 / Week 2 1  FOREWORD This Self-Learning Kit for General Mathematics is designed specifically for Grade 11 students in the Senior High School. Thus, a modest background in grade school mathematics is important, written in a precise, readable, and conventional manner to facilitate students’ understanding of the subject. It is aligned with the BEC of the Department of Education following the prescribed MELCs (Most Essential Learning Competencies. It has the following features proven to be valuable aids to learning Mathematics even at home. What happened This section contains pre-activities like review of the prior knowledge and a pretest on what the learners have learned in their previous discussions. What I Need to Know (Discussion) This section contains definition of terms, different examples of real-life situations as application of functions. It gives examples and the corresponding situations that clearly illustrate the applicability of a mathematical concept. What I have Learned (Evaluation/Post Test) The exercises contained in this section are guaranteed to build mathematical comprehension, skills, and competence. These serve as a diagnostic tool to identify the learners’ areas of strengths and difficulties. 2  OBJECTIVES: At the end of the lesson, the students are expected to: K. represent real-life situations using rational functions and distinguish rational function, rational equation and inequality; S. solve rational equations and inequalities; and A. develop accuracy and patience in solving rational equations and inequalities. LESSON 1 REPRESENTING REAL-LIFE SITUATIONS USING RATIONAL FUNCTIONS I. WHAT HAPPENED PRE-TEST: Complete Me! (𝑥+1) Instruction: Evaluate the function 𝑓(𝑥) = using the values of x in the table. Write 2 your answer in your activity notebook/sheets. x 1 2 3 4 5 f(x) II. WHAT YOU NEED TO KNOW Definition 𝑝(𝑥) A rational function is a function of the form 𝑓(𝑥) = where p(x)and q(x) are polynomial 𝑞(𝑥) functions, and q(x) is not the zero function ( i.e., q(x) ≠ 0). The domain of f(x) is all values of x where q(x)≠ 0. (Debbie Marie B. Verzosa 2016) These are some real-world relationships that can be modeled by rational functions. Unlike polynomial functions, rational functions may contain a variable in the denominator. Example 1. An object is to travel a distance of 10 meters. Express velocity v as a function v(t) of travel time t, in seconds. Solution. The following table of values show v for various values of t. t (seconds) 1 2 4 5 10 v (meters 10 5 2.5 2 1 per second) 3  10 The function 𝑣(𝑡) = can represent 𝑣 as a function of 𝑡. 𝑡 Example 2. The local barangay received a budget of ₱100, 000 to provide medical checkups for the children in the barangay. The amount is to be allotted equally among all children in the barangay. Write an equation or function representing the relationship of allotted amount per child (y-variable) versus the total number of children (x-variable). Fill up the table below with the different allotment amount for different values for the number of children: No. of 10 20 50 100 200 300 500 1 000 children, x Allocated amount, y (100,000) Solution. The equation or function would be 𝑦(𝑥) = 𝑥 No. of 10 20 50 100 200 300 500 1 000 children, x Allocated 10 000 5 000 2 000 1 000 500 333.33 200 100 amount, y III. WHAT HAVE I LEARNED POST TEST: CHALLENGE YOURSELF. ANSWER ME! Do what is asked. Answer in your activity notebook/sheets. Situation 1. 𝑑 Average speed (or velocity) can be computed by the formula 𝑣 = 𝑡. Consider a 100- meter track used for foot races. The speed of a runner can be computed by taking the time for 100 him to run the track and applying it to the formula 𝑠 = since the distance is fixed at 100 𝑡 meters. a. Represent the speed of a runner as a function of the time it takes to run 100 meters in the track. Let x represent the time it takes to run 100 meters. b. Continuing the scenario above, construct a table of values for the speed of a runner against different run times. x 10 12 14 16 18 20 4  s(x) Situation 2. The budget of a school organization is split evenly among its various committees. If they have a budget of ₱60, 000: Construct a function M(n) which would give the amount of money each of the n number of committees would receive. LESSON 2 DISTINGUISHING RATIONAL FUNCTION, RATIONAL EQUATION, AND RATIONAL INEQUALITY I. WHAT HAPPENED PRE-TEST Mathinik Challenge 1 Guess Me! Instructions: Identify the following expression as rational or not. Write your answer in your activity notebook/sheets. 1. 𝑥 2 + 3𝑥 + 2 𝑥+4 2. 1 3𝑥 2 3. 𝑥 2 + 4𝑥 − 3 2 4. √𝑥 + 1 𝑥3 − 1 5. 1 𝑥+2 −2 Mathinik Challenge 2 Match Me! Instruction: Match the following rational equation, rational inequality, and rational function to their corresponding examples. Column A Column B 5  𝑥 2 +2𝑥+3 1. rational equation a) 𝑓(𝑥) = 𝑥+1 2 3 1 2. rational inequality b) − = 𝑥 2𝑥 5 5 2 3. rational function c) 𝑥−3 ≤ 𝑥 II. WHAT YOU NEED TO KNOW DISCUSSION: Review on rational expression. Definition A rational expression is an expression that can be written as a ratio of two polynomials. It can be described as a function where either the numerator, denominator, or both have a variable on it. Rational Equation Rational Inequality Rational Function, Definition An equation An inequality A function of the form involving rational involving rational 𝑝(𝑥) 𝑝(𝑥) f(x)=𝑞(𝑥) or y=𝑞(𝑥) where expression in both expression in both numerator and numerator and p(x)and q(x)are polynomials, denominator. denominator. and q(x) is not the zero Symbol used: = Symbols used: function. , ≤, ≥. Symbol used: = Examples 2 3 1 5 2 𝑥 2 + 2𝑥 + 3 − = ≤ 𝑓(𝑥) = 𝑥 2𝑥 5 𝑥−3 𝑥 5 or 3 1 3𝑥 6 𝑥 2 + 2𝑥 + 3 = ≥ 𝑦= 4 2𝑥 5 2 5 More examples. 1 a) 𝑓(𝑥 ) = 𝑥 It is a rational equation but since the value of the function f is a rational expression then it is a rational function. 2 1 b) 𝑥2 = 𝑥 It is only a rational equation because both sides involve rational expressions. 𝑥+1 1 c) 2𝑥−3 = 𝑥 It is only a rational equation because both sides involve rational expressions. 3 d) 𝑦 = − 𝑥+1 It is a rational equation and a rational function 6  𝑥+1 e) 2𝑥−3 > 0 It is a rational inequality because it is an inequality that contains rational expressions. 1 3 f) 𝑥 ≤ 𝑥+1 It is a rational inequality because it is an inequality that contains rational expressions. III. WHAT HAVE I LEARNED POST TEST CHALLENGE YOURSELF. ANSWER ME! Distinguish the following if it is rational equation, rational inequality or rational function. 2 9 _____________________1. 𝑥 = 𝑥−3 1 _____________________2. 2𝑥+5 1 _____________________3. 𝑓(𝑥) = 𝑥−2 2 9 _____________________4. 7 > 𝑥−3 2 √5𝑥+1 _____________________5. > 7 𝑥−3 1 3 _____________________6. 2𝑥+5 = 𝑥+5 2𝑥−1 _____________________7. 3𝑥+2 < 12 1 _____________________8. 𝑦 = 𝑥−2 2 3 1 _____________________9. 3𝑥 + 4 = 𝑥 _____________________10. 𝑥 2 + 𝑥 + 1 LESSON 3 SOLVING RATIONAL EQUATIONS, AND INEQUALITIES I. WHAT HAPPENED PRE-TEST: Mathinik Challenge 1: Make Me Equal! Find the value of x to make the statement true. Write your answer on your activity notebook/sheets. a) 2𝑥 + 4 = 10 b) 𝑥 + 3 = 2𝑥 + 4 7 c) 𝑥 2 − 1 = 24 d) 𝑥 2 − 2𝑥 + 1 = 0 Mathinik Challenge 2: Put Me In! Put the corresponding symbol (>,