INVERSE FUNCTION
- 1. SEMI-DETAILED LESSON PLAN IN GENERAL MATHEMATICS I. Objectives: At the end of the lesson, the students are expected to: 1. define what is inverse function; 2. find the inverse of a function; and 3. solve problems involving inverse function. II. Subject Matter: Inverse function a. Reference: Oronce,O.(2016).Genearal Mathematics.Rex Book Store,Inc..856 Nicanor Reyes,Manila. https://utexas.instructure.com/courses/1092722/pages/lesson-10-inverse- functions?module_item_id=7237956 b. Materials: visual aids, flash cards, charts c. Teaching strategy: lecture method III. Procedure A. Preparatory Activities 1) Prayer 2) Drill The teacher will have a game about guessing the rule given the constructed table of values. 3) Review The teacher asks the following questions to the students: Last meeting you discussed about one-to-one function, and based on our activity : Between the two given functions, which function represents one-to-one function ? What is one-to-one function then? To determine if the students understood the previous lesson, the teacher will show flash cards to be answered by them. x -2 -1 0 1 2 y -6 -5 -4 -3 -2 x -2 -1 0 1 2 y 2 -4 -6 -4 2
- 2. B. Developmental activities 1) Motivation The teacher will present unarranged phrase to be arranged by the students. 2) Presentation The teacher restate the lesson and the lesson objectives. Our topic for today is to determine the inverse of a one-to-one function. And at the end of the discussion, you, students are expected to define what is inverse function; find the inverse of a function; and solve problems involving inverse of one-to-one function. 3) Discussion The teacher will ask the students how they define the word “INVERSE” and relate it in function. Original function: inverse function: Based on the illustration what do you mean by inverse? Why is it that it states one-to-one function? Why not all function? One-to-one function: function but not one-to-one : x -2 -1 0 1 2 y -6 -5 -4 -3 -2 x -2 -1 0 1 2 y -6 -5 -4 -3 -2 x -2 -1 0 1 2 y -6 -5 -4 -3 -2 x -2 -1 0 1 2 y 4 1 0 1 4 x -6 -5 -4 -3 -2 y -2 -1 0 1 2 x 4 1 0 1 4 y -2 -1 0 1 2 Determine if itisa one-to-one functionornot. 1. {(2,9),(4,5), (11,5)} 2. {(1,1),(9,3), (16,4), (4,2)} 3. 𝑓( 𝑥) = 2𝑥 4. 𝑦 = 𝑥2 + 13 5. {( 𝐾𝑎𝑡ℎ𝑟𝑦𝑛, 𝐷𝑎𝑛𝑖𝑒𝑙),( 𝐿𝑖𝑧𝑎, 𝐸𝑛𝑟𝑖𝑞𝑢𝑒),( 𝐽𝑢𝑙𝑖𝑎, 𝐽𝑜𝑠ℎ𝑢𝑎),(𝐽𝑎𝑛𝑒, 𝐽𝑜𝑠ℎ𝑢𝑎} ENIMRETED EHT ESREVNI FO A ENO-OT-ENO NOITCNUF A relation reversing the process performed by any function f(x) is called inverse of f(x). This means that the every element of the range corresponds to exactly one element of the domain. A function has an inverse if and only if it is one-to-one. Not a valid function
- 3. EXAMPLES: 1. Find the inverse of a function described by the set of ordered pairs {(0, −2),(1,0),(2,2), (3,4)}. Answer: {(−2,0), (0,1),(2,2),(4,3)} 2. Find the inverse of a function 𝑓( 𝑥) = 3𝑥 + 1 Sol. 𝑓( 𝑥) = 3𝑥 + 1 𝑦 = 3𝑥 + 1 𝑥 = 3𝑦 + 1 𝑦 = 𝑥 − 1 3 𝑓−1 (𝑥) = 𝑥 − 1 3 3. Find the inverse of 𝑓( 𝑥) = 5𝑥 + 6 Sol. 𝑓( 𝑥) = 5𝑥 + 6 𝑦 = 5𝑥 + 6 𝑥 = 5𝑦 + 6 𝑦 = 𝑥 − 6 5 𝑓−1 (𝑥) = 𝑥 − 6 5 Based on the given examples how do we find the inverse of one-to-one function? 4. Find the inverse of 𝑓( 𝑥) = 3𝑥2 − 2 if it exists. 5. Find the inverse of 𝑓( 𝑥) = 1 3 𝑥 − 2 and 𝑔( 𝑥) = 3𝑥 + 6 Properties of inverse of a one-to-one function To find the 𝑓−1( 𝑥) 1. Replace 𝑓( 𝑥)with 𝑦 2. Interchange 𝑥 and 𝑦 3. Solve for the new 𝑦 in the equation 4. Replace the new 𝑦 with 𝑓−1( 𝑥) The inverse of 𝑓( 𝑥) is 𝑔(𝑥) and the inverse of 𝑔(𝑥) is ( 𝑥) . Therefore,the inverse of 𝑓( 𝑥) is 𝑓−1(𝑥) and the inverse of 𝑓−1(𝑥) is 𝑓( 𝑥). It does not have an inverse since it is not one-to-one function. Given a one-to-one function 𝑓( 𝑥) and its inverse 𝑓−1(𝑥), then the following are true: 1. The inverse of 𝑓−1(𝑥) is 𝑓( 𝑥) 2. 𝑓( 𝑓−1( 𝑥)) = 𝑥 for all 𝑥 in the domain of 𝑓−1(𝑥). 3. 𝑓−1(𝑓( 𝑥)) = 𝑥 for all 𝑥 in the domain of 𝑓(𝑥).
- 4. C. Concluding Activities 1. Generalization The following questions will be asked to the students What do you mean by inverse function? Are all functions have their inverse functions? How to find the inverse of a one-to-one function? What are the properties of inverse function? 2. Application The teacher will let the students answer the given problems and asks some volunteers to answer the problems. Feedbacks will be made. IV. Evaluation Directions: Answer the given problem. 1. Which among the following functions have an inverse? a . 𝑓( 𝑥) = 2𝑥3 − 5 b. 𝑔( 𝑥) = 3𝑥 − 8 c. ℎ( 𝑥) = 1 𝑥2 d. 𝑘( 𝑥) = | 𝑥| e. 𝑙( 𝑥) = 𝑥2 − 6𝑥 2. Find the inverse of 𝑓( 𝑥) = −𝑥3 + 5 3. Find 𝑓( 𝑥) if 𝑓−1( 𝑥) = 1 𝑥−2 4. The function 𝑦 = 2.54𝑥 represents the conversion of measurement units where y represents the distance in terms of inches given the x distance in centimeters. Find the equation where y represents distance in terms of centimeters given the x distance in inches. How many centimeters are there in 1000 inches? 5. Arnold and Tina are playing a number guessing game. Arnold asks Tina to think of a positive number, triple the number, square the result, then add 7. If Tina’s answer is 43, what was the original function? Use the concept of inverse function in your solution. V. Assignment Prove that the inverse of a linear function is also linear and the two slopes are reciprocals of each other. 1. Bernie wantstoexchange his100 PhilippinepesobilltoUS dollars.He foundoutthat the exchange rate isrepresentedbythe function 𝑦 = 51.16𝑥 where yrepresentsthe amountinPhilippine peso giventhe x US dollars.Findthe equationwhere yrepresentsthe amountof USdollarsgiventhe x Philippine peso.Howmuchisthe equivalentof Bernie’s100 Philippine pesobilltoUS dollars? 2. To convertdegreesCelciustoFahrenheit,the function 𝐹 = 9 5 𝐶 + 32 where 𝐶 isthe temperature in Celcius.Findthe inverse functionconvertingthe temperature inFahrenheittodegreesCelcius.