This lesson plan teaches students about inverse functions. It begins with objectives, materials, and a teaching strategy of lecture. Examples are provided to show how to find the inverse of one-to-one functions by interchanging x and y and solving for the new y. Properties are discussed, such as the inverse of an inverse is the original function. Students are asked to find inverses and solve word problems. The lesson concludes by having students generalize their understanding and complete an evaluation with additional inverse problems.
It is a powerpoint presentation that will help the students to enrich their knowledge about Senior High School subject of General Mathematics. It is comprised about Rational functions and its zeroes. It is also comprised of some examples and exercises to be done for the said topic.
It is a powerpoint presentation that will help the students to enrich their knowledge about Senior High School subject of General Mathematics. It is comprised about Rational functions and its zeroes. It is also comprised of some examples and exercises to be done for the said topic.
This is a presentation by Dada Robert in a Your Skill Boost masterclass organised by the Excellence Foundation for South Sudan (EFSS) on Saturday, the 25th and Sunday, the 26th of May 2024.
He discussed the concept of quality improvement, emphasizing its applicability to various aspects of life, including personal, project, and program improvements. He defined quality as doing the right thing at the right time in the right way to achieve the best possible results and discussed the concept of the "gap" between what we know and what we do, and how this gap represents the areas we need to improve. He explained the scientific approach to quality improvement, which involves systematic performance analysis, testing and learning, and implementing change ideas. He also highlighted the importance of client focus and a team approach to quality improvement.
Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
How to Split Bills in the Odoo 17 POS ModuleCeline George
Bills have a main role in point of sale procedure. It will help to track sales, handling payments and giving receipts to customers. Bill splitting also has an important role in POS. For example, If some friends come together for dinner and if they want to divide the bill then it is possible by POS bill splitting. This slide will show how to split bills in odoo 17 POS.
The Indian economy is classified into different sectors to simplify the analysis and understanding of economic activities. For Class 10, it's essential to grasp the sectors of the Indian economy, understand their characteristics, and recognize their importance. This guide will provide detailed notes on the Sectors of the Indian Economy Class 10, using specific long-tail keywords to enhance comprehension.
For more information, visit-www.vavaclasses.com
Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
Instructions for Submissions thorugh G- Classroom.pptxJheel Barad
This presentation provides a briefing on how to upload submissions and documents in Google Classroom. It was prepared as part of an orientation for new Sainik School in-service teacher trainees. As a training officer, my goal is to ensure that you are comfortable and proficient with this essential tool for managing assignments and fostering student engagement.
How to Create Map Views in the Odoo 17 ERPCeline George
The map views are useful for providing a geographical representation of data. They allow users to visualize and analyze the data in a more intuitive manner.
2024.06.01 Introducing a competency framework for languag learning materials ...Sandy Millin
http://sandymillin.wordpress.com/iateflwebinar2024
Published classroom materials form the basis of syllabuses, drive teacher professional development, and have a potentially huge influence on learners, teachers and education systems. All teachers also create their own materials, whether a few sentences on a blackboard, a highly-structured fully-realised online course, or anything in between. Despite this, the knowledge and skills needed to create effective language learning materials are rarely part of teacher training, and are mostly learnt by trial and error.
Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
This webinar will introduce you to my framework, highlighting the key competencies I identified from my research. It will also show how anybody involved in language teaching (any language, not just English!), teacher training, managing schools or developing language learning materials can benefit from using the framework.
Students, digital devices and success - Andreas Schleicher - 27 May 2024..pptxEduSkills OECD
Andreas Schleicher presents at the OECD webinar ‘Digital devices in schools: detrimental distraction or secret to success?’ on 27 May 2024. The presentation was based on findings from PISA 2022 results and the webinar helped launch the PISA in Focus ‘Managing screen time: How to protect and equip students against distraction’ https://www.oecd-ilibrary.org/education/managing-screen-time_7c225af4-en and the OECD Education Policy Perspective ‘Students, digital devices and success’ can be found here - https://oe.cd/il/5yV
The Roman Empire A Historical Colossus.pdfkaushalkr1407
The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
The Roman Empire’s society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
Culturally, the Romans were eclectic, absorbing and adapting elements from the civilizations they encountered, particularly the Greeks. Roman art, literature, and philosophy reflected this synthesis, creating a rich cultural tapestry. Latin, the Roman language, became the lingua franca of the Western world, influencing numerous modern languages.
Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
Cambridge International AS A Level Biology Coursebook - EBook (MaryFosbery J...
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.