The document discusses function operations such as addition, subtraction, multiplication, and division of functions. It provides examples of performing these operations on specific functions f(x) and g(x). For addition, subtraction and multiplication, the results are simply the corresponding arithmetic operations performed on the individual functions. For division, the domain is restricted to values where g(x) is not equal to 0 to avoid division by zero.
Gives idea about function, one to one function, inverse function, which functions are invertible, how to invert a function and application of inverse functions.
Gives idea about function, one to one function, inverse function, which functions are invertible, how to invert a function and application of inverse functions.
The Sum of Two Functions
The Difference of Two functions
The Product of Two Functions
The Quotient of Two Functions
The Product of A constant and a Function
The Sum of Two Functions
The Difference of Two functions
The Product of Two Functions
The Quotient of Two Functions
The Product of A constant and a Function
* Combine functions using algebraic operations.
* Create a new function by composition of functions.
* Evaluate composite functions.
* Find the domain of a composite function.
* Decompose a composite function into its component functions.
* Model exponential growth and decay
* Use Newton's Law of Cooling
* Use logistic-growth models
* Choose an appropriate model for data
* Express an exponential model in base e
* Construct perpendicular and angle bisectors
* Use bisectors to solve problems
* Identify the circumcenter and incenter of a triangle
* Use triangle segments to solve problems
* Identify, write, and analyze conditional statements
* Write the inverse, converse, and contrapositive of a conditional statement
* Write a counterexample to a fake conjecture
* Find the distance between two points
* Find the midpoint of two given points
* Find the coordinates of an endpoint given one endpoint and a midpoint
* Find the coordinates of a point a fractional distance from one end of a segment
* Connect functions to their graphs
* Graph piecewise-defined functions
* Graph absolute value functions
* Graph greatest-integer functions
* Interpret graphs
* Use the vertical line test to determine a function
* Connect functions to their graphs
* Graph piecewise-defined functions
* Graph absolute value functions
* Graph greatest-integer functions
* Interpret graphs
* Use the vertical line test to determine a function
* Introduce functions and function notation
* Develop skills in constructing and interpreting the graphs of functions
* Learn to apply this knowledge in a variety of situations
* Recognize graphs of common functions.
* Graph functions using vertical and horizontal shifts.
* Graph functions using reflections about the x-axis and the y-axis.
* Graph functions using compressions and stretches.
* Combine transformations.
* Identify intervals on which a function increases, decreases, or is constant
* Use graphs to locate relative maxima or minima
* Test for symmetry
* Identify even or odd functions and recognize their symmetries
* Understand and use piecewise functions
* Solve polynomial equations by factoring
* Solve equations with radicals and check the solutions
* Solve equations with rational exponents
* Solve equations that are quadratic in form
* Solve absolute value equations
* Determine whether a relation or an equation represents a function.
* Evaluate a function.
* Use the vertical line test to identify functions.
* Identify the domain and range of a function from its graph
* Identify intercepts from a function’s graph
* Solve counting problems using the Addition Principle.
* Solve counting problems using the Multiplication Principle.
* Solve counting problems using permutations involving n distinct objects.
* Solve counting problems using combinations.
* Find the number of subsets of a given set.
* Solve counting problems using permutations involving n non-distinct objects.
* Use summation notation.
* Use the formula for the sum of the first n terms of an arithmetic series.
* Use the formula for the sum of the first n terms of a geometric series.
* Use the formula for the sum of an infinite geometric series.
* Solve annuity problems.
* Find the common ratio for a geometric sequence.
* List the terms of a geometric sequence.
* Use a recursive formula for a geometric sequence.
* Use an explicit formula for a geometric sequence.
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.
A Strategic Approach: GenAI in EducationPeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
Francesca Gottschalk - How can education support child empowerment.pptxEduSkills OECD
Francesca Gottschalk from the OECD’s Centre for Educational Research and Innovation presents at the Ask an Expert Webinar: How can education support child empowerment?
Honest Reviews of Tim Han LMA Course Program.pptxtimhan337
Personal development courses are widely available today, with each one promising life-changing outcomes. Tim Han’s Life Mastery Achievers (LMA) Course has drawn a lot of interest. In addition to offering my frank assessment of Success Insider’s LMA Course, this piece examines the course’s effects via a variety of Tim Han LMA course reviews and Success Insider comments.
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.
Biological screening of herbal drugs: Introduction and Need for
Phyto-Pharmacological Screening, New Strategies for evaluating
Natural Products, In vitro evaluation techniques for Antioxidants, Antimicrobial and Anticancer drugs. In vivo evaluation techniques
for Anti-inflammatory, Antiulcer, Anticancer, Wound healing, Antidiabetic, Hepatoprotective, Cardio protective, Diuretics and
Antifertility, Toxicity studies as per OECD guidelines
Introduction to AI for Nonprofits with Tapp NetworkTechSoup
Dive into the world of AI! Experts Jon Hill and Tareq Monaur will guide you through AI's role in enhancing nonprofit websites and basic marketing strategies, making it easy to understand and apply.
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.
June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...Levi Shapiro
Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
This Congress will not stand idly by and allow an environment hostile to Jewish students to persist. The House believes that your institution is in violation of Title VI of the Civil Rights Act, and the inability or
unwillingness to rectify this violation through action requires accountability.
Postsecondary education is a unique opportunity for students to learn and have their ideas and beliefs challenged. However, universities receiving hundreds of millions of federal funds annually have denied
students that opportunity and have been hijacked to become venues for the promotion of terrorism, antisemitic harassment and intimidation, unlawful encampments, and in some cases, assaults and riots.
The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
• The Committee on Ways and Means has been investigating several universities since November 15, 2023, when the Committee held a hearing entitled From Ivory Towers to Dark Corners: Investigating the Nexus Between Antisemitism, Tax-Exempt Universities, and Terror Financing. The Committee followed the hearing with letters to those institutions on January 10, 202
3. Operations on Functions
Given two functions f and g, then for all values of x for
which both fx and gx are defined, we can also define
the following:
Sum
Difference
Product
Quotient
f g x f x g x
f g x f x g x
fg x f x g x
, 0
f xf
x g x
g g x
4. Operations on Functions (cont.)
Example: Let and . Find each
of the following:
a)
b)
c)
d)
2
1f x x 3 5g x x
1f g
3f g
5fg
0
f
g
5. Operations on Functions (cont.)
Example: Let and . Find each
of the following:
a)
b)
c)
d)
2
1f x x 3 5g x x
1f g 1 1gf 2
51 11 3 02 18
3f g
5fg
0
f
g
6. Operations on Functions (cont.)
Example: Let and . Find each
of the following:
a)
b)
c)
d)
2
1f x x 3 5g x x
1f g 1 1gf 2
51 11 3 02 18
3f g 2
3 53 31 410 14
5fg
0
f
g
7. Operations on Functions (cont.)
Example: Let and . Find each
of the following:
a)
b)
c)
d)
2
1f x x 3 5g x x
1f g 1 1gf 2
51 11 3 02 18
3f g 2
3 53 31 410 14
5fg 2
3 5 55 1 02026 52
0
f
g
8. Operations on Functions (cont.)
Example: Let and . Find each
of the following:
a)
b)
c)
d)
2
1f x x 3 5g x x
1f g 1 1gf 2
51 11 3 02 18
3f g 2
3 53 31 410 14
5fg 2
3 5 55 1 02026 52
0
f
g
2
5
0 1
3 0
5
1
9. Operations on Functions (cont.)
Example: Let and . Find
each of the following:
a)
b)
c)
d)
8 9f x x 2 1g x x
f g x
f g x
fg x
f
x
g
10. Operations on Functions (cont.)
Example: Let and . Find
each of the following:
a)
b)
c)
d)
8 9f x x 2 1g x x
f g x 8 9 2 1x x
f g x
fg x
f
x
g
11. Operations on Functions (cont.)
Example: Let and . Find
each of the following:
a)
b)
c)
d)
8 9f x x 2 1g x x
f g x 8 9 2 1x x
f g x 8 9 2 1x x
fg x
f
x
g
12. Operations on Functions (cont.)
Example: Let and . Find
each of the following:
a)
b)
c)
d)
8 9f x x 2 1g x x
f g x 8 9 2 1x x
f g x 8 9 2 1x x
fg x 8 9 2 1x x
f
x
g
13. Operations on Functions (cont.)
Example: Let and . Find
each of the following:
a)
b)
c)
d)
8 9f x x 2 1g x x
f g x 8 9 2 1x x
f g x 8 9 2 1x x
fg x 8 9 2 1x x
f
x
g
8 9
2 1
x
x
14. Operations on Functions (cont.)
Example: Let and . Find
each of the following:
e) What restrictions are on the domain?
8 9f x x 2 1g x x
15. Operations on Functions (cont.)
Example: Let and . Find
each of the following:
e) What restrictions are on the domain?
There are two cases that need restrictions: taking the
square root of a negative number and dividing by zero.
8 9f x x 2 1g x x
16. Operations on Functions (cont.)
Example: Let and . Find
each of the following:
e) What restrictions are on the domain?
There are two cases that need restrictions: taking the
square root of a negative number and dividing by zero.
We address these by making sure the inside of gx > 0:
8 9f x x 2 1g x x
2 1 0
2 1
1
2
x
x
x
So the domain must be
1 1
or ,
2 2
x
