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.
Basic Calculus 11 - Derivatives and Differentiation RulesJuan Miguel Palero
It is a powerpoint presentation that discusses about the lesson or topic of Derivatives and Differentiation Rules. It also encompasses some formulas, definitions and examples regarding the said topic.
* Verify inverse functions.
* Determine the domain and range of an inverse function, and restrict the domain of a function to make it one-to-one.
* Find or evaluate the inverse of a function.
* Use the graph of a one-to-one function to graph its inverse function on the same axes.
Basic Calculus 11 - Derivatives and Differentiation RulesJuan Miguel Palero
It is a powerpoint presentation that discusses about the lesson or topic of Derivatives and Differentiation Rules. It also encompasses some formulas, definitions and examples regarding the said topic.
* Verify inverse functions.
* Determine the domain and range of an inverse function, and restrict the domain of a function to make it one-to-one.
* Find or evaluate the inverse of a function.
* Use the graph of a one-to-one function to graph its inverse function on the same axes.
* 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.
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.
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.
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?
Welcome to TechSoup New Member Orientation and Q&A (May 2024).pdfTechSoup
In this webinar you will learn how your organization can access TechSoup's wide variety of product discount and donation programs. From hardware to software, we'll give you a tour of the tools available to help your nonprofit with productivity, collaboration, financial management, donor tracking, security, and more.
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.
Model Attribute Check Company Auto PropertyCeline George
In Odoo, the multi-company feature allows you to manage multiple companies within a single Odoo database instance. Each company can have its own configurations while still sharing common resources such as products, customers, and suppliers.
Synthetic Fiber Construction in lab .pptxPavel ( NSTU)
Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
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.
2. Concepts & Objectives
Inverse Functions
Review composition of functions
Identify 1-1 functions
Find the inverse of a 1-1 function
3. Composition of Functions
If f and g are functions, then the composite function, or
composition, of g and f is defined by
The domain of g f is the set of all numbers x in the
domain of f such that f x is in the domain of g.
g f x g f x
4. Composition of Functions (cont.)
Example: Let and .
a) Find
b) Find
2 1f x x
4
1
g x
x
2f g
3g f
5. Composition of Functions (cont.)
Example: Let and .
a) Find
b) Find
2 1f x x
4
1
g x
x
2f g
3g f
4
2 4
2 1
g 2 2 4f g f g f
2 4 1 7
3 2 3 1 7f
4 4 1
7
7 1 8 2
g
6. Composition of Functions (cont.)
Example: Let and .
c) Write as one function.
2 1f x x
4
1
g x
x
f g x
7. Composition of Functions (cont.)
Example: Let and .
c) Write as one function.
2 1f x x
4
1
g x
x
f g x
f g x f g x
4
2 1
1x
8
1
1x
8 1 8 1
1 1 1
x x
x x x
9
1
x
x
8. One-to-One Functions
In a one-to-one function, each x-value corresponds to
only one y-value, and each y-value corresponds to only
one x-value. In a 1-1 function, neither the x nor the y can
repeat.
We can also say that f a = f b implies a = b.
A function is a one-to-one function if, for
elements a and b in the domain of f,
a ≠ b implies f a ≠ f b.
10. One-to-One Functions (cont.)
Example: Decide whether is one-to-one.
We want to show that f a = f b implies that a = b:
Therefore, f is a one-to-one function.
3 7f x x
f a f b
3 7 3 7a b
3 3a b
a b
12. One-to-One Functions (cont.)
Example: Decide whether is one-to-one.
This time, we will try plugging in different values:
Although 3 ≠ ‒3, f 3 does equal f ‒3. This means that
the function is not one-to-one by the definition.
2
2f x x
2
3 3 2 11f
2
3 3 2 11f
13. One-to-One Functions (cont.)
Another way to identify whether a function is one-to-one
is to use the horizontal line test, which says that if any
horizontal line intersects the graph of a function in more
than one point, then the function is not one-to-one.
one-to-one not one-to-one
14. Inverse Functions
Some pairs of one-to-one functions undo one another.
For example, if
and
then (for example)
This is true for any value of x. Therefore, f and g are
called inverses of each other.
8 5f x x
5
8
x
g x
8 10 810 5 5f
85 5 80
8
8
8
105g
15. Inverse Functions (cont.)
More formally:
Let f be a one-to-one function. Then g is the inverse
function of f if
for every x in the domain of g,
and for every x in the domain of f.
f g x x
g f x x
16. Inverse Functions (cont.)
Example: Decide whether g is the inverse function of f .
3
1f x x 3
1g x x
17. Inverse Functions (cont.)
Example: Decide whether g is the inverse function of f .
yes
3
1f x x 3
1g x x
3
3
1 1f g x x
1 1x
x
3 3
1 1g f x x
3 3
x
x
18. Inverse Functions (cont.)
If g is the inverse of a function f , then g is written as f -1
(read “f inverse”).
In our previous example, for , 3
1f x x
1 3
1f x x
19. Finding Inverses
Since the domain of f is the range of f -1 and vice versa, if
a set is one-to-one, then to find the inverse, we simply
exchange the independent and dependent variables.
Example: If the relation is one-to-one, find the inverse of
the function.
2,1 , 1,0 , 0,1 , 1,2 , 2,2F not 1-1
3,1 , 0,2 , 2,3 , 4,0G 1-1
1
1,3 , 2,0 , 3,2 , 0,4G
20. Finding Inverses (cont.)
In the same way we did the example, we can find the
inverse of a function by interchanging the x and y
variables.
To find the equation of the inverse of y = f x:
Determine whether the function is one-to-one.
Replace f x with y if necessary.
Switch x and y.
Solve for y.
Replace y with f -1x.
21. Finding Inverses (cont.)
Example: Decide whether each equation defines a one-
to-one function. If so, find the equation of the inverse.
a) 2 5f x x
22. Finding Inverses (cont.)
Example: Decide whether each equation defines a one-
to-one function. If so, find the equation of the inverse.
a) one-to-one
replace f x with y
interchange x and y
solve for y
replace y with f -1x
2 5f x x
2 5y x
2 5x y
2 5y x
5
2
x
y
1 1 5
2 2
f x x
23. Graphing Inverses
Back in Geometry, when we studied reflections, it turned
out that the pattern for reflecting a figure across the line
y = x was to swap the x- and y-values.
It turns out, if we were to graph our inverse functions,
we would see that the inverse is the reflection of the
original function across the line y = x.
This can give you a way to check your work.
, ,x y y x
24. Graphing Inverses
2 5f x x
3
2f x x
1 1 5
2 2
f x x
1 3
2f x x