- To find the inverse of a function, rearrange the original function to solve for x in terms of y and then swap the variables x and y.
- For example, if the original function is xy = 3, rearranging gives y = 3/x and swapping the variables gives the inverse function x = 3/y.
- Some functions are their own inverse, meaning applying the inverse function twice returns the original function.
Factor Theorem and Remainder Theorem. Mathematics10 Project under Mrs. Marissa De Ocampo. Prepared by Danielle Diva, Ronalie Mejos, Rafael Vallejos and Mark Lenon Dacir of 10- Einstein. CNSTHS.
Factor Theorem and Remainder Theorem. Mathematics10 Project under Mrs. Marissa De Ocampo. Prepared by Danielle Diva, Ronalie Mejos, Rafael Vallejos and Mark Lenon Dacir of 10- Einstein. CNSTHS.
Graphs of the Sine and Cosine Functions LectureFroyd Wess
More: www.PinoyBIX.org
Lesson Objectives
Able to plot the different Trigonometric Graphs
Graph of Sine Function (y = f(x) = sinx)
Graph of Cosine Function (y = f(x) = cosx)
Define the Maximum and Minimum value in a graph
Generalized Trigonometric Functions
Graphs of y = sinbx
Graphs of y = sin(bx + c)
Could find the Period of Trigonometric Functions
Could find the Amplitude of Trigonometric Functions
Variations in the Trigonometric Functions
First principle, power rule, derivative of constant term, product rule, quotient rule, chain rule, derivatives of trigonometric functions and their inverses, derivatives of exponential functions and natural logarithmic functions, implicit differentiation, parametric differentiation, L'Hopital's rule
This is meant for university students taking either information technology or engineering courses, this course of differentiation, Integration and limits helps you to develop your problem solving skills and other benefits that come along with it.
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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.
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.
Embracing GenAI - A Strategic ImperativePeter 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.
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.
Macroeconomics- Movie Location
This will be used as part of your Personal Professional Portfolio once graded.
Objective:
Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
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.
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.
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
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
2. Module C3
"Certain images and/or photos on this presentation are the copyrighted property of JupiterImages and are being used with
permission under license. These images and/or photos may not be copied or downloaded without permission from JupiterImages"
3. Functions
xy sin=13
+= xy
One-to-one and many-to-one functions
Each value of x maps to
only one value of y . . .
Consider the following graphs
Each value of x maps to
only one value of y . . .
BUT many other x values
map to that y.
and each y is mapped
from only one x.
and
4. Functions
One-to-one and many-to-one functions
is an example
of a one-to-one function
13
+= xy is an example
of a many-to-one
function
xy sin=
xy sin=13
+= xy
Consider the following graphs
and
6. Functions
Here the many-to-one
function is two-to-one
( except at one point ! )
432
−+= xxy 863 23
+−−= xxxy
Other many-to-one functions are:
This is a many-to-one
function even though it is
one-to-one in some
parts.
It’s always called many-
to-one.
7. Functions
1−±= xy
This is not a
function.
Functions cannot be
one-to-many.
We’ve had one-to-one and many-to-one functions,
so what about one-to-many?
One-to-many relationships do exist BUT, by
definition, these are not functions.
is one-to-many since it gives 2 values of y for
all x values greater than 1.
1,1 ≥−±= xxye.g.
So, for a function, we are sure of the y-value for
each value of x. Here we are not sure.
8. Inverse Functions
SUMMARY
13
+= xy
xy sin=
• A one-to-one function
maps each value of x to
one value of y and each
value of y is mapped
from only one x.
e.g. 13
+= xy
• A many-to-one
function maps each
x to one y but some
y-values will be
mapped from more
than one x.
e.g. xy sin=
9. Inverse Functions
42 += xy
Suppose we want to find the value of y when x = 3 if
We can easily see the answer is 10 but let’s write
out the steps using a flow chart.
We have
To find y for any x, we have
3 6 10
To find x for any y value, we reverse the process.
The reverse function “undoes” the effect of the
original and is called the inverse function.
2× 4+
x 2× 4+
x2 42 +x y=
The notation for the inverse of is)(xf )(1
xf −
10. Inverse Functions
2× 4+
x x2 42 +x
42)( += xxfe.g. 1 For , the flow chart is
2
4−x 2÷
4−x x4−
Reversing the process:
Finding an inverse
The inverse function is
2
4
)(1 −
=− x
xfTip: A useful check on the working is to substitute
any number into the original function and calculate y.
Then substitute this new value into the inverse. It
should give the original number.
Notice that we start with x.
Check:
5=
−
2
414
=+ 4)5(2
=−
)(1
f 14
14e.g. If ,5=x 5 =)(f
11. Inverse Functions
The flow chart method of finding an inverse can
be slow and it doesn’t always work so we’ll now use
another method.
e.g. 1 Find the inverse of xxf 34)( −=
Solution:
xy 34 −=
Rearrange ( to find x ):
Let y = the function:
yx −= 43
Swap x and y:
3
4 −
=x
y
12. Inverse Functions
The flow chart method of finding an inverse can
be slow and it doesn’t always work so we’ll now use
another method.
e.g. 1 Find the inverse of xxf 34)( −=
Solution:
xy 34 −=
Rearrange ( to find x ):
Let y = the function:
yx −= 43
3
4 −
=
Swap x and y:
x
y
3
4 −
=
x
y
13. Inverse Functions
The flow chart method of finding an inverse can
be slow and it doesn’t always work so we’ll now use
another method.
e.g. 1 Find the inverse of xxf 34)( −=
Solution:
xy 34 −=
Rearrange ( to find x ):
Let y = the function:
yx −= 43
3
4 −
=
Swap x and y:
x
y
3
4 −
=
x
y
So,
3
4
)(1 x
xf
−
=−
14. Inverse Functions
e.g. 2 Find the inverse function of 1,
1
3
)( ≠
−
= x
x
xf
)(xf
Notice that the domain excludes the value of x
that would make infinite.
15. Inverse Functions
e.g. 2 Find the inverse function of 1,
1
3
)( ≠
−
= x
x
xf
Solution:
Let y = the function:
1−x
3
=y
There are 2 ways to rearrange to find x:
Either:
16. Inverse Functions
Either:
1−x
3
=
y
e.g. 2 Find the inverse function of 1,
1
3
)( ≠
−
= x
x
xf
1−x
There are 2 ways to rearrange to find x:
Solution:
Let y = the function:
3
=y
17. Inverse Functions
1−x
3
=
y
e.g. 2 Find the inverse function of 1,
1
3
)( ≠
−
= x
x
xf
1−x
There are 2 ways to rearrange to find x:
Solution:
Let y = the function:
Swap x and y: 1
3
+=
x
y
1
3
+=⇒
y
x
3
=y
Either:
18. Inverse Functions
or: )1( −xy 3=1−x
3
=
y
e.g. 2 Find the inverse function of 1,
1
3
)( ≠
−
= x
x
xf
1−x
There are 2 ways to rearrange to find x:
Solution:
Let y = the function:
Swap x and y: 1
3
+=
x
y
1
3
+=⇒
y
x
3
=y
Either:
19. Inverse Functions
or: )1( −xy 3=1−x
3
=
y
e.g. 2 Find the inverse function of 1,
1
3
)( ≠
−
= x
x
xf
1−x
There are 2 ways to rearrange to find x:
Solution:
Let y = the function:
Swap x and y: 1
3
+=
x
y
1
3
+=⇒
y
x
3
=y
Swap x
and y: x
x
y
+
=
3
3=− yyx⇒
yyx += 3⇒
y
y
x
+
=
3
⇒
Either:
20. Inverse Functions
So, for 1,
1
3
)( ≠
−
= x
x
xf
x
x
xf
x
xf
+
=+= −− 3
)(1
3
)( 11
or
Why are these the same?
ANS: x is a common denominator in the 2nd
form
21. Inverse Functions
So, for 1,
1
3
)( ≠
−
= x
x
xf
x
x
xf
x
xf
+
=+= −− 3
)(1
3
)( 11
or
The domain and range are:
)(0 1
xfx −
≠ and 1≠
22. Inverse Functions
The 1st
example we did was for xxf 34)( −=
The inverse was
3
4
)(1 x
xf
−
=−
Suppose we form the compound
function .
)(1
xff −
== −−
))(()( 11
xffxff
3
)34(4 x−−
3
344 x+−
=
x=)(1
xff −
⇒
Can you see why this is true for all functions that
have an inverse?
ANS: The inverse undoes what the function has done.
23. Inverse Functions
xxffxff == −−
)()( 11
The order in which we find the compound function
of a function and its inverse makes no difference.
For all functions which have an inverse,)(xf
24. Inverse Functions
Exercise
Find the inverses of the following functions:
,2)( xxf −= 0≥x
2.
3. 5,
5
2
)( −≠
+
= x
x
xf
,45)( −= xxf1. ∈x
,
1
)(
x
xf = 0≠x
4.
See if you spot
something special about
the answer to this one.
Also, for this, show
xxff =−
)(1
25. Inverse Functions
Rearrange:
Swap x and y:
Let 45 −= xy
xy 54 =+
x
y
=
+
5
4
y
x
=
+
5
4
Since the x-term is
positive I’m going to work
from right to left.
So,
5
4
)(1 +
=− x
xf
Solution: 1. ∈x ,45)( −= xxf
26. Inverse Functions
This is an example
of a self-inverse
function.
Solution: 2. 0≠x,
1
)(
x
xf =
Let
x
y
1
=
Rearrange:
y
x
1
=
Swap x and y:
x
y
1
=
So, ,
1
)(1
x
xf =−
0≠x
)()(1
xfxf =−
27. Inverse Functions
5,
5
2
)( −≠
+
= x
x
xfSolution: 3.
Rearrange:
Swap x and y:
Let
5
2
+
=
x
y
y
x
2
5 =+
5
2
−=
y
x
5
2
−=
x
y
0,5
2
)(1
≠−=−
x
x
xfSo,
29. Inverse Functions
e.g. 3 Find the inverse of 1,
1
32
)( ≠
−
+
= x
x
x
xf
Solution:
Rearrange:
Let y = the function:
Multiply by x – 1 :
Careful! We are trying to find x and it appears
twice in the equation.
32)1( +=−y x x
1
32
−
+
=
x
x
yThe next example is more difficult to rearrange
30. Inverse Functions
32)1( +=−y x x
Careful! We are trying to find x and it appears
twice in the equation.
e.g. 3 Find the inverse of 1,
1
32
)( ≠
−
+
= x
x
x
xf
Solution:
Rearrange:
Multiply by x – 1 :
We must get both x-terms on one side.
Let y = the function:
1
32
−
+
=
x
x
y
31. Inverse Functions
x
2
3
−
+
=
y
y
x 3)2( +=− yy
32 +=− yyx x
32 +=− yyx x
32)1( +=−y x x
e.g. 3 Find the inverse of 1,
1
32
)( ≠
−
+
= x
x
x
xf
Solution:
Multiply by x – 1 :
Remove brackets :
Collect x terms on one side:
Remove the common factor:
Divide by ( y – 2):
Let y = the function:
1
32
−
+
=
x
x
y
Rearrange:
Swap x and y:
32. Inverse Functions
x
2
3
−
+
=
y
y
x 3)2( +=− yy
32 +=− yyx x
32 +=− yyx x
32)1( +=−y x x
e.g. 3 Find the inverse of 1,
1
32
)( ≠
−
+
= x
x
x
xf
Solution:
Multiply by x – 1 :
Remove brackets :
Collect x terms on one side:
Swap x and y:
Remove the common factor:
Divide by ( y – 2):
Let y = the function:
1
32
−
+
=
x
x
y
Rearrange:
So, ,
2
3
)(1
−
+
=−
x
x
xf 2≠x
2
3
−
+
=
x
y
x
33. Inverse Functions
SUMMARY
To find an inverse function:
EITHER:
• Write the given function as a flow chart.
• Reverse all the steps of the flow chart.
OR:
• Step 2: Rearrange ( to find x )
• Step 1: Let y = the function
• Step 3: Swap x and y