* 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.
We define the definite integral as a limit of Riemann sums, compute some approximations, then investigate the basic additive and comparative properties
* 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.
We define the definite integral as a limit of Riemann sums, compute some approximations, then investigate the basic additive and comparative properties
Functions Representations
CMSC 56 | Discrete Mathematical Structure for Computer Science
October 13, 2018
Instructor: Allyn Joy D. Calcaben
College of Arts & Sciences
University of the Philippines Visayas
Properties of Functions
Odd and Even Functions
Periodic Functions
Monotonic Functions
Bounded Functions
Maxima and Minima of Functions
Inverse Function
Sequence and Series
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.
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!
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.
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.
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?
A review of the growth of the Israel Genealogy Research Association Database Collection for the last 12 months. Our collection is now passed the 3 million mark and still growing. See which archives have contributed the most. See the different types of records we have, and which years have had records added. You can also see what we have for the future.
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.
A workshop hosted by the South African Journal of Science aimed at postgraduate students and early career researchers with little or no experience in writing and publishing journal articles.
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
June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...
5.1 Defining and visualizing functions. Dynamic slides.
1. Introduction to set theory and to methodology and philosophy of
mathematics and computer programming
Defining and visualizing functions
An overview
by Jan Plaza
c 2017 Jan Plaza
Use under the Creative Commons Attribution 4.0 International License
Version of November 4, 2017
2. Existence and uniqueness
How to express in first-order logic “there exists unique y s.t. A(y)”
without using the abbreviation ∃!y A(y) ?
3. Existence and uniqueness
How to express in first-order logic “there exists unique y s.t. A(y)”
without using the abbreviation ∃!y A(y) ?
Existence: ∃y A(y).
Uniqueness: ∀y1,y2 (A(y1) ∧ A(y2) → y1 =y2).
4. Existence and uniqueness; Functions
How to express in first-order logic “there exists unique y s.t. A(y)”
without using the abbreviation ∃!y A(y) ?
Existence: ∃y A(y).
Uniqueness: ∀y1,y2 (A(y1) ∧ A(y2) → y1 =y2).
Definition
A function is any binary relation f such that ∀x ∀y1,y2 (xfy1 ∧ xfy2 → y1 =y2),
or equivalently, ∀x (∃y xfy → ∀y1,y2 (xfy1 ∧ xfy2 → y1 =y2)).
5. Existence and uniqueness; Functions
How to express in first-order logic “there exists unique y s.t. A(y)”
without using the abbreviation ∃!y A(y) ?
Existence: ∃y A(y).
Uniqueness: ∀y1,y2 (A(y1) ∧ A(y2) → y1 =y2).
Definition
A function is any binary relation f such that ∀x ∀y1,y2 (xfy1 ∧ xfy2 → y1 =y2),
or equivalently, ∀x (∃y xfy → ∀y1,y2 (xfy1 ∧ xfy2 → y1 =y2)).
This means: if there is y s.t. xfy then such a y is unique.
6. Existence and uniqueness; Functions
How to express in first-order logic “there exists unique y s.t. A(y)”
without using the abbreviation ∃!y A(y) ?
Existence: ∃y A(y).
Uniqueness: ∀y1,y2 (A(y1) ∧ A(y2) → y1 =y2).
Definition
A function is any binary relation f such that ∀x ∀y1,y2 (xfy1 ∧ xfy2 → y1 =y2),
or equivalently, ∀x (∃y xfy → ∀y1,y2 (xfy1 ∧ xfy2 → y1 =y2)).
This means: if there is y s.t. xfy then such a y is unique.
Definition. Let f be a function.
1. Let x ∈ domain(f). The value of f at x , f(x) , is the unique y such that xfy.
2. f maps x to y , denoted f : x → y or x
f
→ y , if xfy.
23. Terminology: total/partial, on
Definition. Let f be a function.
f is on X or f is a (total) function on X if domain(f) = X.
Definition. f is a partial function on X if there is X ⊆X s.t. f is a function onX .
Fact. If f is a partial function on X, then f is a total function on domain(f).
If f is a function and X ⊇ domain(f), then f is a partial function on X.
Fact. Consider these conditions:
1. f is a binary relation,
2. domain(f) ⊆ X,
3. for every x ∈ X, f maps x to at most one value,
4. for every x ∈ X, f maps x to least one value.
Then:
f is a function on X iff f satisfies conditions 1-4;
f is a partial function on X iff f satisfies conditions 1-3.
24. Vertical line tests on the Cartesian plane are the following.
Let G be a subset of the Cartesian plane.
G is the graph of a function from R to R
iff every straight line parallel to the y axis intersects G in exactly one point.
G is the graph of a partial function from R to R
iff every straight line parallel to the y axis intersects G in at most one point.
Vertical line tests in discrete Cartesian diagrams are the following.
Let G be a subset of X × Y in a discrete Cartesian diagram.
G is the graph of a function from X to Y
iff every column in the diagram contains exactly one point of G.
G is the graph of a partial function from X to Y
iff every column in the diagram contains at most one point of G.
25. Terminology: from, to
Definition. Let f be a function.
1. f is to/into Y if range(f) ⊆ Y .
2. f is (a total function) from X to Y , denoted f : X −→ Y or X
f
−→ Y ,
if domain(f)=X and range(f) ⊆ Y .
Definition
f is a partial function from X to Y , denoted f : X −→ Y or X
f
−→ Y ,
if there exists X ⊆ X such that f : X −→ Y .
f : X −→ Y f : X −→ Y
domain(f) ⊆ X domain(f) = X
range(f) ⊆ Y range(f) ⊆ Y
Our definitions imply that the term “to/into Y ” applies to partial functions as well:
for any partial function f, f is to/into Y iff range(f) ⊆ Y .
29. Exercise
Is this a function on R? No.
Is this a partial function on R? Yes.
30. Exercise
Is this a function on R? No.
Is this a partial function on R? Yes.
What is the domain of this function?
31. Exercise
Is this a function on R? No.
Is this a partial function on R? Yes.
What is the domain of this function? R − {0}.
32. Exercise
Is this a function on R? No.
Is this a partial function on R? Yes.
What is the domain of this function? R − {0}.
Is this a function on R − {0}?
33. Exercise
Is this a function on R? No.
Is this a partial function on R? Yes.
What is the domain of this function? R − {0}.
Is this a function on R − {0}? Yes.
34. Exercise
Is this a function on R? No.
Is this a partial function on R? Yes.
What is the domain of this function? R − {0}.
Is this a function on R − {0}? Yes.
Is it correct to say f : R −→ R?
35. Exercise
Is this a function on R? No.
Is this a partial function on R? Yes.
What is the domain of this function? R − {0}.
Is this a function on R − {0}? Yes.
Is it correct to say f : R −→ R? No.
36. Exercise
Is this a function on R? No.
Is this a partial function on R? Yes.
What is the domain of this function? R − {0}.
Is this a function on R − {0}? Yes.
Is it correct to say f : R −→ R? No.
Is it correct to say f : R − {0} −→ R?
37. Exercise
Is this a function on R? No.
Is this a partial function on R? Yes.
What is the domain of this function? R − {0}.
Is this a function on R − {0}? Yes.
Is it correct to say f : R −→ R? No.
Is it correct to say f : R − {0} −→ R? Yes.
38. Exercise
Is this a function on R? No.
Is this a partial function on R? Yes.
What is the domain of this function? R − {0}.
Is this a function on R − {0}? Yes.
Is it correct to say f : R −→ R? No.
Is it correct to say f : R − {0} −→ R? Yes.
Is it correct to say f : R −→ R?
39. Exercise
Is this a function on R? No.
Is this a partial function on R? Yes.
What is the domain of this function? R − {0}.
Is this a function on R − {0}? Yes.
Is it correct to say f : R −→ R? No.
Is it correct to say f : R − {0} −→ R? Yes.
Is it correct to say f : R −→ R? Yes.
40. Exercise
Is this a function on R? No.
Is this a partial function on R? Yes.
What is the domain of this function? R − {0}.
Is this a function on R − {0}? Yes.
Is it correct to say f : R −→ R? No.
Is it correct to say f : R − {0} −→ R? Yes.
Is it correct to say f : R −→ R? Yes.
Is it correct to say f : R − {0} −→ R?
41. Exercise
Is this a function on R? No.
Is this a partial function on R? Yes.
What is the domain of this function? R − {0}.
Is this a function on R − {0}? Yes.
Is it correct to say f : R −→ R? No.
Is it correct to say f : R − {0} −→ R? Yes.
Is it correct to say f : R −→ R? Yes.
Is it correct to say f : R − {0} −→ R? Yes.
42. Example
1. Let f ={ 0, −1 , 1, 0 , 2, 3 }.
f is a function from {0, 1, 2} to Z.
f is also a function from {0, 1, 2} to {−1, 0, 3}.
2. Alternatively we could specify it as:
a function f on {0, 1, 2} s.t. f(0)=−1, f(1)=0, f(2)=3.
3. Alternatively we could specify it as:
a function f on {0, 1, 2} s.t. f : 0 → −1, f : 1 → 0, f : 2 → 3.
4. Alternatively we could specify it as:
a function f on {0, 1, 2} s.t. f(x)=x2 − 1.
5. Let g be a function on {0, 1, 2, 3} s.t. g(x)=x2 − 1.
Although g and f are defined by the same formula,
they are different functions
because they have different domains.
43. Example
1. Expression y =
√
1 − x2 does not define a function on R,
but it defines a function on [−1, 1].
2. Expression y = ±
√
1 − x2 specifies coordinates of points of a unit circle,
however it does not define a function on [−1, 1].
44. Exercise
Consider the formula y =
√
x.
(Recall that for x = 9 we have just y = 3,
because the definition of square root requires it to be non-negative.)
1. Does this formula define a function from R to R?
45. Exercise
Consider the formula y =
√
x.
(Recall that for x = 9 we have just y = 3,
because the definition of square root requires it to be non-negative.)
1. Does this formula define a function from R to R? No.
46. Exercise
Consider the formula y =
√
x.
(Recall that for x = 9 we have just y = 3,
because the definition of square root requires it to be non-negative.)
1. Does this formula define a function from R to R? No.
2. Does this formula define a function from {u ∈ R : u 0} to R?
47. Exercise
Consider the formula y =
√
x.
(Recall that for x = 9 we have just y = 3,
because the definition of square root requires it to be non-negative.)
1. Does this formula define a function from R to R? No.
2. Does this formula define a function from {u ∈ R : u 0} to R? Yes.
48. Exercise
Consider the formula y =
√
x.
(Recall that for x = 9 we have just y = 3,
because the definition of square root requires it to be non-negative.)
1. Does this formula define a function from R to R? No.
2. Does this formula define a function from {u ∈ R : u 0} to R? Yes.
3. Does this formula define a partial function from R to R?
49. Exercise
Consider the formula y =
√
x.
(Recall that for x = 9 we have just y = 3,
because the definition of square root requires it to be non-negative.)
1. Does this formula define a function from R to R? No.
2. Does this formula define a function from {u ∈ R : u 0} to R? Yes.
3. Does this formula define a partial function from R to R? Yes.
50. Exercise
Consider the formula y =
√
x.
(Recall that for x = 9 we have just y = 3,
because the definition of square root requires it to be non-negative.)
1. Does this formula define a function from R to R? No.
2. Does this formula define a function from {u ∈ R : u 0} to R? Yes.
3. Does this formula define a partial function from R to R? Yes.
4. Does this formula define a function from {u ∈ R : u 0} to Q?
51. Exercise
Consider the formula y =
√
x.
(Recall that for x = 9 we have just y = 3,
because the definition of square root requires it to be non-negative.)
1. Does this formula define a function from R to R? No.
2. Does this formula define a function from {u ∈ R : u 0} to R? Yes.
3. Does this formula define a partial function from R to R? Yes.
4. Does this formula define a function from {u ∈ R : u 0} to Q? No.
52. Exercise
Consider the formula y =
√
x.
(Recall that for x = 9 we have just y = 3,
because the definition of square root requires it to be non-negative.)
1. Does this formula define a function from R to R? No.
2. Does this formula define a function from {u ∈ R : u 0} to R? Yes.
3. Does this formula define a partial function from R to R? Yes.
4. Does this formula define a function from {u ∈ R : u 0} to Q? No.
5. Does this formula define a function from {u ∈ R : u 0} to {u ∈ R : u 0}?
53. Exercise
Consider the formula y =
√
x.
(Recall that for x = 9 we have just y = 3,
because the definition of square root requires it to be non-negative.)
1. Does this formula define a function from R to R? No.
2. Does this formula define a function from {u ∈ R : u 0} to R? Yes.
3. Does this formula define a partial function from R to R? Yes.
4. Does this formula define a function from {u ∈ R : u 0} to Q? No.
5. Does this formula define a function from {u ∈ R : u 0} to {u ∈ R : u 0}?
Yes.
54. Exercise
Are the following statements correct?
1. Let f1, f2 be functions.
If domain(f1) = domain(f2) and f1 ⊆ f2, then f1 = f2.
55. Exercise
Are the following statements correct?
1. Let f1, f2 be functions.
If domain(f1) = domain(f2) and f1 ⊆ f2, then f1 = f2. Yes.
56. Exercise
Are the following statements correct?
1. Let f1, f2 be functions.
If domain(f1) = domain(f2) and f1 ⊆ f2, then f1 = f2. Yes.
2. Let R1, R2 be binary relations.
If domain(R1) = domain(R2) and R1 ⊆ R2, then R1 = R2.
57. Exercise
Are the following statements correct?
1. Let f1, f2 be functions.
If domain(f1) = domain(f2) and f1 ⊆ f2, then f1 = f2. Yes.
2. Let R1, R2 be binary relations.
If domain(R1) = domain(R2) and R1 ⊆ R2, then R1 = R2. No.
58. Exercise
Are the following statements correct?
1. Let f1, f2 be functions.
If domain(f1) = domain(f2) and f1 ⊆ f2, then f1 = f2. Yes.
2. Let R1, R2 be binary relations.
If domain(R1) = domain(R2) and R1 ⊆ R2, then R1 = R2. No.
3. Let f1, f2 be partial functions on the same set X.
If f1 ⊆ f2, then f1 = f2.
59. Exercise
Are the following statements correct?
1. Let f1, f2 be functions.
If domain(f1) = domain(f2) and f1 ⊆ f2, then f1 = f2. Yes.
2. Let R1, R2 be binary relations.
If domain(R1) = domain(R2) and R1 ⊆ R2, then R1 = R2. No.
3. Let f1, f2 be partial functions on the same set X.
If f1 ⊆ f2, then f1 = f2. No.
Whenever the answer is negative, provide a counter-example.
60. Note
1. An attempt to define f : Q −→ Z:
f(m
n ) = m + n, where m, n ∈ Z.
This is not correct, because
1
2 = 2
4 but f(1
2) = 1 + 2 = 3=6 = 2 + 4 = f(2
4)
violating ∀x,y1,y2 (xfy1 ∧ xfy2 → y1 =y2).
2. An attempt to define g : Q −→ Z:
let g(0)=1 and
let g(m
n ) = m + n where m, n ∈ Z, m=0, n > 0,
and m and n do not have a common divisor greater than 1.
This is correct.
(We have used a canonical form of rational numbers.)
3. Every rational number has many representations (1
2 = 2
4 = ...).
To be correct, the definition must be
independent of the representation .
61. Exercise
Let Q+ be the set of all positive rational numbers.
Are these attempts to define a function correct?
1. f : Q −→ Q s.t. f(m
n ) = n
m where m, n ∈ Z.
62. Exercise
Let Q+ be the set of all positive rational numbers.
Are these attempts to define a function correct?
1. f : Q −→ Q s.t. f(m
n ) = n
m where m, n ∈ Z. No.
63. Exercise
Let Q+ be the set of all positive rational numbers.
Are these attempts to define a function correct?
1. f : Q −→ Q s.t. f(m
n ) = n
m where m, n ∈ Z. No.
2. f : Q+ −→ Q+ s.t. f(m
n ) = n
m where m, n ∈ Z.
64. Exercise
Let Q+ be the set of all positive rational numbers.
Are these attempts to define a function correct?
1. f : Q −→ Q s.t. f(m
n ) = n
m where m, n ∈ Z. No.
2. f : Q+ −→ Q+ s.t. f(m
n ) = n
m where m, n ∈ Z. Yes.
Show this by proving:
if m1, n1, m2, n2 ∈ Z and m1
n1
, m2
n2
∈ Q+ and m1
n1
= m2
n2
then n1
m1
= n2
m2
.
65. Note. An attempt to to model the concept “mother of”:
1. Persons is a non-empty set,
2. Persons is a finite set,
3. motherOf : Persons −→ Persons
(i.e. every person has a unique mother, who is a person),
4. motherOf is a function such that for every p in its domain:
p=motherOf(p),
p=motherOf(motherOf(p)),
p=motherOf(motherOf(motherOf(p))),
etc. (i.e. there are no cycles).
These conditions are contradictory.
(The same problem occurs with theSupervisorOf in employee databases.)
To see why, try drawing directed graphs
showing a function motherOf on a three-element set Persons.