This document covers representing functions including objectives like defining functions and related terms, determining if a relation is a function, defining piecewise functions, and representing real-life situations
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
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 functions and its operations such as addition, subtraction, multiplication, division and composition. 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.
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 functions and its operations such as addition, subtraction, multiplication, division and composition. 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 learner's module discusses or talks about the topic of Quadratic Functions. It also discusses what is Quadratic Functions. It also shows how to transform or rewrite the equation f(x)=ax2 + bx + c to f(x)= a(x-h)2 + k. It will also show the different characteristics of Quadratic Functions.
(a) Natural Numbers : N = {1,2,3,4,...}
(b) Whole Numbers : W = {0,1,2,3,4, }
(c) Integer Numbers :
or Z = {...–3,–2,–1, 0,1,2,3, },
Z+ = {1,2,3,....}, Z– = {–1,–2,–3, }
Z0 = {± 1, ± 2, ± 3, }
(d) Rational Numbers :
p
Q = { q ; p, q z, q 0 }
(i) R0 : all real numbers except 0 (Zero).
(j) Imaginary Numbers : C = {i,, }
(k) Prime Numbers :
These are the natural numbers greater than 1 which is divisible by 1 and itself only, called prime numbers.
Ex. 2,3,5,7,11,13,17,19,23,29,31,37,41,...
(l) Even Numbers : E = {0,2,4,6, }
(m) Odd Numbers : O = {1,3,5,7, }
Ex. {1,
Note :
5
, –10, 105,
3
22 20
7 , 3
, 0 ....}
The set of the numbers between any two real numbers is called interval.
(a) Close Interval :
(i) In rational numbers the digits are repeated after decimal.
(ii) 0 (zero) is a rational number.
(e) Irrational numbers: The numbers which are not rational or which can not be written in the form of p/q ,called irrational numbers
Ex. { , ,21/3, 51/4, ,e, }
Note:
(i) In irrational numbers, digits are not repeated after decimal.
(ii) and e are called special irrational quantities.
(iii) is neither a rational number nor a irrational number.
(f) Real Numbers : {x, where x is rational and irrational number}
20
[a,b] = { x, a x b }
(b) Open Interval:
(a, b) or ]a, b[ = { x, a < x < b }
(c) Semi open or semi close interval:
[a,b[ or [a,b) = {x; a x < b}
]a,b] or (a,b] = {x ; a < x b}
Let A and B be two given sets and if each element a A is associated with a unique element b B under a rule f , then this relation is called function.
Here b, is called the image of a and a is called the pre- image of b under f.
Note :
(i) Every element of A should be associated with
Ex. R = { 1,1000, 20/6, ,
, –10, –
,.....}
3
B but vice-versa is not essential.
(g) Positive Real Numbers: R+ = (0,)
(h) Negative Real Numbers : R– = (– ,0)
(ii) Every element of A should be associated with a unique (one and only one) element of but
any element of B can have two or more rela- tions in A.
3.1 Representation of Function :
It can be done by three methods :
(a) By Mapping
(b) By Algebraic Method
(c) In the form of Ordered pairs
(A) Mapping :
It shows the graphical aspect of the relation of the elements of A with the elements of B .
Ex. f1:
f2 :
f3 :
f4 :
In the above given mappings rule f1 and f2
shows a function because each element of A is
associated with a unique element of B. Whereas
f3 and f4 are not function because in f 3, element c is associated with two elements of B, and in f4 , b is not associated with any element
of B, which do not follow the definition of function. In f2, c and d are associated with same element, still it obeys the rule of definition of function because it does not tell that every element of A should be associated with different elements of B.
(B) Algebraic Method :
It shows the relation between the elem
MAT-121 COLLEGE ALGEBRAWritten Assignment 32 points eac.docxjessiehampson
MAT-121: COLLEGE ALGEBRA
Written Assignment 3
2 points each except for 5, 6, 9, 15, 16, which are 4 points each as indicated.
SECTION 3.1
Algebraic
For the following exercise, determine whether the relationship represents y as a function of x. If the relationship represents a function then write the relationship as a function of
x
using
f
as the function.
x+y2=5
Consider the relationship 7n-5m=4.
Write the relationship as a function
n
=
k
(
m
).
Evaluate
k
(
5
).
Solve for
k
(
m
) = 7.
Graphical
Given the following graph
Evaluate
f
(4)
Solve for
f
(x) = 4
Numeric
For the following exercise, determine whether the relationship represents a function.
{(0, 5), (-5, 8), (0, -8)}
For the following exercise, use the function
f
represented in table below. (4 points)
x
-18
-12
-6
0
6
12
18
f(x)
24
17
10
3
-4
-11
-18
Answer the following:
Evaluate
f
(-6).
Solve
f
(
x
) = -11
Evaluate
f
(12)
Solve
f
(
x
) = -18
For the following exercise, evaluate the expressions, given functions
f
,
g
, and
h
:
f(x)=4x+2
; g(x)=7-6x; h(x)=7x2-3x+6
f(-1)g(1)h(0) (4 points)
Real-world applications
The number of cubic yards of compost,
C
, needed to cover a garden with an area of
A
square feet is given by
C
=
h
(
A
).
A garden with an area of 5,000 ft2 requires 25 yd3 of compost. Express this information in terms of the function
h
.
Explain the meaning of the statement
h
(2500) = 12.5.
SECTION 3.2
Algebraic
For the following exercise, find the domain and range of each function and state it using interval notation.
f(x)=9-2x5x+13
Numeric
For the following exercise, given each function
f
, evaluate
f
(3),
f
(-2),
f
(1), and f (0). (4 points)
Real-World Applications
The height,
h,
of a projectile is a function of the time,
t,
it is in the air. The height in meters for
t
seconds is given by the function h(t)= -9.8t2+19.6t. What is the domain of the function? What does the domain mean in the context of the problem?
SECTION 3.3
Algebraic
For the following exercise, find the average rate of change of each function on the interval specified in simplest form.
k(x)=23x+1
on [2, 2+h]
Graphical
For the following exercise, use the graph of each function to
estimate
the intervals on which the function is increasing or decreasing.
For the following exercise, find the average rate of change of each function on the interval specified.
g(x)=3x2-23x3 on [1, 3]
Real-World Applications
Near the surface of the moon, the distance that an object falls is a function of time. It is given by d(t)=1.6t2, where
t
is in seconds and d(t) is in meters. If an object is dropped from a certain height, find the average velocity of the object from t = 2 to t = 5.
SECTION 3.4
Algebraic
For the following exercise, determine the domain for each function in interval notation. (4 points)
f(x)=2x+5 and g(x)=4x+9, find f-g, f+g, fg, and fg
For.
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.
Unit 8 - Information and Communication Technology (Paper I).pdfThiyagu K
This slides describes the basic concepts of ICT, basics of Email, Emerging Technology and Digital Initiatives in Education. This presentations aligns with the UGC Paper I syllabus.
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.
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
Acetabularia Information For Class 9 .docxvaibhavrinwa19
Acetabularia acetabulum is a single-celled green alga that in its vegetative state is morphologically differentiated into a basal rhizoid and an axially elongated stalk, which bears whorls of branching hairs. The single diploid nucleus resides in the rhizoid.
2. Lesson Objectives
At the end of the lesson, the students must be
able to:
• define functions and related terms;
• determine if the given relation represents a
function;
• define piece-wise function; and
• represents real-life situations using functions,
including piece-wise functions.
3. Relation
A relation is a set of ordered pairs. The domain
of a relation is the set of first coordinates. The
range is the set of second coordinates.
Example of Relations
1. {(1, 4), (2, 5), (3, 6), (4, 8)}
2. {(4, 2), (4, -2), (9, 3), (9,3)}
3. {(1, a), (1, b), (1, c), (1,d)}
4. Functions
A function is a relation in which each element of
the domain corresponds to exactly one element
of the range.
Examples of Functions
1. {(1, 4), (2, 5), (3, 6), (4, 8)}
2. {(2, 1), (3, 1), (4, 1), (5,1)}
5. Example 1
Determine if the following relations represent a
function.
1. {(q, 0), (w, 1), (e, 2), (t, 3)}
2. {(-1, -2), (0, -2), (1, -2), (2, -2)}
3. {(1, 0), (1,1), (1, 2), (1, -2)}
4. {(x, 3), (y, 4), (z, 3), (w, 4)}
6. Some Types of Functions
Linear Function
A function f is a linear function if f(x) = mx + b,
where m and b are real numbers, and m and f(x) are
not both equal to zero.
Quadratic Function
A quadratic function is any equation of the form
f(x) = ax2+ bx + c where a, b, and c are real numbers
and a ≠ 0.
7. Some Types of Functions
Constant Function
A linear function f is a constant function if
f(x) = mx + b, where m = 0 and b is any real
number. Thus, f(x) = b.
Identity Function
A linear function f is an identity function if f(x) =
mx + b, where m = 1 and b = 0. Thus, f(x) = x.
8. Some Types of Functions
Absolute Value Function
The function f is an absolute value function if for all
real numbers x,
f(x) = x, for x ≥ 0
–x, for x ≤ 0
Piecewise Function
A piecewise function or a compound function is a
function defined by multiple sub-functions, where
each sub-function applies to a certain interval of
the main function's domain.
9. Example 2
To sell more T-shirts, the class needs to charge a
lower price as indicated in the following table:
The price for which you can sell x printed T-shirts is
called the price function p(x). p(x) represents each
data point in the table.
10. Solution to Example 2
Step 1:
Find the slope m of the line using the slope formula m = y2 – y1 / x2 – x1
Step 2:
Write the linear equation with two variables by substituting the values
of m and (x1, y1) to the formula y – y1 = m(x – x1)—the point-slope form
of a linear equation.
y – y1 = m(x – x1)
y – 540 = −15 (x − 500)
y – 540 = − 15 x + 100
y = − 15 x + 640
y = 640 – 0.2x
Thus, the price function is p(x) = 640 – 0.2x.
11. Example 3
Find the dimensions of the largest rectangular
garden that can be enclosed by 60 m of fencing.
12. Solution to Example 3
Let x and y denote the lengths of the sides of the garden.
Then the area A = xy must be given its maximum value.
Express A in terms of a single variable, either x or y. The
total perimeter is 60 meters.
2x + 2y = 60
x + y = 30
y = 30 – x
Hence,
A = xy
A = x(30 – x)
A= 30x – x2
13. Solution to Example 3
Write this equation in the vertex form by completing the
square.
A = –(x2 – 30x + 225) + 225
A = –(x – 15)2 + 225
The maximum area is 225 square meters.
Since x = 15 (the width) and 30 – x = 15 (the length), the
dimension that gives the maximum area is 15 meters by
15 meters.
14. Example 4
Sketch the graph of the given piecewise
function. What is f(– 4)? What is f(2)?
f(x) = x + 2, if x ≥ 0
–x2+ 2, if x < 0
15. Solution to Example 4
To the right of the y-axis, the graph is a line
that has a slope of 1 and y-intercept of 2. To
the left of the y-axis, the graph of the function
is a parabola that opens downward and whose
vertex is (0, 2).
To sketch the graph of the
function, you can lightly draw
both graphs. Then darken the
portion of the graph that
represents the function.
16. Solution to Example 4
To find the value of the function when x = – 4,
use the second equation.
f(– 4) = – (– 4)2+ 2 = – 16 + 2 = – 14
To find the value of the function when x = 2,
use the first equation.
f(2) = 2 + 2 = 4
17. Exercise A
Determine whether or not each relation is a
function. Give the domain and range of each
relation.
1. {(2, 3), (4, 5), (6, 6)}
2. {(5, 1), (5, 2), (5, 3)}
3. {(6, 7), (6, 8), (7, 7), (7, 8)}
18. Exercise B
Tell whether the function described in each of
the following is a linear function, a constant
function, an identity function, an absolute value
function, ora piecewise function.
1. f(x) = 3x − 7
2. g(x) = 12
3. f(x) = 3, if x > −5
-6, if x < −5
19. Exercise B
Tell whether the function described in each of
the following is a linear function, a constant
function, an identity function, an absolute value
function, or a piecewise function.
4. 5.
20. Exercise C
A zumba instructor charges according to the number of
participants. If there are 15 participants or below, the
instructor charges ₱500.00 for each participant per
month. If the number of participants is between 15 and
30, he charges ₱400.00 for each participant per month. If
there are 30 participants or more, he charges ₱350.00 for
each participant per month.
1. Write the piecewise function that describes what the
instructor charges.
2. Graph the function.