This document defines and lists several common parent functions including: constant, linear, quadratic, cubic, absolute value, greatest integer, square root, cube root, exponential, logarithmic, reciprocal, rational, and trigonometric functions. The parent functions are basic building blocks used to model real world phenomena through transformations and combinations.
The document provides exercises on composition of functions. It gives the definitions of various functions f(x) and g(x) and asks to calculate f(x)+g(x), f(x)-g(x), f(x)*g(x), fog(x), gof(x), fof(x), and gog(x) for different functions f(x) and g(x). It provides 30 problems to calculate the composition of the given functions through addition, subtraction, multiplication and composition of functions.
This document discusses graphing absolute value functions. It provides examples of graphing various absolute value functions, including f(x)=|x|^2, f(x)=2|x|-1, f(x)=|x|^2-1, f(x)=3|x|^2-4, f(x)=|cos(x)|, f(x)=2|x|-3-1, and shows how to write a piecewise function definition for f(x)=|x|^2. The graphs are V-shaped and symmetric about the y-axis, with vertices at the points where the absolute value terms are equal to zero.
Composite functions refer to combining two functions, where the output of the inner function is used as the input of the outer function. This is denoted as f(g(x)), where g(x) is evaluated first, and then substituted into f(x). Examples using tables and graphs are provided to demonstrate evaluating composite functions.
This document discusses how to find the sum and differences of functions. The sum of two functions can be found by adding the y-coordinates of each function. For example, if f(x) = 2x + 3 and g(x) = x^2 - x - 5, then h(x) = (f + g)x is found by adding the y-coordinates of f(x) and g(x). The difference of two functions is found by subtracting the y-coordinates.
This document discusses graphing composite functions. It provides examples of composing two functions f(x) and g(x), such as finding (f ? g)(x) and (g ? f)(x), and graphing the resulting composite functions. The document emphasizes determining the domains of the composite functions by considering the restrictions on the domains of the original functions. It also gives examples of finding equations for composite functions f(g(x)) and g(f(x)) and stating their domains and ranges.
Composite functions refer to combining two functions where the output of one acts as the input of the other. The notation used is (f ? g)(x) which reads "f composed with g of x", where the inner function g(x) is evaluated first, then substituted into the outer function f(x). Examples show how to evaluate composite functions by first substituting the inner function and then evaluating the outer function.
The document defines inverse functions and provides examples. An inverse function f-1(x) undoes the original function f(x) so that f-1(f(x)) = x. For a function to have an inverse, it must be one-to-one meaning each output of f(x) corresponds to only one input x. The document gives examples of linear functions that are invertible and the function y=x2 that is not invertible because it is not one-to-one. It also states that if a function f(x) is one-to-one on its domain, then it has an inverse function and the domain of f(x) is equal to the range of the
A function transforms inputs (domain) to outputs (range). The inverse of a function reverses the inputs and outputs, so the domain of the inverse is the original function's range, and the range of the inverse is the original function's domain. To have an inverse, a function must be one-to-one, meaning each input maps to a single unique output. The graph of an inverse is a reflection of the original function's graph across the line y=x.
This document introduces function notation and how to evaluate functions. It explains that f(x) represents the output of the function f for a given input x. Examples show different functions defined using this notation, such as f(x) = 2x + 1. The document also demonstrates how to evaluate functions for given x-values, find x-values for which a function is a certain amount, and graph functions using this notation. It compares translating and shifting graphs of functions.
The function F(x) = 2x + 3 is a linear function that increases at a rate of 2 units for every 1 unit increase in x, with a y-intercept of 3. The function f(x) = x^2 + 2x - 3 is a quadratic function that is symmetric around the y-axis, with its vertex at the point (-1,0) and intersecting the x-axis at x = -3 and x = 0.
1) A quadratic function is an equation of the form f(x) = ax^2 + bx + c, where a ¡Ù 0. Its graph is a parabola.
2) The vertex of a parabola is the point where it intersects its axis of symmetry. If a > 0, the parabola opens upward and the vertex is a minimum. If a < 0, it opens downward and the vertex is a maximum.
3) The standard form of a quadratic equation is f(x) = a(x - h)^2 + k, where the vertex is (h, k) and the axis of symmetry is x = h.
The document proves that map f (map g xs) = map (f . g) xs for any functions f and g and list xs. It does this by:
1) Starting from the base case that mapping an empty list results in an empty list
2) Assuming map f (map g xs) = map (f . g) xs is true for some list xs
3) Showing through substitutions and definitions that this holds true for lists of the form x:xs as well
4) Concluding that the original statement is true for all lists xs
The document summarizes key concepts about composition functions:
1. Composition functions are not commutative.
2. They are associative.
3. Identity functions leave other functions unchanged when composed.
The document also provides examples of determining the component functions of composition functions and finding inverse functions.
The document discusses inverse functions. It defines an inverse function as switching the x's and y's in a function's ordered pairs. If this inversion results in another function, it is called the inverse function. The domain of the original function becomes the range of the inverse function, and vice versa. It provides examples of finding the expressions for inverse functions and checking if they are correct inverses. It also discusses the differences between one-to-one and many-to-one functions, and uses the horizontal line test to determine which is being graphed. Exercises are provided to find specific inverse functions.
1. The document discusses function notation, evaluating functions, and finding the domain from the equation of a function.
2. It provides examples of different types of function notation including f(x)=x^2+1 and {(x,y)|y=x^2+1}.
3. Evaluating a function involves substitution, and the document gives an example of evaluating f(x)=x^2+3x-1 when x=-2.
4. To find the domain from the equation, one looks at what type of equation it is such as linear, absolute value, quadratic, radical, fractional, or other. The domain is the set of inputs that can be substituted into the
The document provides examples of evaluating composite functions. It gives the steps to find (f ? g)(x) by substituting the expression for g(x) into f(x) and simplifying. Examples are provided of finding (f ? g)(x) and (g ? f)(x) for various functions f(x) and g(x), as well as evaluating composite functions at given values.
This study guide covers topics in precalculus including:
1) Examples of natural numbers, integers, rational numbers, and real numbers.
2) Composition of functions including finding f(g(x)) and g(f(x)) for given functions f(x) and g(x).
3) Finding inverses of functions and verifying inverses by composition.
4) Operations on complex numbers including plotting, finding modulus, distance, midpoint, addition, subtraction, multiplication, and division.
5) Graphing functions, classifying function types, and stating domains and ranges.
6) Finding limits of functions as x approaches values.
This document provides an overview of functions from chapter 1 of an additional mathematics module. It defines key terms like domain, codomain, range, and discusses different types of relations including one-to-one, many-to-one, and many-to-many. It also covers function notation, evaluating functions, composite functions, and provides examples of calculating images and objects of functions. The chapter aims to introduce students to the fundamental concepts of functions through definitions, diagrams, and practice exercises.
1) The document discusses the generalization of derivatives from single-variable calculus to multivariable calculus. In multivariable calculus, a function f(x,y) is differentiable at a point (a,b) if the graph near that point is indistinguishable from a plane.
2) The derivative of a function of two variables f(x,y) at a point (a,b) is represented by the Jacobian matrix Jf(a,b), which can be written as the gradient vector grad f(a,b).
3) The directional derivative of f(x,y) at (a,b) in the direction of a unit vector u is
This document contains solutions to mathematical problems involving functions. It defines several functions and solves for their domains, ranges, and other properties. Some key points extracted:
1) It defines functions for the areas of isosceles triangles and spheres in terms of their variables.
2) It analyzes properties of various functions like whether they are injective, surjective, or both.
3) It finds the domains and ranges of multiple functions by solving equations or looking at discontinuities.
1. The document discusses composite functions, which involve combining two functions f(x) and g(x) where the output of one is used as the input of the other. It provides examples of evaluating composite functions using tables and graphs.
2. Key steps for evaluating composite functions are: 1) Substitute the inner function into the outer function and 2) Simplify the expression. Order matters as f(g(x)) and g(f(x)) may have different values.
3. Examples are worked through to find composite functions given basic functions like f(x) = x + 1 and g(x) = 2x as well as more complex rational functions.
The document discusses continuity of functions. A function is continuous at a point c if the limit of the function as x approaches c exists and equals the value of the function at c. Polynomial and rational functions are continuous everywhere in their domains. A function can have discontinuities where it is not defined or where the limit does not equal the function value. Discontinuities are either removable, where redefining the function can make it continuous, or non-removable, where an asymptote prevents continuity. A function is continuous on a closed interval if it is continuous on the interior and the one-sided limits equal the function values at the endpoints.
This document provides definitions and properties for trigonometric functions including sine, cosine, and tangent. It defines the domains and ranges of sine, cosine, and tangent. Examples of trigonometric ratios are given for common angles like 30, 45, 60, and 90 degrees. Trigonometric identities are also listed, such as the sine and cosine of sums and differences of angles.
1. The document discusses functions and their notation. It defines functions as mappings from inputs to outputs and provides examples of function notation.
2. It also discusses different types of relations between inputs and outputs such as one-to-one, one-to-many, many-to-one, and many-to-many relations.
3. The document explains concepts related to functions such as composite functions, inverse functions, and how to determine composite and inverse functions through examples.
This document discusses continuity of functions, including definitions of continuity at a point and on an interval. It introduces the intermediate value theorem and extreme value theorem for continuous functions on closed intervals. It also states that differentiability at a point implies continuity at that point.
This includes the overall cultivation practices of Rose prepared by:
Kushal Lamichhane (AKL)
Instructor
Shree Gandhi Adarsha Secondary School
Kageshowri Manohara-09, Kathmandu, Nepal
A function transforms inputs (domain) to outputs (range). The inverse of a function reverses the inputs and outputs, so the domain of the inverse is the original function's range, and the range of the inverse is the original function's domain. To have an inverse, a function must be one-to-one, meaning each input maps to a single unique output. The graph of an inverse is a reflection of the original function's graph across the line y=x.
This document introduces function notation and how to evaluate functions. It explains that f(x) represents the output of the function f for a given input x. Examples show different functions defined using this notation, such as f(x) = 2x + 1. The document also demonstrates how to evaluate functions for given x-values, find x-values for which a function is a certain amount, and graph functions using this notation. It compares translating and shifting graphs of functions.
The function F(x) = 2x + 3 is a linear function that increases at a rate of 2 units for every 1 unit increase in x, with a y-intercept of 3. The function f(x) = x^2 + 2x - 3 is a quadratic function that is symmetric around the y-axis, with its vertex at the point (-1,0) and intersecting the x-axis at x = -3 and x = 0.
1) A quadratic function is an equation of the form f(x) = ax^2 + bx + c, where a ¡Ù 0. Its graph is a parabola.
2) The vertex of a parabola is the point where it intersects its axis of symmetry. If a > 0, the parabola opens upward and the vertex is a minimum. If a < 0, it opens downward and the vertex is a maximum.
3) The standard form of a quadratic equation is f(x) = a(x - h)^2 + k, where the vertex is (h, k) and the axis of symmetry is x = h.
The document proves that map f (map g xs) = map (f . g) xs for any functions f and g and list xs. It does this by:
1) Starting from the base case that mapping an empty list results in an empty list
2) Assuming map f (map g xs) = map (f . g) xs is true for some list xs
3) Showing through substitutions and definitions that this holds true for lists of the form x:xs as well
4) Concluding that the original statement is true for all lists xs
The document summarizes key concepts about composition functions:
1. Composition functions are not commutative.
2. They are associative.
3. Identity functions leave other functions unchanged when composed.
The document also provides examples of determining the component functions of composition functions and finding inverse functions.
The document discusses inverse functions. It defines an inverse function as switching the x's and y's in a function's ordered pairs. If this inversion results in another function, it is called the inverse function. The domain of the original function becomes the range of the inverse function, and vice versa. It provides examples of finding the expressions for inverse functions and checking if they are correct inverses. It also discusses the differences between one-to-one and many-to-one functions, and uses the horizontal line test to determine which is being graphed. Exercises are provided to find specific inverse functions.
1. The document discusses function notation, evaluating functions, and finding the domain from the equation of a function.
2. It provides examples of different types of function notation including f(x)=x^2+1 and {(x,y)|y=x^2+1}.
3. Evaluating a function involves substitution, and the document gives an example of evaluating f(x)=x^2+3x-1 when x=-2.
4. To find the domain from the equation, one looks at what type of equation it is such as linear, absolute value, quadratic, radical, fractional, or other. The domain is the set of inputs that can be substituted into the
The document provides examples of evaluating composite functions. It gives the steps to find (f ? g)(x) by substituting the expression for g(x) into f(x) and simplifying. Examples are provided of finding (f ? g)(x) and (g ? f)(x) for various functions f(x) and g(x), as well as evaluating composite functions at given values.
This study guide covers topics in precalculus including:
1) Examples of natural numbers, integers, rational numbers, and real numbers.
2) Composition of functions including finding f(g(x)) and g(f(x)) for given functions f(x) and g(x).
3) Finding inverses of functions and verifying inverses by composition.
4) Operations on complex numbers including plotting, finding modulus, distance, midpoint, addition, subtraction, multiplication, and division.
5) Graphing functions, classifying function types, and stating domains and ranges.
6) Finding limits of functions as x approaches values.
This document provides an overview of functions from chapter 1 of an additional mathematics module. It defines key terms like domain, codomain, range, and discusses different types of relations including one-to-one, many-to-one, and many-to-many. It also covers function notation, evaluating functions, composite functions, and provides examples of calculating images and objects of functions. The chapter aims to introduce students to the fundamental concepts of functions through definitions, diagrams, and practice exercises.
1) The document discusses the generalization of derivatives from single-variable calculus to multivariable calculus. In multivariable calculus, a function f(x,y) is differentiable at a point (a,b) if the graph near that point is indistinguishable from a plane.
2) The derivative of a function of two variables f(x,y) at a point (a,b) is represented by the Jacobian matrix Jf(a,b), which can be written as the gradient vector grad f(a,b).
3) The directional derivative of f(x,y) at (a,b) in the direction of a unit vector u is
This document contains solutions to mathematical problems involving functions. It defines several functions and solves for their domains, ranges, and other properties. Some key points extracted:
1) It defines functions for the areas of isosceles triangles and spheres in terms of their variables.
2) It analyzes properties of various functions like whether they are injective, surjective, or both.
3) It finds the domains and ranges of multiple functions by solving equations or looking at discontinuities.
1. The document discusses composite functions, which involve combining two functions f(x) and g(x) where the output of one is used as the input of the other. It provides examples of evaluating composite functions using tables and graphs.
2. Key steps for evaluating composite functions are: 1) Substitute the inner function into the outer function and 2) Simplify the expression. Order matters as f(g(x)) and g(f(x)) may have different values.
3. Examples are worked through to find composite functions given basic functions like f(x) = x + 1 and g(x) = 2x as well as more complex rational functions.
The document discusses continuity of functions. A function is continuous at a point c if the limit of the function as x approaches c exists and equals the value of the function at c. Polynomial and rational functions are continuous everywhere in their domains. A function can have discontinuities where it is not defined or where the limit does not equal the function value. Discontinuities are either removable, where redefining the function can make it continuous, or non-removable, where an asymptote prevents continuity. A function is continuous on a closed interval if it is continuous on the interior and the one-sided limits equal the function values at the endpoints.
This document provides definitions and properties for trigonometric functions including sine, cosine, and tangent. It defines the domains and ranges of sine, cosine, and tangent. Examples of trigonometric ratios are given for common angles like 30, 45, 60, and 90 degrees. Trigonometric identities are also listed, such as the sine and cosine of sums and differences of angles.
1. The document discusses functions and their notation. It defines functions as mappings from inputs to outputs and provides examples of function notation.
2. It also discusses different types of relations between inputs and outputs such as one-to-one, one-to-many, many-to-one, and many-to-many relations.
3. The document explains concepts related to functions such as composite functions, inverse functions, and how to determine composite and inverse functions through examples.
This document discusses continuity of functions, including definitions of continuity at a point and on an interval. It introduces the intermediate value theorem and extreme value theorem for continuous functions on closed intervals. It also states that differentiability at a point implies continuity at that point.
This includes the overall cultivation practices of Rose prepared by:
Kushal Lamichhane (AKL)
Instructor
Shree Gandhi Adarsha Secondary School
Kageshowri Manohara-09, Kathmandu, Nepal
How to Manage Different Customer Addresses in Odoo 18 AccountingCeline George
?
A business often have customers with multiple locations such as office, warehouse, home addresses and this feature allows us to associate with different addresses with each customer streamlining the process of creating sales order invoices and delivery orders.
Paper 107 | From Watchdog to Lapdog: Ishiguro¡¯s Fiction and the Rise of ¡°Godi...Rajdeep Bavaliya
?
Dive into a captivating analysis where Kazuo Ishiguro¡¯s nuanced fiction meets the stark realities of post?2014 Indian journalism. Uncover how ¡°Godi Media¡± turned from watchdog to lapdog, echoing the moral compromises of Ishiguro¡¯s protagonists. We¡¯ll draw parallels between restrained narrative silences and sensationalist headlines¡ªare our media heroes or traitors? Don¡¯t forget to follow for more deep dives!
M.A. Sem - 2 | Presentation
Presentation Season - 2
Paper - 107: The Twentieth Century Literature: From World War II to the End of the Century
Submitted Date: April 4, 2025
Paper Name: The Twentieth Century Literature: From World War II to the End of the Century
Topic: From Watchdog to Lapdog: Ishiguro¡¯s Fiction and the Rise of ¡°Godi Media¡± in Post-2014 Indian Journalism
[Please copy the link and paste it into any web browser to access the content.]
Video Link: https://youtu.be/kIEqwzhHJ54
For a more in-depth discussion of this presentation, please visit the full blog post at the following link: https://rajdeepbavaliya2.blogspot.com/2025/04/from-watchdog-to-lapdog-ishiguro-s-fiction-and-the-rise-of-godi-media-in-post-2014-indian-journalism.html
Please visit this blog to explore additional presentations from this season:
Hashtags:
#GodiMedia #Ishiguro #MediaEthics #WatchdogVsLapdog #IndianJournalism #PressFreedom #LiteraryCritique #AnArtistOfTheFloatingWorld #MediaCapture #KazuoIshiguro
Keyword Tags:
Godi Media, Ishiguro fiction, post-2014 Indian journalism, media capture, Kazuo Ishiguro analysis, watchdog to lapdog, press freedom India, media ethics, literature and media, An Artist of the Floating World
Vitamin and nutritional deficiency occurs when the body does not receive enough essential nutrients, such as vitamins and minerals, needed for proper functioning. This can lead to various health problems, including weakened immunity, stunted growth, fatigue, poor wound healing, cognitive issues, and increased susceptibility to infections and diseases. Long-term deficiencies can cause serious and sometimes irreversible health complications.
HistoPathology Ppt. Arshita Gupta for Diplomaarshitagupta674
?
Hello everyone please suggest your views and likes so that I uploaded more study materials
In this slide full HistoPathology according to diploma course available like fixation
Tissue processing , staining etc
List View Components in Odoo 18 - Odoo ºÝºÝߣsCeline George
?
In Odoo, there are many types of views possible like List view, Kanban view, Calendar view, Pivot view, Search view, etc.
The major change that introduced in the Odoo 18 technical part in creating views is the tag <tree> got replaced with the <list> for creating list views.
ECONOMICS, DISASTER MANAGEMENT, ROAD SAFETY - STUDY MATERIAL [10TH]SHERAZ AHMAD LONE
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This study material for Class 10th covers the core subjects of Economics, Disaster Management, and Road Safety Education, developed strictly in line with the JKBOSE textbook. It presents the content in a simplified, structured, and student-friendly format, ensuring clarity in concepts. The material includes reframed explanations, flowcharts, infographics, and key point summaries to support better understanding and retention. Designed for classroom teaching and exam preparation, it aims to enhance comprehension, critical thinking, and practical awareness among students.
LAZY SUNDAY QUIZ "A GENERAL QUIZ" JUNE 2025 SMC QUIZ CLUB, SILCHAR MEDICAL CO...Ultimatewinner0342
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? Lazy Sunday Quiz | General Knowledge Trivia by SMC Quiz Club ¨C Silchar Medical College
Presenting the Lazy Sunday Quiz, a fun and thought-provoking general knowledge quiz created by the SMC Quiz Club of Silchar Medical College & Hospital (SMCH). This quiz is designed for casual learners, quiz enthusiasts, and competitive teams looking for a diverse, engaging set of questions with clean visuals and smart clues.
? What is the Lazy Sunday Quiz?
The Lazy Sunday Quiz is a light-hearted yet intellectually rewarding quiz session held under the SMC Quiz Club banner. It¡¯s a general quiz covering a mix of current affairs, pop culture, history, India, sports, medicine, science, and more.
Whether you¡¯re hosting a quiz event, preparing a session for students, or just looking for quality trivia to enjoy with friends, this PowerPoint deck is perfect for you.
? Quiz Format & Structure
Total Questions: ~50
Types: MCQs, one-liners, image-based, visual connects, lateral thinking
Rounds: Warm-up, Main Quiz, Visual Round, Connects (optional bonus)
Design: Simple, clear slides with answer explanations included
Tools Needed: Just a projector or screen ¨C ready to use!
? Who Is It For?
College quiz clubs
School or medical students
Teachers or faculty for classroom engagement
Event organizers needing quiz content
Quizzers preparing for competitions
Freelancers building quiz portfolios
? Why Use This Quiz?
Ready-made, high-quality content
Curated with lateral thinking and storytelling in mind
Covers both academic and pop culture topics
Designed by a quizzer with real event experience
Usable in inter-college fests, informal quizzes, or Sunday brain workouts
? About the Creators
This quiz has been created by Rana Mayank Pratap, an MBBS student and quizmaster at SMC Quiz Club, Silchar Medical College. The club aims to promote a culture of curiosity and smart thinking through weekly and monthly quiz events.
? SEO Tags:
quiz, general knowledge quiz, trivia quiz, ºÝºÝߣShare quiz, college quiz, fun quiz, medical college quiz, India quiz, pop culture quiz, visual quiz, MCQ quiz, connect quiz, science quiz, current affairs quiz, SMC Quiz Club, Silchar Medical College
? Reuse & Credit
You¡¯re free to use or adapt this quiz for your own events or sessions with credit to:
SMC Quiz Club ¨C Silchar Medical College & Hospital
Curated by: Rana Mayank Pratap
Photo chemistry Power Point Presentationmprpgcwa2024
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Photochemistry is the branch of chemistry that deals with the study of chemical reactions and processes initiated by light.
Photochemistry involves the interaction of light with molecules, leading to electronic excitation. Energy from light is transferred to molecules, initiating chemical reactions.
Photochemistry is used in solar cells to convert light into electrical energy.
It is used Light-driven chemical reactions for environmental remediation and synthesis. Photocatalysis helps in pollution abatement and environmental cleanup. Photodynamic therapy offers a targeted approach to treating diseases It is used in Light-activated treatment for cancer and other diseases.
Photochemistry is used to synthesize complex organic molecules.
Photochemistry contributes to the development of sustainable energy solutions.
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June 25 ISSIP Event - slides in process
20250618 PPre-Event Presentation Summary - Progress Update with Board Series June 25
ISSIP Website Upcoming Events Description: https://issip.org/event/semi-annual-issip-progress-call/
Register here (even if you cannot attend live online, all who register will get link to recording and slides post-event): https://docs.google.com/forms/d/e/1FAIpQLSdThrop1rafOCo4PQkYiS2XApclJuMjYONEHRMGBsceRdcQqg/viewform
This pre-event presentation: /slideshow/june-2025-progress-update-with-board-call_in-process-pptx/280718770
This pre-event recording: https://youtu.be/Shjgd5o488o
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