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This document defines functions and their key properties such as domain, range, and inverse functions. It provides examples of determining whether a function is one-to-one and finding its inverse. Composite functions are discussed, including ensuring the range of the inner function is contained within the domain of the outer function. Self-inverse functions are introduced, where the inverse of a function is equal to the function itself.

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JC Vectors summary

1. The document provides revision on various topics in vectors including ratio theorem, scalar and vector products, lines, planes, perpendiculars, reflections, angles, distances, direction cosines, and geometric meanings.
2. Key concepts covered include using scalar and vector products to find angles between lines, planes, and determining if lines or planes are parallel/perpendicular.
3. Methods for finding the foot of a perpendicular from a point to a line or plane, reflecting lines and planes, and determining relationships between lines and planes are summarized.

Functions 1 - Math Academy - JC H2 maths A levels

Slides for Functions lecture 1.
JC H2 Maths
A levels Singapore
www.mathacademy.sg
Copyright 2015 Math Academy

Inverse Function.pptx

This document provides information about inverse functions including:
- The inverse of a function is formed by reversing the coordinates of each ordered pair. A function has an inverse only if it is one-to-one.
- The domain of the inverse function is the range of the original function, and the range of the inverse is the domain of the original.
- To find the inverse of a one-to-one function, replace f(x) with y, interchange x and y, then solve for y in terms of x and replace y with f^-1(x).
- Several examples are provided to demonstrate finding the inverse of different one-to-one functions by following the given steps.

Form 4 formulae and note

This document contains notes and formulae on additional mathematics for Form 4. It covers topics such as functions, quadratic equations, quadratic functions, indices and logarithms, coordinate geometry, statistics, circular measure, differentiation, solutions of triangles, and index numbers. The key points covered include the definition of functions, the formula for the sum and product of roots of a quadratic equation, the axis of symmetry and nature of roots of quadratic functions, and common differentiation rules.

Function of several variables

In this lecture, we will discuss:
Functions of one variable
Functions of SeveraL VariabLes
Limits for Functions of Two Variables
Partial Derivatives of a Function of Two Variables

Chapter 10 - Limit and Continuity

This document summarizes Chapter 10 from a mathematics textbook. The chapter covers limits and continuity. It introduces limits, such as one-sided limits and limits at infinity. It defines continuity as a function being continuous at a point if the limit exists and is equal to the function value. Discontinuities can occur if a limit does not exist or is infinite. The chapter applies limits and continuity to solve inequalities involving polynomials and rational functions. Examples are provided to illustrate key concepts like evaluating limits, identifying discontinuities, and using continuity to solve nonlinear inequalities.

JC Vectors summary

1. The document provides revision on various topics in vectors including ratio theorem, scalar and vector products, lines, planes, perpendiculars, reflections, angles, distances, direction cosines, and geometric meanings.
2. Key concepts covered include using scalar and vector products to find angles between lines, planes, and determining if lines or planes are parallel/perpendicular.
3. Methods for finding the foot of a perpendicular from a point to a line or plane, reflecting lines and planes, and determining relationships between lines and planes are summarized.

Functions 1 - Math Academy - JC H2 maths A levels

Slides for Functions lecture 1.
JC H2 Maths
A levels Singapore
www.mathacademy.sg
Copyright 2015 Math Academy

Inverse Function.pptx

This document provides information about inverse functions including:
- The inverse of a function is formed by reversing the coordinates of each ordered pair. A function has an inverse only if it is one-to-one.
- The domain of the inverse function is the range of the original function, and the range of the inverse is the domain of the original.
- To find the inverse of a one-to-one function, replace f(x) with y, interchange x and y, then solve for y in terms of x and replace y with f^-1(x).
- Several examples are provided to demonstrate finding the inverse of different one-to-one functions by following the given steps.

Form 4 formulae and note

This document contains notes and formulae on additional mathematics for Form 4. It covers topics such as functions, quadratic equations, quadratic functions, indices and logarithms, coordinate geometry, statistics, circular measure, differentiation, solutions of triangles, and index numbers. The key points covered include the definition of functions, the formula for the sum and product of roots of a quadratic equation, the axis of symmetry and nature of roots of quadratic functions, and common differentiation rules.

Function of several variables

In this lecture, we will discuss:
Functions of one variable
Functions of SeveraL VariabLes
Limits for Functions of Two Variables
Partial Derivatives of a Function of Two Variables

Chapter 10 - Limit and Continuity

This document summarizes Chapter 10 from a mathematics textbook. The chapter covers limits and continuity. It introduces limits, such as one-sided limits and limits at infinity. It defines continuity as a function being continuous at a point if the limit exists and is equal to the function value. Discontinuities can occur if a limit does not exist or is infinite. The chapter applies limits and continuity to solve inequalities involving polynomials and rational functions. Examples are provided to illustrate key concepts like evaluating limits, identifying discontinuities, and using continuity to solve nonlinear inequalities.

Caso Practico Funciones Exponenciales

Este documento presenta información sobre funciones exponenciales. Introduce las funciones exponenciales y sus aplicaciones en matemáticas, administración de empresas y ciencias naturales. Luego, define las funciones exponenciales, sus dominios y rangos, y proporciona ejemplos de trazar gráficas de funciones exponenciales y resumir sus propiedades. Finalmente, discute las transformaciones de funciones exponenciales a través de traslaciones, reflexiones, estiramientos y contracciones.

Lesson 11: The Chain Rule

The chain rule allows us to differentiate a composition of functions. It's a complicated peel-the-onion process, but it can be learned.

Pack de rubrica de matematica

Este documento presenta cuatro temas sobre conjuntos y lógica proposicional. El primer tema define tres subconjuntos A, B y C de un conjunto referencial Re y pide tabular los subconjuntos y elaborar un diagrama de Venn. El segundo tema evalúa las proposiciones simples y compuestas dadas sobre el conjunto Re. El tercer tema pide elaborar diagramas de Venn para operaciones entre los conjuntos A, B y C. El cuarto tema usa álgebra proposicional para demostrar una equivalencia entre subconjuntos.

Functions

1. A function is a special type of binary relation where each element of the domain (A) is uniquely mapped to an element of the codomain (B).
2. Functions can be defined using set builder notation, ordered pairs, or function notation. Common functions include identity, constant, and composite functions.
3. Partial functions drop the requirement that each domain element maps to a range element. Injections map distinct domain elements to distinct range elements while surjections map onto the entire range. Bijections are both injections and surjections.

Applied Calculus: Continuity and Discontinuity of Function

This document provides an overview of continuity of functions including:
1. The three conditions for a function f(x) to be continuous at a point x=c.
2. Examples of determining if functions are continuous or discontinuous at given points.
3. Definitions of a function being continuous at endpoints and across an entire interval.
4. Theorems regarding continuity of algebraic combinations of functions and polynomials.
5. The two types of discontinuity - removable and non-removable - and the concept of continuous extensions.
6. Examples of determining continuous extensions of functions at given points.

Complex Numbers 1 - Math Academy - JC H2 maths A levels

The document provides lessons on complex numbers. It defines a complex number as being of the form z = x + iy, where x and y are real numbers. It discusses operations like addition, subtraction, multiplication and division of complex numbers. It also defines the complex conjugate and gives some examples of performing operations on complex numbers.

Introductory maths analysis chapter 12 official

This chapter discusses additional topics in differentiation including:
- Derivatives of logarithmic and exponential functions
- Elasticity of demand
- Implicit differentiation
- Logarithmic differentiation
- Newton's method for approximating roots
- Finding higher-order derivatives directly and implicitly.
Examples are provided for each topic to illustrate the differentiation techniques.

integration by parts

Integration by Parts allows us to integrate some products using the formula:
dv du
u dx = uv - v dx
dx dx
This formula means that we differentiate one factor to get du/dx and integrate the other factor to get v. We then substitute these into the formula to evaluate the integral. For example, to integrate xsinx we would have u = x and dv/dx = sinx, then substitute into the formula. This technique can be used to evaluate integrals of products that cannot be integrated by other methods.

Derivacion implicita

Este documento presenta el concepto de derivación implícita. Explica que una función se define implícitamente cuando está dada por una ecuación en lugar de una expresión explícita. Muestra cómo derivar funciones implícitas mediante la derivación de ambos lados de la ecuación que la define. Resuelve dos ejemplos para ilustrar el método.

Introductory maths analysis chapter 13 official

This document is a chapter from an introductory mathematical analysis textbook. It covers curve sketching, including how to find relative and absolute extrema, determine concavity, use the second derivative test, identify asymptotes, and apply concepts of maxima and minima. The chapter contains learning objectives, an outline of topics, examples of applying techniques to sketch curves and solve optimization problems, and instructional content to introduce these curve sketching concepts.

Odd and even functions

The document defines even, odd, and neither functions based on their symmetry properties. An even function is symmetric about the y-axis, such that f(-x) = f(x). An odd function is symmetric about the origin, such that f(-x) = -f(x). A function is neither even nor odd if it does not satisfy those properties. Examples are provided to demonstrate how to determine if a given function is even, odd, or neither.

Relations & functions.pps

The document discusses Cartesian products, domains, ranges, and co-domains of relations and functions through examples and definitions. It explains that the Cartesian product of sets A and B, written as A×B, is the set of all ordered pairs (a,b) where a is an element of A and b is an element of B. It also defines what constitutes a relation between two sets and provides examples of relations and functions, discussing their domains and ranges. Arrow diagrams are presented to illustrate various functions along with questions and their solutions related to relations and functions.

Sintitul 3

1) Factorizar un polinomio significa descomponerlo en dos o más polinomios llamados factores, cuya multiplicación da el polinomio original. 2) Existen varios métodos para factorizar polinomios, incluyendo el método del factor común, el método de las identidades, el método de las aspas y el método de los divisores binomios. 3) Cada método tiene reglas específicas para descomponer polinomios de diferentes formas en sus factores.

Function in Mathematics

This is about Some features of function,
Variation of Function,
Explain of function,
Interval ,
Domain,
Use of Function

Finite elements : basis functions

1. The document discusses the basis functions used in 1D and 2D finite element analysis.
2. For 1D elements, it presents the linear, quadratic and cubic basis functions derived from imposing interpolation conditions on linear, quadratic and cubic shape functions respectively.
3. For 2D elements, it derives the linear and quadratic basis functions for triangular and rectangular elements by transforming the element coordinates and imposing interpolation conditions on linear and quadratic Ansatz shape functions.

Functions

The document discusses functions and evaluating functions. It provides examples of determining if a given equation is a function using the vertical line test and evaluating functions by substituting values into the function equation. It also includes examples of evaluating composite functions using flow diagrams to illustrate the steps of evaluating each individual function.

Integration by Parts & by Partial Fractions

The document provides an overview of calculus and analytical geometry presented by Group D. It discusses various topics related to integration including the history and development of integration, the definition and purpose of integration, common integration rules and formulas like integration by parts and integration by partial fractions, and real-life applications of integration in fields like engineering, medicine, and physics. Examples are provided to demonstrate how to use integration techniques like integration by parts and integration by partial fractions to evaluate definite integrals.

Calculus of variation problems

This presentation gives example of "Calculus of Variations" problems that can be solved analytical. "Calculus of Variations" presentation is prerequisite to this one.
For comments please contact me at solo.hermelin@gmail.com.
For more presentations on different subjects visit my website at http://www.solohermelin.com.

Logarithms

The document discusses functions and their properties. It defines a function as a rule that assigns exactly one output value to each input value in its domain. Functions can be represented graphically, numerically in a table, or with an algebraic rule. The domain of a function is the set of input values, while the range is the set of output values. Basic operations like addition, subtraction, multiplication and division can be performed on functions in the same way as real numbers. Composition of functions is defined as evaluating one function using the output of another as the input.

Functions

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.

Caso Practico Funciones Exponenciales

Este documento presenta información sobre funciones exponenciales. Introduce las funciones exponenciales y sus aplicaciones en matemáticas, administración de empresas y ciencias naturales. Luego, define las funciones exponenciales, sus dominios y rangos, y proporciona ejemplos de trazar gráficas de funciones exponenciales y resumir sus propiedades. Finalmente, discute las transformaciones de funciones exponenciales a través de traslaciones, reflexiones, estiramientos y contracciones.

Lesson 11: The Chain Rule

The chain rule allows us to differentiate a composition of functions. It's a complicated peel-the-onion process, but it can be learned.

Pack de rubrica de matematica

Este documento presenta cuatro temas sobre conjuntos y lógica proposicional. El primer tema define tres subconjuntos A, B y C de un conjunto referencial Re y pide tabular los subconjuntos y elaborar un diagrama de Venn. El segundo tema evalúa las proposiciones simples y compuestas dadas sobre el conjunto Re. El tercer tema pide elaborar diagramas de Venn para operaciones entre los conjuntos A, B y C. El cuarto tema usa álgebra proposicional para demostrar una equivalencia entre subconjuntos.

Functions

1. A function is a special type of binary relation where each element of the domain (A) is uniquely mapped to an element of the codomain (B).
2. Functions can be defined using set builder notation, ordered pairs, or function notation. Common functions include identity, constant, and composite functions.
3. Partial functions drop the requirement that each domain element maps to a range element. Injections map distinct domain elements to distinct range elements while surjections map onto the entire range. Bijections are both injections and surjections.

Applied Calculus: Continuity and Discontinuity of Function

This document provides an overview of continuity of functions including:
1. The three conditions for a function f(x) to be continuous at a point x=c.
2. Examples of determining if functions are continuous or discontinuous at given points.
3. Definitions of a function being continuous at endpoints and across an entire interval.
4. Theorems regarding continuity of algebraic combinations of functions and polynomials.
5. The two types of discontinuity - removable and non-removable - and the concept of continuous extensions.
6. Examples of determining continuous extensions of functions at given points.

Complex Numbers 1 - Math Academy - JC H2 maths A levels

The document provides lessons on complex numbers. It defines a complex number as being of the form z = x + iy, where x and y are real numbers. It discusses operations like addition, subtraction, multiplication and division of complex numbers. It also defines the complex conjugate and gives some examples of performing operations on complex numbers.

Introductory maths analysis chapter 12 official

This chapter discusses additional topics in differentiation including:
- Derivatives of logarithmic and exponential functions
- Elasticity of demand
- Implicit differentiation
- Logarithmic differentiation
- Newton's method for approximating roots
- Finding higher-order derivatives directly and implicitly.
Examples are provided for each topic to illustrate the differentiation techniques.

integration by parts

Integration by Parts allows us to integrate some products using the formula:
dv du
u dx = uv - v dx
dx dx
This formula means that we differentiate one factor to get du/dx and integrate the other factor to get v. We then substitute these into the formula to evaluate the integral. For example, to integrate xsinx we would have u = x and dv/dx = sinx, then substitute into the formula. This technique can be used to evaluate integrals of products that cannot be integrated by other methods.

Derivacion implicita

Este documento presenta el concepto de derivación implícita. Explica que una función se define implícitamente cuando está dada por una ecuación en lugar de una expresión explícita. Muestra cómo derivar funciones implícitas mediante la derivación de ambos lados de la ecuación que la define. Resuelve dos ejemplos para ilustrar el método.

Introductory maths analysis chapter 13 official

This document is a chapter from an introductory mathematical analysis textbook. It covers curve sketching, including how to find relative and absolute extrema, determine concavity, use the second derivative test, identify asymptotes, and apply concepts of maxima and minima. The chapter contains learning objectives, an outline of topics, examples of applying techniques to sketch curves and solve optimization problems, and instructional content to introduce these curve sketching concepts.

Odd and even functions

The document defines even, odd, and neither functions based on their symmetry properties. An even function is symmetric about the y-axis, such that f(-x) = f(x). An odd function is symmetric about the origin, such that f(-x) = -f(x). A function is neither even nor odd if it does not satisfy those properties. Examples are provided to demonstrate how to determine if a given function is even, odd, or neither.

Relations & functions.pps

The document discusses Cartesian products, domains, ranges, and co-domains of relations and functions through examples and definitions. It explains that the Cartesian product of sets A and B, written as A×B, is the set of all ordered pairs (a,b) where a is an element of A and b is an element of B. It also defines what constitutes a relation between two sets and provides examples of relations and functions, discussing their domains and ranges. Arrow diagrams are presented to illustrate various functions along with questions and their solutions related to relations and functions.

Sintitul 3

1) Factorizar un polinomio significa descomponerlo en dos o más polinomios llamados factores, cuya multiplicación da el polinomio original. 2) Existen varios métodos para factorizar polinomios, incluyendo el método del factor común, el método de las identidades, el método de las aspas y el método de los divisores binomios. 3) Cada método tiene reglas específicas para descomponer polinomios de diferentes formas en sus factores.

Function in Mathematics

This is about Some features of function,
Variation of Function,
Explain of function,
Interval ,
Domain,
Use of Function

Finite elements : basis functions

1. The document discusses the basis functions used in 1D and 2D finite element analysis.
2. For 1D elements, it presents the linear, quadratic and cubic basis functions derived from imposing interpolation conditions on linear, quadratic and cubic shape functions respectively.
3. For 2D elements, it derives the linear and quadratic basis functions for triangular and rectangular elements by transforming the element coordinates and imposing interpolation conditions on linear and quadratic Ansatz shape functions.

Functions

The document discusses functions and evaluating functions. It provides examples of determining if a given equation is a function using the vertical line test and evaluating functions by substituting values into the function equation. It also includes examples of evaluating composite functions using flow diagrams to illustrate the steps of evaluating each individual function.

Integration by Parts & by Partial Fractions

The document provides an overview of calculus and analytical geometry presented by Group D. It discusses various topics related to integration including the history and development of integration, the definition and purpose of integration, common integration rules and formulas like integration by parts and integration by partial fractions, and real-life applications of integration in fields like engineering, medicine, and physics. Examples are provided to demonstrate how to use integration techniques like integration by parts and integration by partial fractions to evaluate definite integrals.

Calculus of variation problems

This presentation gives example of "Calculus of Variations" problems that can be solved analytical. "Calculus of Variations" presentation is prerequisite to this one.
For comments please contact me at solo.hermelin@gmail.com.
For more presentations on different subjects visit my website at http://www.solohermelin.com.

Caso Practico Funciones Exponenciales

Caso Practico Funciones Exponenciales

Lesson 11: The Chain Rule

Lesson 11: The Chain Rule

Pack de rubrica de matematica

Pack de rubrica de matematica

Functions

Functions

Applied Calculus: Continuity and Discontinuity of Function

Applied Calculus: Continuity and Discontinuity of Function

Complex Numbers 1 - Math Academy - JC H2 maths A levels

Complex Numbers 1 - Math Academy - JC H2 maths A levels

Introductory maths analysis chapter 12 official

Introductory maths analysis chapter 12 official

integration by parts

integration by parts

Derivacion implicita

Derivacion implicita

Exercice fonctions réciproques

Exercice fonctions réciproques

Introductory maths analysis chapter 13 official

Introductory maths analysis chapter 13 official

Odd and even functions

Odd and even functions

Exercice sur logarithme népérien propose par le prof

Exercice sur logarithme népérien propose par le prof

Relations & functions.pps

Relations & functions.pps

Sintitul 3

Sintitul 3

Function in Mathematics

Function in Mathematics

Finite elements : basis functions

Finite elements : basis functions

Functions

Functions

Integration by Parts & by Partial Fractions

Integration by Parts & by Partial Fractions

Calculus of variation problems

Calculus of variation problems

Logarithms

The document discusses functions and their properties. It defines a function as a rule that assigns exactly one output value to each input value in its domain. Functions can be represented graphically, numerically in a table, or with an algebraic rule. The domain of a function is the set of input values, while the range is the set of output values. Basic operations like addition, subtraction, multiplication and division can be performed on functions in the same way as real numbers. Composition of functions is defined as evaluating one function using the output of another as the input.

Functions

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.

functions.pdf

The document defines and provides notation for real-valued functions with real variables. It defines domain, codomain, range, and graph of a function. It describes even, odd, and periodic functions as well as one-to-one and onto functions. It also defines function composition, inverse functions, and how to create new functions from old ones through vertical and horizontal translations, reflections, stretching, and compressing.

Functions and graphs

The document discusses various concepts related to functions and graphs:
1) It defines what a function is and provides examples of different types of functions such as identity, constant, polynomial, and rational functions.
2) It explains the terminology used in functions such as domain, co-domain, pre-image, and image.
3) It discusses the properties of one-to-one, onto, and bijective functions and provides examples of each. The concepts of inverse and composition of functions are also introduced.
4) Useful mathematical functions like floor, ceiling, and round are defined. Concepts related to limits, continuity, differentiability of functions and their graphs are explained briefly.
5

7.3 power functions and function operations

A power function has the form y=ax^b where a is a real number and b is a rational number. Functions can be combined using addition, subtraction, multiplication, division, and composition. When combining functions, the domain is the set of values where both functions are defined. Composition involves applying one function to the output of another.

Functions

The document defines key concepts related to functions including: domain, codomain, range, one-to-one (injective) functions, increasing/decreasing functions, onto (surjective) functions, bijective functions, inverse functions, function composition, and floor and ceiling functions. It provides examples to illustrate these concepts and determine if specific functions have given properties.

Calculus 1 Lecture Notes (Functions and Their Graphs)

Calculus 1 Lecture Notes
Functions and Their Graphs

functions limits and continuity

This document provides an overview of functions, limits, and continuity. It defines key concepts such as domain and range of functions, and examples of standard real functions. It also covers even and odd functions, and how to calculate limits, including left and right hand limits. Methods for evaluating algebraic limits using substitution, factorization, and rationalization are presented. The objectives are to understand functions, domains, ranges, and how to evaluate limits of functions.

AS LEVEL Function (CIE) EXPLAINED WITH EXAMPLE AND DIAGRAMS

1. A function is a relationship between inputs and outputs where each input corresponds to exactly one output. The domain of a function is the set of possible inputs, and the range is the set of possible outputs.
2. For a function to be one-to-one, each output must correspond to only one input. This can be tested using the horizontal line test - drawing horizontal lines on the graph. Restricting the domain can make non-one-to-one functions one-to-one.
3. The inverse of a function undoes the input-output relationship by switching the domain and range. Only one-to-one functions have inverses. The graph of an inverse function passes the vertical

Functions limits and continuity

This document discusses functions, limits, and continuity. It begins by defining functions, domains, ranges, and some standard real functions like constant, identity, modulus, and greatest integer functions. It then covers limits of functions including one-sided limits and properties of limits. Examples are provided to illustrate evaluating limits using substitution and factorization methods. The overall objectives are to understand functions, domains, ranges, limits of functions and methods to evaluate limits.

237654933 mathematics-t-form-6

homework help,online homework help,online tutors,online tutoring,research paper help,do my homework,
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Inverse functions

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

The Algebric Functions

This document discusses algebraic functions and their properties. Some key points:
- When working with real numbers, you cannot divide by zero or take the square root or even root of a negative number. These restrictions limit the domain of a function.
- To find the domain of a rational function, set the denominator equal to zero and exclude those values.
- The domain of a sum, difference, product, or quotient of functions f and g consists of values that are in the domains of both f and g, except for quotients where the denominator cannot be zero.
- Composition of functions means applying one function to the output of another. The domain of f∘g consists of values where g(x) is

Basic Calculus.docx

The document provides an overview of basic calculus concepts including functions, trigonometric functions, operations on functions, and composite functions. It defines a function as a rule that assigns exactly one output value to each input value. It also defines domain as the set of permissible input values and range as the set of corresponding output values. Examples are provided to demonstrate evaluating functions, finding limits, graphing functions, and performing operations like addition, multiplication, and composition on functions.

Functions by mstfdemirdag

This document discusses key concepts related to functions including:
- The domain of a function is the set of all real numbers for which the expression is defined as a real number.
- Two functions are equal if and only if their expressions and domains are equal.
- An even function satisfies f(-x) = f(x) and an odd function satisfies f(-x) = -f(x). Graphs of even functions are symmetric to the y-axis and odds are symmetric to the origin.
- Composition of functions f o g is defined as (f o g)(x) = f(g(x)). Inverse functions satisfy y = f(x) if and only if x =

7 functions

This document defines functions and relations. It discusses identifying the domain and range of functions and relations, evaluating functions, and performing operations on functions such as addition, subtraction, multiplication, division, and composition. It also covers graphing functions, including piecewise functions, absolute value functions, greatest and least integer functions. Key examples are provided to illustrate how to identify domains and ranges, evaluate functions, perform operations on functions, and graph different types of functions.

Math presentation on domain and range

This document defines key concepts related to functions including domain, co-domain, and range. It provides examples of determining the domain and range of various functions. The domain of a function is the set of inputs, the co-domain is the set of all possible outputs, and the range is the set of actual outputs. Examples show how to use inequalities to find the domain by determining when a function gives real values and how to manipulate equations to find the range.

Unit 1.4

The document discusses building new functions from existing functions through algebraic combinations, composition, and implicit definitions. It provides examples of combining two functions using addition, subtraction, multiplication, and division. Composition of functions is defined as applying one function to the output of another. Implicitly defined functions relate sets of ordered pairs as relations that can define multiple functions. Examples are provided for composing, decomposing, and graphing implicitly defined functions.

Inverse Functions

1) The document discusses finding the inverse of functions by interchanging the x and y variables and solving for y. It provides examples of finding the inverses of f(x)=3x-7 and g(x)=2x^3+1.
2) It also discusses verifying inverses by checking if the composition of a function and its inverse equals x. And finding inverses of functions with restricted domains, including an example of f(x)=sqrt(x+4).
3) Finally, it discusses the relationship between a function being one-to-one and having an inverse function, both algebraically and graphically.

Higher Maths 1.2.1 - Sets and Functions

The document discusses key concepts in set theory and functions, including:
- Sets can contain numbers, elements, and be represented using curly brackets.
- Venn diagrams use overlapping circles to show logical connections between sets.
- A function has a domain (input) and range (output), where each input is mapped to a unique output.
- Composite functions combine other functions by substituting one into another.
- Inverse functions reverse the input and output of a function if it exists.
- Common functions that can be graphed include linear, quadratic, trigonometric, cubic, exponential and logarithmic functions.

Logarithms

Logarithms

Functions

Functions

functions.pdf

functions.pdf

Functions and graphs

Functions and graphs

7.3 power functions and function operations

7.3 power functions and function operations

Functions

Functions

Calculus 1 Lecture Notes (Functions and Their Graphs)

Calculus 1 Lecture Notes (Functions and Their Graphs)

functions limits and continuity

functions limits and continuity

AS LEVEL Function (CIE) EXPLAINED WITH EXAMPLE AND DIAGRAMS

AS LEVEL Function (CIE) EXPLAINED WITH EXAMPLE AND DIAGRAMS

Functions limits and continuity

Functions limits and continuity

237654933 mathematics-t-form-6

237654933 mathematics-t-form-6

Inverse functions

Inverse functions

The Algebric Functions

The Algebric Functions

Basic Calculus.docx

Basic Calculus.docx

Functions by mstfdemirdag

Functions by mstfdemirdag

7 functions

7 functions

Math presentation on domain and range

Math presentation on domain and range

Unit 1.4

Unit 1.4

Inverse Functions

Inverse Functions

Higher Maths 1.2.1 - Sets and Functions

Higher Maths 1.2.1 - Sets and Functions

Sec 3 A Maths Notes Indices

1. The document discusses solving exponential equations with one, two, or three terms using properties of exponents such as changing bases to the same term and equating powers.
2. Examples are provided for solving two-term exponential equations by making the bases equal and equations with three terms by substituting variables, changing bases to the same term, and equating powers.
3. Solving exponential equations as products using properties such as treating exponents as multipliers is also demonstrated through examples.

Sec 4 A Maths Notes Maxima Minima

The document contains two examples of maximum and minimum problems involving differentiation.
Example 1 asks the reader to find the minimum volume of a cone given that a sphere must fit inside it. It is found that the minimum volume occurs when the radius of the cone is 28.577 cm.
Example 2 involves finding the maximum volume of a cylinder inscribed in a sphere. The maximum volume is calculated to be 104,000 cm3, occurring when the height of the cylinder is 28.5 cm.
The document provides guidance on solving maximum and minimum problems using differentiation, illustrated through these two examples involving geometric shapes.

Sec 3 E Maths Notes Coordinate Geometry

This document provides examples and explanations of using the distance formula and equations of lines in coordinate geometry. It defines the distance formula and shows how to calculate the distance between two points with given coordinates. It also demonstrates how to determine the gradient and y-intercept of a line given its equation, find the equation of a line given the gradient and a point or two points, and find values related to lines parallel or intersecting given lines.

Sec 3 A Maths Notes Indices

1) The document provides examples of solving exponential equations with various methods depending on whether the equation has two terms, three or more terms, or involves indices as products or quotients.
2) Key steps include splitting equations, letting one term equal a variable, raising both sides to the same power, and changing all terms to have the same base before equating exponents.
3) Examples range from simple equations like 82=x to more complex ones involving subtraction, addition, and multiplication of terms with different bases and exponents like (23)3=x+2-x.

Sec 2 Maths Notes Change Subject

The document provides examples and techniques for changing the subject of a formula. It demonstrates flipping both sides of an equation, multiplying or dividing both sides by the same term, and isolating the term to be made the subject by collecting like terms on one side of the equation and leaving the term by itself on the other side. Common mistakes discussed include incorrectly flipping terms individually rather than both sides of the equation and prematurely making denominators the same.

Sec 1 Maths Notes Equations

1) The document provides steps for solving simple linear equations with no fractions and fractional equations.
2) For linear equations with no fractions, the steps are to expand if there is a bracket, group like terms to one side, and then solve for the variable.
3) For fractional equations, the steps are to multiply both sides by the common denominator to clear the fractions, then group like terms and solve.

Vectors2

The document discusses various topics related to vectors and planes:
1. It explains the vector product in three forms - mathematical calculation, 3D picture, and in terms of sine. It provides an example to calculate the vector product of two vectors.
2. It discusses the different forms of the equation of a plane - parametric, scalar product, and Cartesian forms. It provides examples to write the equation of a plane in these different forms.
3. It explains how to find the foot of the perpendicular from a point to a plane. It provides examples to find the foot and shortest distance.
4. It discusses how to find acute angles between lines, planes, and a line and plane. Examples

Math academy-partial-fractions-notes

This document provides an introduction to partial fractions. It defines key terms like polynomials, rational functions, and proper and improper fractions. It then outlines the three main cases for splitting a fraction into partial fractions: (1) a linear factor (ax+b), (2) a repeated linear factor, and (3) a quadratic factor (ax^2+bx+c). For each case, it provides an example of how to write the fraction as a sum of partial fractions. It concludes by emphasizing two important checks: (1) the fraction must be proper, and (2) the denominator must be completely factorized before attempting to write it as partial fractions.

Recurrence

This document provides examples of recurrence relations and their solutions. It begins by defining convergence of sequences and limits. It then provides examples of recurrence relations, solving them using algebraic and graphical methods. One example finds the 6th term of a sequence defined by a recurrence relation to be 2.3009. Another example solves a recurrence relation algebraically to express the general term un in terms of n. The document emphasizes using graphical methods like sketching graphs to prove properties of sequences defined by recurrence relations.

Probability 2 - Math Academy - JC H2 maths A levels

The document provides information on probability concepts including Venn diagrams, union and intersection of events, useful probability formulas, mutually exclusive vs independent events, and examples testing these concepts. Specifically, it defines union as "taking everything in A and B", intersection as "taking common parts in A and B", provides formulas for probability of unions and intersections, and shows how to determine if events are mutually exclusive or independent using the probability of their intersection. It also includes worked examples testing concepts like mutually exclusive events and independence.

Probability 1 - Math Academy - JC H2 maths A levels

The document discusses conditional probability and provides examples. It defines conditional probability P(A|B) as the probability of event A occurring given that event B has already occurred. An example calculates probabilities for drawing marbles from a bag. Another example finds probabilities for selecting chocolates with different flavors from a box containing chocolates of various flavors. Formulas and step-by-step workings are provided for calculating conditional probabilities.

Sec 3 A Maths Notes Indices

Sec 3 A Maths Notes Indices

Sec 4 A Maths Notes Maxima Minima

Sec 4 A Maths Notes Maxima Minima

Sec 3 E Maths Notes Coordinate Geometry

Sec 3 E Maths Notes Coordinate Geometry

Sec 3 A Maths Notes Indices

Sec 3 A Maths Notes Indices

Sec 2 Maths Notes Change Subject

Sec 2 Maths Notes Change Subject

Sec 1 Maths Notes Equations

Sec 1 Maths Notes Equations

Vectors2

Vectors2

Math academy-partial-fractions-notes

Math academy-partial-fractions-notes

Recurrence

Recurrence

Probability 2 - Math Academy - JC H2 maths A levels

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Probability 1 - Math Academy - JC H2 maths A levels

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A Strategic Approach: GenAI in Education

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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.

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How to Fix the Import Error in the Odoo 17

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- 1. FUNCTIONS 1 Function Basics 1.1 Set notations [a, b] = {x ∈ R : a ≤ x ≤ b} (a, b] = {x ∈ R : a < x ≤ b} (a, b) = {x ∈ R : a < x < b} (a, ∞) = {x ∈ R : x > a} R = set of real numbers R+ = set of positive real numbers 1.2 Rule, domain and range A function f, is defined by its rule and domain. f : x → x + 2 rule , x ∈ [0, ∞). domain f(x) = x + 2, x ≥ 0. X f(X) 1 2 3 3 4 5 f Domain Range The domain Df , is the set of all possible x values. The range Rf , is the set of all possible y values. Remark: When we state a function, we must always state both its rule and domain. www.MathAcademy.sg 1 © 2019 Math Academy
- 2. Finding Range Sketch the graph to find Rf . Rf is the range of y-values that the graph takes. Example 1. Find the range of the following: (a) f : x → x2 − 1, x ∈ [−1, 2], −1 1 2 −1 3 x y ∴ Rf = (b) g : x → ex + 1, x ∈ R. 1 x y ∴ Rg = 2 Horizontal line test A function is said to be 1 − 1 if for every y ∈ Rf , there is only ONE x such that f(x) = y. −1 1 1 x y It is not 1-1 since the line y = 1 cuts the graph twice. x y It is 1-1 since every horizontal line cuts the graph at most once. Horizontal Line Test f is NOT a 1 − 1 function if there is a horizontal line that cuts the graph at MORE THAN ONE point. f IS a 1 − 1 function if any horizontal line y = a, where a ∈ Rf , cuts the graph AT ONLY ONE point. Differentiation test for 1-1 f is 1 − 1 if f′(x) > 0 for ALL x in the domain (strictly increasing functions) or f′(x) < 0 for ALL x in the domain (strictly decreasing functions) www.MathAcademy.sg 2 © 2019 Math Academy
- 3. Example 2 (Different techniques to answer 1-1 questions). [2012/RVHS/I/6modified] The function f is defined by f : x → | − x2 − 2x + 3|, x ∈ R. (a) With the aid of a diagram, explain why f is not 1-1. [2] (b) If the domain of f is restricted to the set {x ∈ R : x ≥ k}, state with a reason the least value of k for which the function is 1-1. [2] (c) By considering the derivative of f(x), prove that f is a one-one function for the domain you have found in (b). [2] Solution: (a) Show not 1-1 through graph Sketch the graph, give the equation of a SPECIFIC horizontal line that cuts the graph in at least 2 points. −3 1 3 y = f(x) x y Solution: (b) Show 1-1 through graph Sketch the graph, explain that any horizontal line cuts the graph at only 1 point. Least value of k is 1. When x ≥ 1, any horizontal line y = a for a ∈ [0, ∞), cuts the graph of y = f(x) at only 1 point, hence it is 1-1. (c) Show 1-1 through differentiation Show that f′(x) is either > 0 or < 0 for all x ∈ Df . Note: This method cannot be used to show that a function is not 1-1. For x ≥ 1, f(x) = x2 + 2x − 3. Since f′(x) > 0 for x ≥ 1, f is a strictly increasing function, and hence it is 1-1. www.MathAcademy.sg 3 © 2019 Math Academy
- 4. 3 Inverse functions Df Rf x y f f(x) = y f−1 f−1(y) = x Properties of inverse function 1. For f−1 to exist, f must be a 1-1 function. 2. Df−1 = Rf . 3. Rf−1 = Df . 4. (f−1)−1 = f. Geometrical relationship between a function and its inverse (i) The graph of f−1 is the reflection of the graph f about the line y = x. (ii) (a, b) lies on f ⇔ (b, a) lies on f−1. (iii) x = k is an asymptote of f ⇔ y = k is an asymptote of f−1 Remark: The notation f−1 stands for the inverse function of f. It is not the same as 1 f . www.MathAcademy.sg 4 © 2019 Math Academy
- 5. Example 3 (Cambridge N2008/II/4). The function f is defined by f : x → x2 − 8x + 17 for x > 4. (i) Sketch the graph of y = f(x). Your sketch should indicate the position of the graph in relation to the origin. (ii) Show that the inverse function f−1 exists and find f−1(x) in similar form. (iii) On the same diagram as in part (i), sketch the graph of y = f−1. (iv) Write down the equation of the line in which the graph of y = f(x) must be reflected in order to obtain the graph of y = f−1, and hence find the exact solution of the equation f(x) = f−1(x). Solution: (i) 4 1 y = f(x) x y (ii) Showing inverse exists To show f−1 exists, we only need to show that f is 1-1. Every horizontal line y = a for a > 1 cuts the graph of y = f(x) at only 1 point, hence it is 1-1 and f−1 exists. [We first make x the subject:] ∴ f−1(x) = √ x − 1 + 4, x > 1. [Change x to f−1x and y to x.] [Note: You must state the domain of f−1 !] www.MathAcademy.sg 5 © 2019 Math Academy
- 6. (iii) Geometrical relationship between a function and its inverse (i) The graph of f−1 is the reflection of the graph f about the line y = x. (ii) (a, b) lies on f ⇔ (b, a) lies on f−1. (iii) x = k is an asymptote of f ⇔ y = k is an asymptote of f−1 1 4 1 4 y = f(x) y = f−1(x) y = x x y (iv) It must be reflected along the line y = x. Since f and f−1 intersect at the line y = x, finding exact solution of the equation f(x) = f−1(x) is equivalent to finding f(x) = x x2 − 8x + 17 = x x2 − 9x + 17 = 0 x = 9 ± √ 92 − 4(1)(17) 2 = 9 + √ 13 2 or 9 − √ 13 2 (rej as it is not in Df ) www.MathAcademy.sg 6 © 2019 Math Academy
- 7. Example 4 (2010/MI/Prelim/I/2c). Function g is defined by g : x → x2 − 3x for x ∈ R, If the domain of g is restricted to the set {x ∈ R : x ≥ a}, find the least value of a for which g−1 exists. Hence, find g−1 and state its domain. [4] [a = 1.5, g−1(x) = 3 2 + √ x + 9 4, x ≥ −9 4] www.MathAcademy.sg 7 © 2019 Math Academy
- 8. Example 5 (2014/PJC/Prelim/I/7modified). It is given that f(x) = { 1 1+ √ x for 0 ≤ x < 4, −1 for 4 ≤ x < 5, and that f(x + 5) = f(x) for all real values of x. (i) Find f(24) and f(30). (ii) Sketch the graph of y = f(x) for −5 ≤ x < 12. Solution: (i) (ii) −5 5 10 12 1 -1 1 3 y x 4 www.MathAcademy.sg 8 © 2019 Math Academy
- 9. 4 Composite Functions 4.1 Domain, Range, Rule Let f and g be the following functions: g : x → x + 1 for x ≥ −1, f : x → x − 2 for x ≥ 0. Lets investigate the composite function fg(x). -1 0 1 2 0 1 2 3 ... ... g Dg Rg What happens now if we change the domain of f to the following? g : x → x + 1 for x ≥ −1, f : x → x − 2 for x ≥ 1. -1 0 1 2 0 1 2 3 ... ... g Dg Rg At the intermediate stage, the number 0 will have no place to map to! This is an undesired situation. www.MathAcademy.sg 9 © 2019 Math Academy
- 10. fg Diagram for function fg(x) Dg Rfg Df Rg Properties of Composite Functions (i) Domain fg = domain g. (ii) Composite function fg exists ⇔ Rg ⊆ Df Example 6. Consider the following functions: f : x → x2 x ∈ R g : x → 1 x − 3 x ∈ R, x ̸= 3 Find the composition functions (i)fg, (ii) gf. Solution: (i) fg(x) = f (g(x)) = f ( 1 x − 3 ) = ( 1 x − 3 )2 Domain of fg(x) = domain g = R {3}. (ii) gf(x) = g (f(x)) = g(x2 ) = 1 x2 − 3 Domain of gf(x) = domain f = R. www.MathAcademy.sg 10 © 2019 Math Academy
- 11. Example 7. Two functions, f, g are defined by f : x → x2 , x ≥ 0 g : x → 2x + 1, 0 ≤ x ≤ 1 (a) Show that the composite function fg exist. Find (b) its rule and domain and (c) corresponding range. Solution: (a) To show that fg exists, we need to show that Rg ⊆ Df . 1 1 3 y = g(x) x y Since Dg = [0, 1], from the graph of g, we observe that Rg = [1, 3]. Rg = [1, 3] Df = [0, ∞) ⇒ Rg ⊆ Df Therefore, fg exists. (b) f(g(x)) = f(2x + 1) = (2x + 1)2 Dfg = Dg = [0, 1] (c) To find Rfg, Step 1. Draw the graph of f. Step 2. For the domain, set it as Range g, which in this case is [1, 3] Step 3. Find the range under this new domain. −1 1 3 1 9 y = f(x) x y Therefore, Rfg = [1, 9]. www.MathAcademy.sg 11 © 2019 Math Academy
- 12. Example 8 (2009/NYJC/Prelim/I/2a modified). The functions f and g are defined by f : x → x2 − 1, x ∈ R, g : x → √ x + 4, x ∈ R+ . (i) Show that the composite function fg exists. (ii) Define fg in similar form. (iii) Find the range of fg. [ii) fg(x) = x + 3, x ∈ R+ iii) (3,∞)] www.MathAcademy.sg 12 © 2019 Math Academy
- 13. 4.2 ff−1 (x) and f−1 f(x) A function given by f(x) = x is called an identity function. It maps any value to itself. f−1f Diagram for function f−1f(x) a b a x ff−1(x) = f−1f(x) = x Dff−1 = Df−1 . Df−1f = Df . Key Observations (a) Both ff−1(x) and f−1f(x) are equal to x. (b) ff−1(x) and f−1f(x) have different domains. (Sketch the correct domain on a graph) Example 9. Let the function f be defined by f : x → x + 4, x ∈ [0, ∞). Sketch the graphs of the functions ff−1 and, f−1f on 2 separate graphs. www.MathAcademy.sg 13 © 2019 Math Academy
- 14. Example 10. Consider the following functions: g : x → 1 x − 3 x ∈ R, x ̸= 3 g−1 h : x → 1 x , x ≥ 0 Find the function h. Solution: h = gg−1 h = g(g−1 h) = g ( 1 x ) = 1 1 x − 3 = x 1 − 3x Domain h = domain g−1 h = [0, ∞) Example 11. Consider the following functions: fg : x → x2 − 6x + 14 x ∈ R+ g−1 : x → x + 5, x ∈ R Find the function f. [f(x) = x2 + 4x + 9, x ∈ R] www.MathAcademy.sg 14 © 2019 Math Academy
- 15. Example 12 (Self-inverse Functions). The functions f and g are defined as follows: f : x → 5 − x 1 − x , x ∈ R, x ̸= 1. (i) Explain why f has an inverse, f−1, and show that f−1 = f. (ii) Evaluate f51(4). Solution: (i) f(x) = 5−x 1−x = 1 + 4 1−x . f′ (x) = 4(−1)(1 − x)−2 (−1) = 4 (1 − x)2 Since f′(x) > 0 for all x ∈ R, x ̸= 1, f is 1-1 and hence has an inverse. y = 1 + 4 1 − x y − 1 = 4 1 − x 1 y − 1 = 1 − x 4 4 y − 1 = 1 − x 1 + 4 1 − y = x ∴ f−1 (x) = 1 + 4 1 − x 1 1 x y From the graph of f, Rf = R{1}. ∴ Df−1 = R{1} = Df . Hence f = f−1. (ii) f(x) = f−1 (x) f2 (x) = x f3 (x) = f ( f2 (x) ) = f(x) f4 (x) = f ( f3 (x) ) = f(f(x)) = f2 (x) = x ... ∴ f51 (x) = f(x) ∴ f51 (4) = f(4) = 5 − 4 1 − 4 = −1 3 www.MathAcademy.sg 15 © 2019 Math Academy