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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.

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Operation on functions

The document discusses operations that can be performed on functions, including addition, subtraction, multiplication, division, and composition. It provides examples of evaluating each type of operation on functions by first evaluating the individual functions at a given value or variable and then performing the indicated operation on the results. Composition involves evaluating the inner function first and substituting its result into the outer function.

Integration SPM

The document contains examples of indefinite integrals of various functions:
1) Finding the antiderivatives of polynomials like 4x3 + 3x - 2.
2) Finding an antiderivative involving a differential equation like dy/dx = 4x3 - 4x.
3) Evaluating integrals involving rational functions like ∫(3 - 2/x2 + 6x3)dx.
4) Finding antiderivatives of expressions involving radicals like ∫((x2+3)2/x2)dx.
5) Solving differential equations and evaluating integrals using substitution.

Remainder theorem

The remainder theorem states that when a polynomial f(x) is divided by (x - c), the remainder is equal to f(c). This is based on the concept of synthetic division, where a polynomial is divided into a quotient and remainder. Some examples are worked out to demonstrate evaluating a polynomial f(x) at a value c to find the remainder when divided by (x - c), in accordance with the remainder theorem.

Sequences, Series, and the Binomial Theorem

An infinite sequence is a function whose domain is the set of natural numbers, while a finite sequence has a domain of natural numbers up to some limit. A sequence can be described by its general term, which gives a rule for calculating each term based on its position in the sequence. The sum of the terms of a sequence is called a series, which is finite if it includes a finite number of terms and infinite if it includes all terms.

MIT Math Syllabus 10-3 Lesson 6: Equations

This document provides information about equations, including definitions, properties, and steps to solve different types of equations. It defines an equation as a statement about the equality of two expressions. Equations can be solved to find all values that satisfy the equation. The key properties of equality, including addition/subtraction and multiplication/division properties, allow equivalent equations to be formed in order to solve equations. The document discusses linear equations, absolute value equations, formulas, and using equations to model real-world situations.

Introduction to Logarithm

FellowBuddy.com is an innovative platform that brings students together to share notes, exam papers, study guides, project reports and presentation for upcoming exams.
We connect Students who have an understanding of course material with Students who need help.
Benefits:-
# Students can catch up on notes they missed because of an absence.
# Underachievers can find peer developed notes that break down lecture and study material in a way that they can understand
# Students can earn better grades, save time and study effectively
Our Vision & Mission – Simplifying Students Life
Our Belief – “The great breakthrough in your life comes when you realize it, that you can learn anything you need to learn; to accomplish any goal that you have set for yourself. This means there are no limits on what you can be, have or do.”
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Graphing polynomials

This document discusses how to graph polynomial functions and find local extrema. It provides instructions on making a table of values, plotting points, connecting them with a smooth curve, and checking end behavior based on the degree and leading coefficient. Extrema are defined as local maxima or minima, where the graph changes from increasing to decreasing. Examples are given to demonstrate graphing polynomials and finding turning points that indicate local extrema.

Operation on functions

The document discusses operations that can be performed on functions, including addition, subtraction, multiplication, division, and composition. It provides examples of evaluating each type of operation on functions by first evaluating the individual functions at a given value or variable and then performing the indicated operation on the results. Composition involves evaluating the inner function first and substituting its result into the outer function.

Integration SPM

The document contains examples of indefinite integrals of various functions:
1) Finding the antiderivatives of polynomials like 4x3 + 3x - 2.
2) Finding an antiderivative involving a differential equation like dy/dx = 4x3 - 4x.
3) Evaluating integrals involving rational functions like ∫(3 - 2/x2 + 6x3)dx.
4) Finding antiderivatives of expressions involving radicals like ∫((x2+3)2/x2)dx.
5) Solving differential equations and evaluating integrals using substitution.

Remainder theorem

The remainder theorem states that when a polynomial f(x) is divided by (x - c), the remainder is equal to f(c). This is based on the concept of synthetic division, where a polynomial is divided into a quotient and remainder. Some examples are worked out to demonstrate evaluating a polynomial f(x) at a value c to find the remainder when divided by (x - c), in accordance with the remainder theorem.

Sequences, Series, and the Binomial Theorem

An infinite sequence is a function whose domain is the set of natural numbers, while a finite sequence has a domain of natural numbers up to some limit. A sequence can be described by its general term, which gives a rule for calculating each term based on its position in the sequence. The sum of the terms of a sequence is called a series, which is finite if it includes a finite number of terms and infinite if it includes all terms.

MIT Math Syllabus 10-3 Lesson 6: Equations

This document provides information about equations, including definitions, properties, and steps to solve different types of equations. It defines an equation as a statement about the equality of two expressions. Equations can be solved to find all values that satisfy the equation. The key properties of equality, including addition/subtraction and multiplication/division properties, allow equivalent equations to be formed in order to solve equations. The document discusses linear equations, absolute value equations, formulas, and using equations to model real-world situations.

Introduction to Logarithm

FellowBuddy.com is an innovative platform that brings students together to share notes, exam papers, study guides, project reports and presentation for upcoming exams.
We connect Students who have an understanding of course material with Students who need help.
Benefits:-
# Students can catch up on notes they missed because of an absence.
# Underachievers can find peer developed notes that break down lecture and study material in a way that they can understand
# Students can earn better grades, save time and study effectively
Our Vision & Mission – Simplifying Students Life
Our Belief – “The great breakthrough in your life comes when you realize it, that you can learn anything you need to learn; to accomplish any goal that you have set for yourself. This means there are no limits on what you can be, have or do.”
Like Us - https://www.facebook.com/FellowBuddycom

Graphing polynomials

This document discusses how to graph polynomial functions and find local extrema. It provides instructions on making a table of values, plotting points, connecting them with a smooth curve, and checking end behavior based on the degree and leading coefficient. Extrema are defined as local maxima or minima, where the graph changes from increasing to decreasing. Examples are given to demonstrate graphing polynomials and finding turning points that indicate local extrema.

Inverse trig functions

The document discusses inverse trigonometric functions and how to define their inverses by restricting the domains of the trig functions. It explains that the sine function's inverse is defined on [-1,1] and the cosine function's inverse is defined on [0,π]. Similarly, the tangent function's inverse is defined on (-π/2, π/2). Graphs and examples of the inverse sine, cosine, and tangent functions are provided.

2011 p5-math-sa1-mgs

The document is a 15-page mathematics exam for Primary 5 students in Methodist Girls' School consisting of multiple choice and written response questions testing concepts like fractions, ratios, percentages, geometry, word problems, and more. It includes instructions to candidates, questions ranging from 1 to 5 marks, and spaces to show working and write answers. The exam covers topics commonly found in Primary 5 mathematics curricula.

Types of function

The document discusses different types of functions including:
1) Surjective functions where the range equals the co-domain.
2) Injective functions where distinct inputs have distinct outputs.
3) Bijective functions which are both injective and surjective.
It also discusses even and odd functions, inverses, composites, and examples of calculating different functions.

General term

The document provides examples of finding the general term of different sequences. For each sequence, it identifies the pattern between the terms and determines the formula for the nth term. The general terms provided are:
1) f(n) = 2n for the sequence 2, 4, 6, 8,...
2) f(n) = 2n - 1 for the sequence 1, 3, 5, 7,...
3) f(n) = 2n for the sequence 2, 4, 8, 16,...
4) f(n) = n^2 for the sequence 1, 4, 9, 16,...
It also provides guidelines for finding the general term such as looking for a common difference,

Arithmetic progression

1. An arithmetic progression (AP) is a sequence of numbers where the difference between successive terms is constant. This constant difference is called the common difference.
2. To define an AP, we need to know the first term (a1) and the common difference (d). The nth term of an AP can be calculated as an = a1 + (n-1)d.
3. Megha's annual salary increases by Rs. 1,000 each year, forming an AP. Her salary after 23 years can be calculated as a23 = Rs. 10,000 + 22(Rs. 1,000) = Rs. 32,000

Mathematics 8 Systems of Linear Inequalities

This learner's module discusses and help the students about the topic Systems of Linear Inequalities. It includes definition, examples, applications of Systems of Linear Inequalities.

6.5 determinant x

The document defines the determinant of a square matrix. For a 1x1 matrix with value k, the determinant is defined to be k. For a 2x2 matrix with values a, b, c, d, the determinant is defined as ad - bc. This definition is motivated geometrically as representing the signed area of the parallelogram formed by the vector points (a,b) and (c,d). It is also motivated algebraically in that a system of equations has a unique solution if and only if the determinant of the coefficient matrix is non-zero. Cramer's rule is presented for solving systems of linear equations.

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

Fundamental theorem of arithmatic

This document contains slides on number theory concepts including:
1) The fundamental theorem of arithmetic and its proof
2) Applications of the fundamental theorem including finding the total number of divisors and sum of divisors of a number
3) A proof that there are infinitely many prime numbers
4) Determining whether a given number is prime or composite
The document provides explanations, proofs, and examples related to these key number theory topics.

Module 3 quadratic functions

This module covers quadratic functions and equations. Students will learn to determine the zeros of quadratic functions by relating them to the roots of quadratic equations. They will also learn to find the roots of quadratic equations using factoring, completing the square, and the quadratic formula. The module aims to help students derive quadratic functions given certain conditions like the zeros, a table of values, or a graph.

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.

Absolute Value Equations

The document discusses solving absolute value equations. It explains that absolute value is the distance a number is from 0, and provides examples. It then states that when solving absolute value equations, two separate equations must be created to account for the number inside the absolute value being positive or negative. Steps are provided for solving sample absolute value equations.

4.1 inverse functions t

This document provides examples and exercises on inverse functions. It first shows an example of finding the inverse of the function f(x) = 2x-3/(x+2). It gives the steps to solve for x in terms of y and obtain the inverse function f^-1(x) = -2x-3/(x-2). It then asks the reader to verify that f(f^-1(x)) = x. The exercises provide 20 functions and ask the reader to find their inverses and verify the inverses are correct. It also gives graphs of 8 functions and asks the reader to determine properties of the inverse graphs, including their domains and ranges and any fixed or end points.

Chapter1 functions

This document provides information about functions from Additional Mathematics Module Form 4. It defines relations and different ways to represent them using arrow diagrams, ordered pairs, and graphs. It explains the concepts of domain, codomain, range, objects and images. Functions are defined as a special type of relation where each object has only one image. Function notation and evaluation are demonstrated through examples. Composite functions are introduced as the combined effect of two functions gf(x). Examples are provided to show how to determine one function given information about the composite function.

Hyperbolas

The document provides instructions for graphing two hyperbolas based on their standard equations. It first gives the equations x^2 - y^2/4 = 1 and y^2 - x^2/9 = 1 and explains how to locate and draw the vertices, foci, asymptotes, and hyperbola shape for each one. It also reviews key properties of hyperbolas such as their transverse axis and the relationship between the foci and constant difference of distances along the hyperbola.

Equiations and inequalities

Here are the key steps to solve word problems involving linear equations:
1. Read the problem carefully and identify the important details.
2. Define variables to represent unknown quantities.
3. Write a mathematical expression relating the variables based on the context of the problem.
4. Form an equation and solve it using proper order of operations.
5. Check that the solution makes sense in the context of the original problem.

Higher order derivatives

Higher order derivatives are obtained by repeatedly taking the derivative of a function or its derivatives. The order of a derivative refers to how many times differentiation has been performed. To find a higher order derivative, one simply takes the derivative of the existing derivative. For example, to get the third derivative f'''(x) of a function f(x), one would take the derivative of the second derivative f''(x). Higher derivatives provide important information about the curvature and flexibility of a function at different points.

Derivatives

This document provides information about derivatives and their applications:
1. It defines the derivative as the limit of the difference quotient, and explains how to calculate derivatives using first principles. It also covers rules for finding derivatives of sums, products, quotients, exponentials, and logarithmic functions.
2. Higher order derivatives are introduced, with examples of how to take second and third derivatives.
3. Applications of derivatives like finding velocity and acceleration from a position-time function are demonstrated. Maximum/minimum values and how to find local and absolute extrema are also discussed with an example.

Exponents)

This document discusses exponents and exponential functions. It defines what exponents are, how they are read, and some of their key properties like the product rule for exponents. It also examines exponential functions where the base is raised to the power of x. The graphs of exponential functions where the base is greater than 1, between 0 and 1, and equal to 1 are explored. The graphs are always increasing, have a horizontal asymptote at y=0, and a y-intercept of (0,1). Exponential equations are also briefly covered.

584 fundamental theorem of calculus

The document summarizes the Fundamental Theorem of Calculus, which establishes a connection between computing integrals (areas under curves) and computing derivatives. It shows that if f is continuous on an interval [a,b] and F is defined by integrating f, then F' = f. Graphs and examples are provided to justify this theorem geometrically and demonstrate its applications to computing derivatives and integrals.

Inverse trig functions

The document discusses inverse trigonometric functions and how to define their inverses by restricting the domains of the trig functions. It explains that the sine function's inverse is defined on [-1,1] and the cosine function's inverse is defined on [0,π]. Similarly, the tangent function's inverse is defined on (-π/2, π/2). Graphs and examples of the inverse sine, cosine, and tangent functions are provided.

2011 p5-math-sa1-mgs

The document is a 15-page mathematics exam for Primary 5 students in Methodist Girls' School consisting of multiple choice and written response questions testing concepts like fractions, ratios, percentages, geometry, word problems, and more. It includes instructions to candidates, questions ranging from 1 to 5 marks, and spaces to show working and write answers. The exam covers topics commonly found in Primary 5 mathematics curricula.

Types of function

The document discusses different types of functions including:
1) Surjective functions where the range equals the co-domain.
2) Injective functions where distinct inputs have distinct outputs.
3) Bijective functions which are both injective and surjective.
It also discusses even and odd functions, inverses, composites, and examples of calculating different functions.

General term

The document provides examples of finding the general term of different sequences. For each sequence, it identifies the pattern between the terms and determines the formula for the nth term. The general terms provided are:
1) f(n) = 2n for the sequence 2, 4, 6, 8,...
2) f(n) = 2n - 1 for the sequence 1, 3, 5, 7,...
3) f(n) = 2n for the sequence 2, 4, 8, 16,...
4) f(n) = n^2 for the sequence 1, 4, 9, 16,...
It also provides guidelines for finding the general term such as looking for a common difference,

Arithmetic progression

1. An arithmetic progression (AP) is a sequence of numbers where the difference between successive terms is constant. This constant difference is called the common difference.
2. To define an AP, we need to know the first term (a1) and the common difference (d). The nth term of an AP can be calculated as an = a1 + (n-1)d.
3. Megha's annual salary increases by Rs. 1,000 each year, forming an AP. Her salary after 23 years can be calculated as a23 = Rs. 10,000 + 22(Rs. 1,000) = Rs. 32,000

Mathematics 8 Systems of Linear Inequalities

This learner's module discusses and help the students about the topic Systems of Linear Inequalities. It includes definition, examples, applications of Systems of Linear Inequalities.

6.5 determinant x

The document defines the determinant of a square matrix. For a 1x1 matrix with value k, the determinant is defined to be k. For a 2x2 matrix with values a, b, c, d, the determinant is defined as ad - bc. This definition is motivated geometrically as representing the signed area of the parallelogram formed by the vector points (a,b) and (c,d). It is also motivated algebraically in that a system of equations has a unique solution if and only if the determinant of the coefficient matrix is non-zero. Cramer's rule is presented for solving systems of linear equations.

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

Fundamental theorem of arithmatic

This document contains slides on number theory concepts including:
1) The fundamental theorem of arithmetic and its proof
2) Applications of the fundamental theorem including finding the total number of divisors and sum of divisors of a number
3) A proof that there are infinitely many prime numbers
4) Determining whether a given number is prime or composite
The document provides explanations, proofs, and examples related to these key number theory topics.

Module 3 quadratic functions

This module covers quadratic functions and equations. Students will learn to determine the zeros of quadratic functions by relating them to the roots of quadratic equations. They will also learn to find the roots of quadratic equations using factoring, completing the square, and the quadratic formula. The module aims to help students derive quadratic functions given certain conditions like the zeros, a table of values, or a graph.

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.

Absolute Value Equations

The document discusses solving absolute value equations. It explains that absolute value is the distance a number is from 0, and provides examples. It then states that when solving absolute value equations, two separate equations must be created to account for the number inside the absolute value being positive or negative. Steps are provided for solving sample absolute value equations.

4.1 inverse functions t

This document provides examples and exercises on inverse functions. It first shows an example of finding the inverse of the function f(x) = 2x-3/(x+2). It gives the steps to solve for x in terms of y and obtain the inverse function f^-1(x) = -2x-3/(x-2). It then asks the reader to verify that f(f^-1(x)) = x. The exercises provide 20 functions and ask the reader to find their inverses and verify the inverses are correct. It also gives graphs of 8 functions and asks the reader to determine properties of the inverse graphs, including their domains and ranges and any fixed or end points.

Chapter1 functions

This document provides information about functions from Additional Mathematics Module Form 4. It defines relations and different ways to represent them using arrow diagrams, ordered pairs, and graphs. It explains the concepts of domain, codomain, range, objects and images. Functions are defined as a special type of relation where each object has only one image. Function notation and evaluation are demonstrated through examples. Composite functions are introduced as the combined effect of two functions gf(x). Examples are provided to show how to determine one function given information about the composite function.

Hyperbolas

The document provides instructions for graphing two hyperbolas based on their standard equations. It first gives the equations x^2 - y^2/4 = 1 and y^2 - x^2/9 = 1 and explains how to locate and draw the vertices, foci, asymptotes, and hyperbola shape for each one. It also reviews key properties of hyperbolas such as their transverse axis and the relationship between the foci and constant difference of distances along the hyperbola.

Equiations and inequalities

Here are the key steps to solve word problems involving linear equations:
1. Read the problem carefully and identify the important details.
2. Define variables to represent unknown quantities.
3. Write a mathematical expression relating the variables based on the context of the problem.
4. Form an equation and solve it using proper order of operations.
5. Check that the solution makes sense in the context of the original problem.

Higher order derivatives

Higher order derivatives are obtained by repeatedly taking the derivative of a function or its derivatives. The order of a derivative refers to how many times differentiation has been performed. To find a higher order derivative, one simply takes the derivative of the existing derivative. For example, to get the third derivative f'''(x) of a function f(x), one would take the derivative of the second derivative f''(x). Higher derivatives provide important information about the curvature and flexibility of a function at different points.

Derivatives

This document provides information about derivatives and their applications:
1. It defines the derivative as the limit of the difference quotient, and explains how to calculate derivatives using first principles. It also covers rules for finding derivatives of sums, products, quotients, exponentials, and logarithmic functions.
2. Higher order derivatives are introduced, with examples of how to take second and third derivatives.
3. Applications of derivatives like finding velocity and acceleration from a position-time function are demonstrated. Maximum/minimum values and how to find local and absolute extrema are also discussed with an example.

Inverse trig functions

Inverse trig functions

2011 p5-math-sa1-mgs

2011 p5-math-sa1-mgs

Types of function

Types of function

Functions

Functions

General term

General term

Arithmetic progression

Arithmetic progression

Mathematics 8 Systems of Linear Inequalities

Mathematics 8 Systems of Linear Inequalities

6.5 determinant x

6.5 determinant x

Functions 1 - Math Academy - JC H2 maths A levels

Functions 1 - Math Academy - JC H2 maths A levels

Fundamental theorem of arithmatic

Fundamental theorem of arithmatic

Module 3 quadratic functions

Module 3 quadratic functions

Form 4 formulae and note

Form 4 formulae and note

Absolute Value Equations

Absolute Value Equations

4.1 inverse functions t

4.1 inverse functions t

Chapter1 functions

Chapter1 functions

Hyperbolas

Hyperbolas

Equiations and inequalities

Equiations and inequalities

5.4 mutually exclusive events

5.4 mutually exclusive events

Higher order derivatives

Higher order derivatives

Derivatives

Derivatives

Exponents)

This document discusses exponents and exponential functions. It defines what exponents are, how they are read, and some of their key properties like the product rule for exponents. It also examines exponential functions where the base is raised to the power of x. The graphs of exponential functions where the base is greater than 1, between 0 and 1, and equal to 1 are explored. The graphs are always increasing, have a horizontal asymptote at y=0, and a y-intercept of (0,1). Exponential equations are also briefly covered.

584 fundamental theorem of calculus

The document summarizes the Fundamental Theorem of Calculus, which establishes a connection between computing integrals (areas under curves) and computing derivatives. It shows that if f is continuous on an interval [a,b] and F is defined by integrating f, then F' = f. Graphs and examples are provided to justify this theorem geometrically and demonstrate its applications to computing derivatives and integrals.

Algebra 2 Unit 5 Lesson 7

This document discusses inverses of functions. It provides examples of finding the inverse of various functions by switching the x and y coordinates, solving for y, and determining if the inverse is a function. Key points made are: to find the inverse change the coordinate pair; a function and its inverse are reflections over y=x; when composing a function with its inverse, you get back the original function. Examples are worked through and conclusions are drawn about the domains and ranges of inverses.

Exercise #10 notes

This document provides examples for graphing reciprocals of functions. It explains that for reciprocals, smaller numbers on the x-axis become larger on the y-axis, and vice versa. An example graphs the reciprocal of f(x)=1/x^2. It notes that reciprocals can be written in the form f(x)=k/x+h, where k and h are determined from the original function. Further examples graph the reciprocals of f(x)=2/x+4 and f(x)=sin(x).

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.

gfg

This document provides an introduction to symbolic math in MATLAB. It discusses differentiation and integration of functions using symbolic operators. Differentiation is defined as finding the rate of change of a function with respect to a variable. Integration finds the original function given its derivative. The document provides examples of differentiating and integrating simple functions in MATLAB's symbolic toolbox and exercises for the reader to practice.

Pc12 sol c04_ptest

This document contains practice problems and solutions for combining functions. It includes:
1. Multiple choice questions about compositions of functions.
2. Explicit equations for compositions and composite functions using given functions f(x), g(x), h(x), and k(x).
3. Graphing composite functions and determining their domains.
4. Evaluating composite functions for given values of x.
5. Writing composite functions as sums or compositions of simpler functions.

F.Komposisi

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.

Algebra 2 Unit 5 Lesson 2

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

Lesson 51

The document summarizes operations and composition of functions. It defines a function as a relationship between inputs x and outputs y where each x has a single y value. It describes operations on functions such as addition, subtraction and multiplication by adding, subtracting or multiplying the outputs of two functions with the same input. Composition of functions f and g is defined as f(g(x)) where the output of g is used as the input for f. An example shows finding the composition (f∘g)(x) and (g∘f)(x) which are usually not equal. Students are assigned exercises to practice these concepts.

Exercise #11 notes

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.

Lesson 15: Inverse Functions and Logarithms

The inverse of a function "undoes" the effect of the function. We look at the implications of that property in the derivative, as well as logarithmic functions, which are inverses of exponential functions.

Pc12 sol c04_review

This document provides solutions to review problems involving combining functions through addition, subtraction, multiplication, division, and composition. Some key examples include:
- Sketching the graphs of f(x) + g(x), f(x) - g(x), f(x) * g(x), and f(x) / g(x) given the graphs of f(x) and g(x)
- Writing explicit equations for combinations of functions and determining their domains and ranges
- Evaluating composite functions like f(g(x)) and g(f(x)) given definitions of f(x) and g(x)
- Determining if two functions are inverses using their compositions

Exponential functions

The document discusses exponential functions of the form f(x) = ax, where a is the base. It defines exponential functions and provides examples of evaluating them. The key aspects of exponential graphs are that they increase rapidly as x increases and have a horizontal asymptote of y = 0 if a > 1 or y = 0 if 0 < a < 1. Examples are given of sketching graphs of exponential functions and stating their domains and ranges. The graph of the natural exponential function f(x) = ex is also discussed.

2.2 Polynomial Function Notes

The document discusses polynomial functions, including how to graph common polynomials, find zeros of polynomials, and write polynomials given their roots. It provides examples of matching polynomial equations to their graphs, finding the real zeros of polynomials by factoring, and writing polynomials when given the roots. The document also covers how to use a graphing calculator to find the zeros of polynomials.

Chapter 3

1) A quadratic function is an equation of the form f(x) = ax^2 + bx + c, where a ≠ 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.

Day 5 examples

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.

Mc ty-explogfns-2009-1

Exponential and logarithm functions are important in both theory and practice. They examine the graphs of exponential functions f(x)=ax where a>0 and logarithm functions f(x)=loga(x) where a>0. It is important to practice these functions so their properties become intuitive. Key properties include exponential functions where a>1 increase rapidly for positive x and 0<a<1 increase for decreasing negative x, and both pass through (0,1). The natural logarithm function f(x)=ln(x) is particularly important.

Lesson 15: Inverse Functions and Logarithms

The inverse of a function "undoes" the effect of the function. We look at the implications of that property in the derivative, as well as logarithmic functions, which are inverses of exponential functions.

Functions

This chapter discusses 4 concepts: (1) function relations involving domains and ranges, (2) absolute value functions, (3) composite functions using substitution, and (4) inverse functions by making "x" equal to "y" and solving for y. Examples are provided for each concept to demonstrate finding outputs and inverses of simple functions.

Exponents)

Exponents)

584 fundamental theorem of calculus

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Logarithms

Logarithms

gfg

gfg

Pc12 sol c04_ptest

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Functions

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This document provides guidance for mathematics teachers to improve student performance in Additional Mathematics for the SPM 2009 exam. It identifies common weaknesses and mistakes by student category (very weak to excellent). Suggestions are given to rectify issues for different topics in Paper 1 and Paper 2, such as functions, quadratic equations, vectors, and integration. For weaker students, the focus is on getting partial marks. For stronger students, emphasis is placed on careless mistakes. Teachers are advised to provide targeted practice addressing specific weaknesses.

Skills In Add Maths

This document provides examples of solving equations, expanding and factorizing expressions, solving simultaneous equations, working with indices and logarithms. It includes over 100 problems across these topics for students to practice. The problems range in complexity from basic single-step equations to multi-part logarithmic expressions and systems of simultaneous equations.

Add10kelantan

1. This document appears to be an answer key or marking scheme for a mathematics exam with 25 questions. It provides the answers, workings, or mark schemes for each question on the exam.
2. For each question, it lists the number of marks awarded for the full or partial answers. The total marks are tallied at the end.
3. The document contains detailed mathematical solutions and workings for the questions, evaluating answers for correctness according to set schemes.

Soalan ptk tambahan

1. Dokumen tersebut membincangkan perancangan dan pelaksanaan pengajaran pembelajaran di dalam bilik darjah.
2. Beberapa aspek utama yang dibincangkan termasuk objektif pengajaran, penggunaan sumber dan teknik pengajaran, serta kemahiran guru.
3. Dokumen ini memberikan panduan kepada guru dalam merancang dan melaksanakan proses pengajaran dan pembelajaran yang berkesan.

Attachments 2012 04_1

Attachments 2012 04_1

Janjang aritmetik

Janjang aritmetik

Teknik Peningkatan Prestasi

Teknik Peningkatan Prestasi

Skills In Add Maths

Skills In Add Maths

Add10kelantan

Add10kelantan

Add10sabah

Add10sabah

Add10terengganu

Add10terengganu

Add10perak

Add10perak

Add10ns

Add10ns

Add10johor

Add10johor

Strategi pengajaran pembelajaran

Strategi pengajaran pembelajaran

Soalan ptk tambahan

Soalan ptk tambahan

Refleksi

Refleksi

Perancangan pengajaran pembelajaran

Perancangan pengajaran pembelajaran

Penilaian

Penilaian

Pengurusan bilik darjah

Pengurusan bilik darjah

Pengurusan murid

Pengurusan murid

Penguasaan mata pelajaran

Penguasaan mata pelajaran

Penggunaan sumber dalam p & p

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Pemulihan dan pengayaan

Pemulihan dan pengayaan

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𝐃𝐢𝐬𝐜𝐮𝐬𝐬 𝐭𝐡𝐞 𝐄𝐏𝐏 𝐂𝐮𝐫𝐫𝐢𝐜𝐮𝐥𝐮𝐦 𝐢𝐧 𝐭𝐡𝐞 𝐏𝐡𝐢𝐥𝐢𝐩𝐩𝐢𝐧𝐞𝐬:
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𝐄𝐱𝐩𝐥𝐚𝐢𝐧 𝐭𝐡𝐞 𝐍𝐚𝐭𝐮𝐫𝐞 𝐚𝐧𝐝 𝐒𝐜𝐨𝐩𝐞 𝐨𝐟 𝐚𝐧 𝐄𝐧𝐭𝐫𝐞𝐩𝐫𝐞𝐧𝐞𝐮𝐫:
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Denis is a dynamic and results-driven Chief Information Officer (CIO) with a distinguished career spanning information systems analysis and technical project management. With a proven track record of spearheading the design and delivery of cutting-edge Information Management solutions, he has consistently elevated business operations, streamlined reporting functions, and maximized process efficiency.
Certified as an ISO/IEC 27001: Information Security Management Systems (ISMS) Lead Implementer, Data Protection Officer, and Cyber Risks Analyst, Denis brings a heightened focus on data security, privacy, and cyber resilience to every endeavor.
His expertise extends across a diverse spectrum of reporting, database, and web development applications, underpinned by an exceptional grasp of data storage and virtualization technologies. His proficiency in application testing, database administration, and data cleansing ensures seamless execution of complex projects.
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A wound is a break in the integrity of the skin or tissues, which may be associated with disruption of the structure and function.
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Healing can occur in two ways: Regeneration and Repair
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Syllabus
Chapter-1
Introduction to objective, scope and outcome the subject
Chapter 2
Introduction: Scope and Specialization of Civil Engineering, Role of civil Engineer in Society, Impact of infrastructural development on economy of country.
Chapter 3
Surveying: Object Principles & Types of Surveying; Site Plans, Plans & Maps; Scales & Unit of different Measurements.
Linear Measurements: Instruments used. Linear Measurement by Tape, Ranging out Survey Lines and overcoming Obstructions; Measurements on sloping ground; Tape corrections, conventional symbols. Angular Measurements: Instruments used; Introduction to Compass Surveying, Bearings and Longitude & Latitude of a Line, Introduction to total station.
Levelling: Instrument used Object of levelling, Methods of levelling in brief, and Contour maps.
Chapter 4
Buildings: Selection of site for Buildings, Layout of Building Plan, Types of buildings, Plinth area, carpet area, floor space index, Introduction to building byelaws, concept of sun light & ventilation. Components of Buildings & their functions, Basic concept of R.C.C., Introduction to types of foundation
Chapter 5
Transportation: Introduction to Transportation Engineering; Traffic and Road Safety: Types and Characteristics of Various Modes of Transportation; Various Road Traffic Signs, Causes of Accidents and Road Safety Measures.
Chapter 6
Environmental Engineering: Environmental Pollution, Environmental Acts and Regulations, Functional Concepts of Ecology, Basics of Species, Biodiversity, Ecosystem, Hydrological Cycle; Chemical Cycles: Carbon, Nitrogen & Phosphorus; Energy Flow in Ecosystems.
Water Pollution: Water Quality standards, Introduction to Treatment & Disposal of Waste Water. Reuse and Saving of Water, Rain Water Harvesting. Solid Waste Management: Classification of Solid Waste, Collection, Transportation and Disposal of Solid. Recycling of Solid Waste: Energy Recovery, Sanitary Landfill, On-Site Sanitation. Air & Noise Pollution: Primary and Secondary air pollutants, Harmful effects of Air Pollution, Control of Air Pollution. . Noise Pollution Harmful Effects of noise pollution, control of noise pollution, Global warming & Climate Change, Ozone depletion, Greenhouse effect
Text Books:
1. Palancharmy, Basic Civil Engineering, McGraw Hill publishers.
2. Satheesh Gopi, Basic Civil Engineering, Pearson Publishers.
3. Ketki Rangwala Dalal, Essentials of Civil Engineering, Charotar Publishing House.
4. BCP, Surveying volume 1

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- 1. TOPIC : FUNCTIONS Absolute Value Functions Eg: f (x) = x − 2 Function Notations 1) Find the possible values of x • f :x → x+2 if f(x) = 3 CONCEPTS MAP (means the function f maps x onto ” x + 2 Solution x-2 =3 , x=5 • f ( x) = x + 2 is read as ’ f of x is x + 2 - (x - 2) = 3 , x = -1 RELATIONS • x is the object , f (x) is the image 2) Sketch the Graph of Examples: a) If 5 is the object , find the image f (x) = x − 2 for the Solution : f ( 5) = 5 + 2 = 7, domain -1 ≤ x ≤ 6 How to represent Types Of Relations Solution: Relations b) Given f (3 y) = 11 , find y. x = -1 , f(x) = 3 1. One-to-one relation Solution: f ( 3y) = 3y + 2 = 11, x = 5 , f(x) = 3 1) Arrow diagram 3y=9 , f(x) = x-3 = 0, x = 3 y=3 f (x) a• •1 2 4 3 5 F 3 b• 6 7 U Composite Functions •2 N • If a function f is followed by a function g , x c• •3 we obtain the composite function g f . -1 0 3 6 2. One-to-many relation C 3) Corresponding f g d• •4 T • Range: x f (x) gf 0 ≤ f(x) ≤ 3 2) Ordered Pairs 1 4 I gf (a,1) (b,2), (c,2), 3 5 O Inverse functions (f-1) (d,3) 6 • In general gf ≠ fg . N • Concept: f(x) = y , Then, f –1 (y) = x 3 3) Graph S • How to determine composite function: • Eg: 3. Many-to-one relation Example : Given f : x x +1 Given f : x 2x + 1. Find f –1 2 × g : x 2x Solution: 1 × × 7 6 Determine i.) f g ii ) f 2 y −1 y = 2x + 1 , x = s 9 Solutions; 2 × 10 i.) fg (x) = f (g (x) (ii) f 2 (x) = f f (x) x −1 11 14 = f ( 2x ) = f (x +1) So ; f –1 (x) = a b c d 2 = 2x + 1 = (x +1) +1 = x + 2 •Note: Only one – to – one functions will Objects : a, b, c, d Eg.• Given f : x 2x and give one – to – one inverse functions. 4. Many-to-many relation Domain : {a, b, c, d} g f: x 3x - 1, find g. • ff-1(x) = f -1f (x) = x Solution: g (f(x) = 3 x -1 Eg.• Given f : x 2x and Codomain : {1, 2, 3,4} 1 5 g( 2x) = 3x - 1 f g: x x + 3, find g. f(x) = 2x+1 x −1 Solution: f (g(x) ) = x + 3 f −1( x ) = y 2 Images : 1,2,3 7 Lets 2x = y, x= , 2 g(x) = x + 3 2 2 ax + b − dx + b Range : { 1, 2, 3 } x+3 f ( x) = f ( x) = y 3x g ( x) = cx + d cx − a g(y) = 3( ) -1,g(x) = −1 2 2 2