This document provides examples and exercises on determining composite functions from given functions. It includes:
- Examples of determining possible functions f and g for composite functions like y = (x + 4)2 and y = √x + 5.
- A table sketching and finding domains and ranges for various composite functions like y = f(f(x)) and y = f(g(x)) given functions f(x) and g(x).
- Exercises to determine composite functions f(g(x)) and possible functions f, g, and h for more complex functions like y = x2 - 6x + 5.
- Questions about restrictions on variables and domains for composite functions
This document contains sample solutions to checkpoint questions about combining functions. It provides the steps to:
1) Sketch the graph of y = f(x)/g(x) given the graphs of y = f(x) and y = g(x)
2) Write explicit equations for functions like g(x), h(x), and k(x) that satisfy an equation like f(x) = g(x) - h(x) - k(x)
3) Determine the domain and range of functions formed by combining basic functions using operations like addition, subtraction, multiplication, and division.
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
1. The document provides graphs and equations for functions f(x), g(x), and their combinations. It asks the reader to sketch graphs, determine domains and ranges, and solve related problems.
2. The key combinations are addition, subtraction, multiplication, and division of f(x) and g(x). Their domains and ranges are identified from the original function graphs.
3. For a combination like f(x) + g(x), the domain is the same as the more restrictive of the two original functions, while the range includes all outputs equal to or greater than the original function ranges.
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.
1. The document provides examples of evaluating composite functions using tables and graphs of component functions f(x) and g(x).
2. It gives the explicit equations and domains/ranges for several composite functions formed from basic polynomials and rational functions.
3. The examples show how to determine the composition f(g(x)) or g(f(x)) by applying each function in sequence based on their definitions.
1. The document defines relations and functions. It provides examples of relations including r1, r2, r3, r4, and r5.
2. Functions are defined as mappings from a domain A to a range B. Examples of one-to-one, many-to-one, and onto functions are given.
3. Different types of functions are described including constant, linear, quadratic, polynomial, rational, absolute value, step, and periodic functions. Examples are provided for each type.
(1) The document defines four functions: f(x)=2x-6, g(x)=-3x+5, h(x)=x^2-1, k(x)=(2x+5)^2-1. It then defines operations on functions such as addition, subtraction, multiplication, and composition.
(2) Examples are given of calculating the sum, difference, and product of two functions, as well as the composite function g∘f. The domain and range of the composite functions are discussed.
(3) The inverse of a function is defined. Examples inverse functions are calculated from relations provided in the text.
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.
This document contains sample solutions to checkpoint questions about combining functions. It provides the steps to:
1) Sketch the graph of y = f(x)/g(x) given the graphs of y = f(x) and y = g(x)
2) Write explicit equations for functions like g(x), h(x), and k(x) that satisfy an equation like f(x) = g(x) - h(x) - k(x)
3) Determine the domain and range of functions formed by combining basic functions using operations like addition, subtraction, multiplication, and division.
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
1. The document provides graphs and equations for functions f(x), g(x), and their combinations. It asks the reader to sketch graphs, determine domains and ranges, and solve related problems.
2. The key combinations are addition, subtraction, multiplication, and division of f(x) and g(x). Their domains and ranges are identified from the original function graphs.
3. For a combination like f(x) + g(x), the domain is the same as the more restrictive of the two original functions, while the range includes all outputs equal to or greater than the original function ranges.
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.
1. The document provides examples of evaluating composite functions using tables and graphs of component functions f(x) and g(x).
2. It gives the explicit equations and domains/ranges for several composite functions formed from basic polynomials and rational functions.
3. The examples show how to determine the composition f(g(x)) or g(f(x)) by applying each function in sequence based on their definitions.
1. The document defines relations and functions. It provides examples of relations including r1, r2, r3, r4, and r5.
2. Functions are defined as mappings from a domain A to a range B. Examples of one-to-one, many-to-one, and onto functions are given.
3. Different types of functions are described including constant, linear, quadratic, polynomial, rational, absolute value, step, and periodic functions. Examples are provided for each type.
(1) The document defines four functions: f(x)=2x-6, g(x)=-3x+5, h(x)=x^2-1, k(x)=(2x+5)^2-1. It then defines operations on functions such as addition, subtraction, multiplication, and composition.
(2) Examples are given of calculating the sum, difference, and product of two functions, as well as the composite function g∘f. The domain and range of the composite functions are discussed.
(3) The inverse of a function is defined. Examples inverse functions are calculated from relations provided in the text.
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.
The document is a math worksheet containing calculus problems involving functions. It includes 21 problems involving operations on functions such as composition, inversion and transformations of function graphs. The problems involve determining expressions for composed functions, inverses, graphs of related functions obtained through transformations of an original function graph. The document also provides answers to the problems.
The document discusses curve sketching of functions by analyzing their derivatives. It provides:
1) A checklist for graphing a function which involves finding where the function is positive/negative/zero, its monotonicity from the first derivative, and concavity from the second derivative.
2) An example of graphing the cubic function f(x) = 2x^3 - 3x^2 - 12x through analyzing its derivatives.
3) Explanations of the increasing/decreasing test and concavity test to determine monotonicity and concavity from a function's derivatives.
1. The function f(x) is defined as 2 + x^2 and g(x) is defined as 1 + x^2.
2. It is given that f(g(x)) = 3 + 2(g(x) - 1) + (g(x) - 1)^2.
3. Substituting g(x) = 1 + x^2 into the equation for f(g(x)) yields f(x) = 2 + x^2.
The document defines and provides examples of several types of functions including:
1) Constant functions where f(x) = a for all values of x.
2) Linear functions of the form f(x) = ax + b.
3) Quadratic functions of the form f(x) = ax2 + bx + c.
4) Polynomial functions which are the sum of terms with variables raised to various powers.
This document provides information on key concepts related to derivatives including:
1. Critical numbers and how to find them using the first derivative test
2. How the first derivative relates to intervals of increasing and decreasing functions
3. How to determine local maxima and minima using the first derivative test
4. How to find absolute maxima and minima on a closed interval
5. How to determine concavity using the second derivative test and identify inflection points. Worked examples are provided to demonstrate each concept.
(1) This document discusses random variables and stochastic processes. It defines key concepts such as random variables, probability mass functions, cumulative distribution functions, discrete and continuous random variables.
(2) It provides examples of defining random variables for experiments involving coin tosses and ball drawings. It illustrates how to determine the probability mass function and cumulative distribution function of discrete random variables.
(3) The document also discusses continuous random variables and their probability density functions. It introduces the concepts of joint probability distributions for two random variables and how to find marginal and conditional probabilities.
This document discusses differentiation and defines the derivative. It begins by formally defining the derivative as a limit and then provides formulas to find the derivatives of simple functions like constants, linear functions, and power functions. It also covers numerical derivatives, implicit differentiation, and higher-order derivatives. Examples are provided to illustrate each concept.
This document provides 98 examples of functions and their derivatives. The functions include polynomials, trigonometric functions like sine, cosine, tangent, inverse trigonometric functions, exponential functions, logarithmic functions, and combinations of these functions.
1. The limit as x approaches 4 of x4-16 is 0. When factored, the expression becomes (x-4)(x+4)(x2+4) which equals 0 as x approaches 4.
2. The limit as x approaches infinity of x7-x2+1 is 1. When factored, the leading terms are x7 for both the top and bottom expressions, which equals 1 as x approaches infinity.
3. The limit as x approaches -1 of x2-1 is 0. When factored, the expression becomes (x+1)(x-1) which equals 0 as the factors are 0 when x is -1.
M A T H E M A T I C A L M E T H O D S J N T U M O D E L P A P E R{Wwwguest3f9c6b
This document contains 8 sets of mathematical methods problems for an examination. Each set contains 8 multi-part problems related to topics like linear algebra, differential equations, interpolation, curve fitting, Fourier series, and more. The problems are intended for engineering students and test their understanding of key concepts and ability to apply various mathematical techniques to solve problems.
The document contains 23 math problems involving equations, inequalities, geometry concepts like angles and lengths of lines, limits, and other algebraic expressions. The problems cover a wide range of math topics including functions, polynomials, systems of equations, trigonometry, and calculus.
1. The document provides examples of graphing systems of inequalities on a coordinate plane. It contains 7 problems where students are asked to shade the region satisfying 3 given inequalities on a graph.
2. The problems involve skills like drawing lines representing linear equations, identifying the region between lines, and determining the intersecting area that satisfies all inequalities simultaneously.
3. Feedback is provided on the answers with notes on common mistakes like drawing lines as solid instead of dashed.
This document contains solutions to exercises from a pre-calculus textbook on radical functions.
1) It provides tables, graphs and explanations for various radical functions such as √x, √x+3, and their relation to other functions.
2) Students are asked to sketch graphs of radical functions based on given quadratic, cubic or other functions, and identify domains and ranges.
3) Radical equations are solved by graphing related functions and finding the x-intercept(s).
The document contains 10 math problems involving graphing functions and inequalities on Cartesian planes. The problems involve sketching graphs of functions, finding coordinates that satisfy equations, drawing lines to solve equations, and shading regions defined by inequalities. Tables are used to list x and y values satisfying equations.
The document discusses using functions to find output values from input values. It provides examples of functions in the form of y=fx(x) and has students complete function tables and plot points for various functions. It discusses how the Rube Goldberg cartoon from the beginning uses an input, output, and rule to demonstrate a function.
This document provides an overview of functions from chapter 1 of an additional mathematics module. It defines key terms like domain, codomain, range, and discusses different types of relations including one-to-one, many-to-one, and many-to-many. It also covers function notation, evaluating functions, composite functions, and provides examples of calculating images and objects of functions. The chapter aims to introduce students to the fundamental concepts of functions through definitions, diagrams, and practice exercises.
1. The document discusses functions and relations through examples and questions.
2. It covers finding the value of functions, solving equations involving functions, and evaluating composite functions.
3. Key concepts covered include domain, codomain, range, one-to-one, many-to-one, one-to-many and many-to-many relations.
This document contains a review of graph transformations including translations, reflections, stretches, and compressions. It provides examples of transforming the graphs of functions by sketching the original graph and the images resulting from various combinations of transformations. It also includes determining equations to describe the transformed graphs.
This document provides 24 two-step equations to solve. The solutions are provided in curly brackets next to each equation. The equations involve addition, subtraction, multiplication, division, and operations with variables.
This document appears to be a math test containing 8 multi-part questions testing various calculus concepts such as: product rule, quotient rule, optimization, Lagrange multipliers, chain rule, implicit differentiation, and a reference sheet of important derivatives. The test covers topics including finding maxima/minima, normal lines, vectors, work, investments, gradients, tangent planes, and conic sections.
The document contains solutions to exam review problems for a Pre-Calculus 40s course. It includes over 150 problems, each labeled with a number and featuring: the problem statement, the solution, and sometimes additional notes. The problems cover a wide range of Pre-Calculus topics and concepts.
This document contains examples and solutions for permutations and combinations problems from pre-calculus. It includes 12 problems covering topics like counting the total outcomes of removing coins from bags, determining the number of possible Braille patterns, and finding coefficients in binomial expansions using Pascal's triangle. The document uses formulas, diagrams, and step-by-step workings to explain the solutions.
The document is a math worksheet containing calculus problems involving functions. It includes 21 problems involving operations on functions such as composition, inversion and transformations of function graphs. The problems involve determining expressions for composed functions, inverses, graphs of related functions obtained through transformations of an original function graph. The document also provides answers to the problems.
The document discusses curve sketching of functions by analyzing their derivatives. It provides:
1) A checklist for graphing a function which involves finding where the function is positive/negative/zero, its monotonicity from the first derivative, and concavity from the second derivative.
2) An example of graphing the cubic function f(x) = 2x^3 - 3x^2 - 12x through analyzing its derivatives.
3) Explanations of the increasing/decreasing test and concavity test to determine monotonicity and concavity from a function's derivatives.
1. The function f(x) is defined as 2 + x^2 and g(x) is defined as 1 + x^2.
2. It is given that f(g(x)) = 3 + 2(g(x) - 1) + (g(x) - 1)^2.
3. Substituting g(x) = 1 + x^2 into the equation for f(g(x)) yields f(x) = 2 + x^2.
The document defines and provides examples of several types of functions including:
1) Constant functions where f(x) = a for all values of x.
2) Linear functions of the form f(x) = ax + b.
3) Quadratic functions of the form f(x) = ax2 + bx + c.
4) Polynomial functions which are the sum of terms with variables raised to various powers.
This document provides information on key concepts related to derivatives including:
1. Critical numbers and how to find them using the first derivative test
2. How the first derivative relates to intervals of increasing and decreasing functions
3. How to determine local maxima and minima using the first derivative test
4. How to find absolute maxima and minima on a closed interval
5. How to determine concavity using the second derivative test and identify inflection points. Worked examples are provided to demonstrate each concept.
(1) This document discusses random variables and stochastic processes. It defines key concepts such as random variables, probability mass functions, cumulative distribution functions, discrete and continuous random variables.
(2) It provides examples of defining random variables for experiments involving coin tosses and ball drawings. It illustrates how to determine the probability mass function and cumulative distribution function of discrete random variables.
(3) The document also discusses continuous random variables and their probability density functions. It introduces the concepts of joint probability distributions for two random variables and how to find marginal and conditional probabilities.
This document discusses differentiation and defines the derivative. It begins by formally defining the derivative as a limit and then provides formulas to find the derivatives of simple functions like constants, linear functions, and power functions. It also covers numerical derivatives, implicit differentiation, and higher-order derivatives. Examples are provided to illustrate each concept.
This document provides 98 examples of functions and their derivatives. The functions include polynomials, trigonometric functions like sine, cosine, tangent, inverse trigonometric functions, exponential functions, logarithmic functions, and combinations of these functions.
1. The limit as x approaches 4 of x4-16 is 0. When factored, the expression becomes (x-4)(x+4)(x2+4) which equals 0 as x approaches 4.
2. The limit as x approaches infinity of x7-x2+1 is 1. When factored, the leading terms are x7 for both the top and bottom expressions, which equals 1 as x approaches infinity.
3. The limit as x approaches -1 of x2-1 is 0. When factored, the expression becomes (x+1)(x-1) which equals 0 as the factors are 0 when x is -1.
M A T H E M A T I C A L M E T H O D S J N T U M O D E L P A P E R{Wwwguest3f9c6b
This document contains 8 sets of mathematical methods problems for an examination. Each set contains 8 multi-part problems related to topics like linear algebra, differential equations, interpolation, curve fitting, Fourier series, and more. The problems are intended for engineering students and test their understanding of key concepts and ability to apply various mathematical techniques to solve problems.
The document contains 23 math problems involving equations, inequalities, geometry concepts like angles and lengths of lines, limits, and other algebraic expressions. The problems cover a wide range of math topics including functions, polynomials, systems of equations, trigonometry, and calculus.
1. The document provides examples of graphing systems of inequalities on a coordinate plane. It contains 7 problems where students are asked to shade the region satisfying 3 given inequalities on a graph.
2. The problems involve skills like drawing lines representing linear equations, identifying the region between lines, and determining the intersecting area that satisfies all inequalities simultaneously.
3. Feedback is provided on the answers with notes on common mistakes like drawing lines as solid instead of dashed.
This document contains solutions to exercises from a pre-calculus textbook on radical functions.
1) It provides tables, graphs and explanations for various radical functions such as √x, √x+3, and their relation to other functions.
2) Students are asked to sketch graphs of radical functions based on given quadratic, cubic or other functions, and identify domains and ranges.
3) Radical equations are solved by graphing related functions and finding the x-intercept(s).
The document contains 10 math problems involving graphing functions and inequalities on Cartesian planes. The problems involve sketching graphs of functions, finding coordinates that satisfy equations, drawing lines to solve equations, and shading regions defined by inequalities. Tables are used to list x and y values satisfying equations.
The document discusses using functions to find output values from input values. It provides examples of functions in the form of y=fx(x) and has students complete function tables and plot points for various functions. It discusses how the Rube Goldberg cartoon from the beginning uses an input, output, and rule to demonstrate a function.
This document provides an overview of functions from chapter 1 of an additional mathematics module. It defines key terms like domain, codomain, range, and discusses different types of relations including one-to-one, many-to-one, and many-to-many. It also covers function notation, evaluating functions, composite functions, and provides examples of calculating images and objects of functions. The chapter aims to introduce students to the fundamental concepts of functions through definitions, diagrams, and practice exercises.
1. The document discusses functions and relations through examples and questions.
2. It covers finding the value of functions, solving equations involving functions, and evaluating composite functions.
3. Key concepts covered include domain, codomain, range, one-to-one, many-to-one, one-to-many and many-to-many relations.
This document contains a review of graph transformations including translations, reflections, stretches, and compressions. It provides examples of transforming the graphs of functions by sketching the original graph and the images resulting from various combinations of transformations. It also includes determining equations to describe the transformed graphs.
This document provides 24 two-step equations to solve. The solutions are provided in curly brackets next to each equation. The equations involve addition, subtraction, multiplication, division, and operations with variables.
This document appears to be a math test containing 8 multi-part questions testing various calculus concepts such as: product rule, quotient rule, optimization, Lagrange multipliers, chain rule, implicit differentiation, and a reference sheet of important derivatives. The test covers topics including finding maxima/minima, normal lines, vectors, work, investments, gradients, tangent planes, and conic sections.
The document contains solutions to exam review problems for a Pre-Calculus 40s course. It includes over 150 problems, each labeled with a number and featuring: the problem statement, the solution, and sometimes additional notes. The problems cover a wide range of Pre-Calculus topics and concepts.
This document contains examples and solutions for permutations and combinations problems from pre-calculus. It includes 12 problems covering topics like counting the total outcomes of removing coins from bags, determining the number of possible Braille patterns, and finding coefficients in binomial expansions using Pascal's triangle. The document uses formulas, diagrams, and step-by-step workings to explain the solutions.
The document discusses the benefits of exercise for mental health. Regular physical activity can help reduce anxiety and depression and improve mood and cognitive function. Exercise causes chemical changes in the brain that may help protect against mental illness and improve symptoms.
This document is a record of dates, containing six identical entries of "September 18, 2014" with no other text or context provided. Each entry is on its own line and labeled with "18th sept 2014" and a number.
This document contains solutions to exercises from a pre-calculus textbook involving inverse functions and relations. Some of the key questions answered include:
- Sketching the graphs of functions and their inverses after transformations like reflections
- Finding equations that represent the inverse of various given functions
- Determining whether pairs of functions are inverses of each other by comparing their equations
- Restricting domains of functions to make their inverses functions as well
- Finding coordinates of points on inverse relations after translations
- Sketching graphs of inverses based on restrictions of the domain of the original relation
This document is a record of dates, containing six identical entries of "September 18, 2014" with no other text or context provided. Each entry is on its own line and labeled with "18th sept 2014" and a number.
This four sentence document repeats the date September 22, 2014 four times without providing any additional context or information. The document states the same date, September 22, 2014, in each of its four sentences without elaborating on the significance of the date or including any other details.
This document contains a series of text messages exchanged on February 11, 2013. The messages discuss meeting at 1:40 PM and previously meeting at 1:16 PM the same day to discuss an unspecified event occurring on February 11, 2013.
This document contains solutions to exercises involving radian measure. It includes sketches of angles in standard position, conversions between degrees and radians, calculations of trigonometric ratios, determining arc lengths and sector areas of circles based on central angles and radii, and calculations of angular velocity and linear distance traveled given rotational information.
Review exam questions extra practice perms and combsGarden City
This document contains 17 exam questions about permutations, combinations, and expansions of binomial expressions. The questions cover topics like arranging people in rows, choosing committees from groups, expanding expressions like (a - b)^n, and finding coefficients of terms in expansions. The questions test skills in counting arrangements, choosing groups subject to conditions, evaluating binomial coefficients, and performing expansions of binomial expressions.
This document contains solutions to exercises involving permutations. It explains how to calculate the number of permutations of different objects using factorial notation. For example, it shows that there are 6! = 720 ways to arrange 7 distinct keys on a circular key ring, since any key can be chosen as the first key. It also provides examples of using permutations to solve problems involving arranging letters in words, choosing music to perform, and placing video games on a shelf by genre.
The document is a record of dates from September 17, 2014. It contains 20 entries, each listing the date September 17, 2014. The document functions as a log or record of the single date of September 17, 2014 recorded 20 separate times.
This document discusses solving radical equations by factoring the expression under the radical sign and then solving for x. It involves identifying the vertex of a parabola, factoring an expression of the form x^2 - bx - c, and solving the resulting equation by taking the square root of both sides to find the values where the radical is defined.
This document contains solutions to exercises from a pre-calculus lesson on trigonometric ratios for any angle in standard position. It provides the values of trigonometric ratios such as sine, cosine, tangent, cosecant, secant and cotangent for various angles in a table. It also shows sample calculations for determining coterminal angles and expressions to represent all coterminal angles for given angles.
This document contains sample solutions to checkpoint questions about combining functions. It provides the steps to:
1) Sketch the graph of y = f(x)/g(x) given the graphs of y = f(x) and y = g(x)
2) Write explicit equations for functions like g(x), h(x), and k(x) that satisfy an equation like f(x) = g(x) - h(x) - k(x)
3) Determine the domain and range of functions formed by combining basic functions using operations like addition, subtraction, multiplication, and division.
This document contains solutions to checkpoint questions about transforming graphs of functions. It includes examples of translating graphs by shifting them horizontally and vertically based on changes to the x and y variables in the function. It also contains an example of reflecting a graph across the x-axis. The questions require sketching the transformed graphs on grids and writing the equations of the transformed functions based on the given transformations.
This document provides examples and exercises for combining functions algebraically. It gives examples of combining two functions using addition, subtraction, multiplication, division, and composition. For each combination, it provides the explicit equation and determines the domain and range. It asks the reader to write explicit equations and determine domains and ranges for various combinations of functions, such as f(x) = x2 - 4 and g(x) = x - 1. The document also explores how the commutative, associative, and distributive properties apply when combining functions algebraically.
This document provides examples and exercises for combining functions algebraically. It gives examples of writing explicit equations for combinations of functions using addition, subtraction, multiplication, division, and composition. For each combination, it provides the steps to write the explicit equation and determines the domain. It also gives examples finding explicit equations for functions f(x) and g(x) such that their combination equals a given function h(x). The document provides the work and reasoning for each example.
The document defines the derivative and discusses differentiation of basic functions:
- The derivative of a function f at x is defined as the limit of (f(x+h)-f(x))/h as h approaches 0.
- The derivative of a constant function is always 0.
- The derivative of a linear function f(x)=mx+b is the slope m.
- The derivative of a power function f(x)=xn is nxn-1.
This document contains solutions to exercises about translating graphs of functions. It includes:
1) Examples of translating the graph of y = |x| by different amounts, and writing the equations of the translated graphs.
2) Describing how graphs are translated based on the equations of the form y = f(x - h) or y - k = f(x).
3) Sketching translated graphs on grids and stating their domains and ranges.
4) An example of finding the coordinates of the image of a point on a graph after it is translated.
5) Discussion of how vertical and horizontal asymptotes are affected by translation, and writing the equations of asymptotes for a
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.
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.
The inverse of a function is obtained by reflecting the graph of the original function over the line y=x. The inverse of a function f(x) is written as f^-1(x). For a function to have an inverse, it must be one-to-one, meaning each x-value only corresponds to a single y-value. To check if a function is one-to-one, apply the horizontal line test - if no horizontal line intersects the graph at more than one point, it is one-to-one and will have an inverse function.
This document discusses several topics in calculus of several variables:
- Functions of several variables and their partial derivatives
- Maxima and minima of functions of several variables
- Double integrals and constrained maxima/minima using Lagrange multipliers
It provides examples of computing partial derivatives of functions, interpreting them geometrically, and using partial derivatives to determine rates of change. Level curves are also discussed as a way to sketch graphs of functions of two variables.
This document contains a multi-part math worksheet involving graphing and solving quadratic equations. It includes:
1) Graphing quadratic functions and identifying their properties.
2) Finding the standard form of parabolas given three points.
3) Factoring quadratic expressions.
4) Solving quadratic equations by factoring, taking square roots, or graphing.
5) Word problems modeling real-world situations with quadratic functions and equations.
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.
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.
This document provides instructions for graphing linear equations. It begins with examples of solving linear equations algebraically. Students are then introduced to key properties of linear equations: they contain two variables and graph as straight lines. The document demonstrates graphing various linear equations by plotting their solution sets as points and connecting them with a straight line. It concludes by asking students to reflect on similarities and differences between the graphed linear equations.
The document summarizes key concepts about composition functions:
1. Composition functions are not commutative.
2. They are associative.
3. Identity functions leave other functions unchanged when composed.
The document also provides examples of determining the component functions of composition functions and finding inverse functions.
The document provides guidelines for graphing polynomial and rational functions. It discusses the key features of graphs of quadratic, cubic, quartic and quintic polynomials. It then discusses how to graph rational functions by identifying intercepts, asymptotes, discontinuities and using sign analysis to determine the positive and negative portions of the graph. An example rational function is graphed as an illustration.
The document provides examples of combining functions using various operations like addition, subtraction, multiplication, division, and composition. It then defines four functions j(x), k(x), and l(x) and asks to write out equations combining these functions using the different operations. Finally, it introduces two new functions m(x) and n(x) and asks to write their compositions and find their domains.
To multiply polynomials, you can use the distributive property and properties of exponents. When multiplying monomials, group terms with the same bases and add their exponents. When multiplying binomials, use FOIL or distribute one binomial over the other. For polynomials with more than two terms, you can distribute or use a rectangle model to systematically multiply each term.
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.
Here are the steps to solve this compound interest problem:
* Principal (P) = $1000
* Annual interest rate (r) = 9% = 0.09
* Number of interest periods per year (n) = 12 (monthly)
* Interest is compounded monthly
* Calculate amount after 5 years, 10 years, and 15 years using the compound interest formula:
5 years: A = 1000(1 + 0.09/12)^(12*5) = $1581.53
10 years: A = 1000(1 + 0.09/12)^(12*10) = $2501.04
15 years: A = 1000(1 + 0.09/12
This document contains 3 short entries dated October 06, 2014 that are all labeled "6th october 2014". The document appears to be a log or record with multiple brief entries made on the same date.
This document is a list of dates, all occurring on October 3rd, 2014. Each entry repeats the date and contains a page number. There are 9 total entries in the list, each with the same date but incrementing page numbers from 1 through 9.
This document appears to be a log of dates from October 1st, 2014. It contains four entries all with the date October 1st, 2014 listed. The document provides a brief record of dates but does not include any other contextual information.
This document is a series of 8 entries all with the date of September 30, 2014. Each entry contains only the date with no other text or information provided.
The document is dated September 25, 2014. It appears to be a brief one paragraph document that does not provide much context or details. The date is the only substantive information given.
The document is dated September 25, 2014. It appears to be a brief one paragraph document that does not provide much context or details. The date is the only substantive information given.
This document is a record of events from September 24, 2014. It consists of 7 entries all with the same date of September 24, 2014 listed at the top, suggesting some type of daily log or journal was being kept for that date.
This document is a series of 7 entries all dated September 23, 2014 without any other notable information provided. Each entry simply states the date of September 23, 2014.
This document is a log of dates from September 16, 2014. It contains 5 entries all with the same date of September 16, 2014 listed in various formats including 16th sept 2014 and September 16, 2014.
This 3 sentence document simply repeats the date "September 11, 2014" three times on three different lines. It does not provide any other context or information.
The document is a list of dates, all occurring on September 9th, 2014. Each entry repeats the date 10 times, once for each numbered line. The sole purpose of the document is to repeatedly record the same date, September 9th, 2014, across 10 lines.
This document is a series of 7 entries all dated September 23, 2014 without any other notable information provided. Each entry simply states the date of September 23, 2014.
This four sentence document repeats the date September 22, 2014 four times without providing any additional context or information. The document states the same date, September 22, 2014, in each of its four sentences without elaborating on the significance of the date or including any other details.
This document is a log of dates from September 16, 2014. It contains 5 entries all with the same date of September 16, 2014 listed in various formats including 16th sept 2014 and September 16, 2014.
This 3 sentence document simply repeats the date "September 11, 2014" three times on three different lines. It does not provide any other context or information.
The document is a list of dates, all being September 9th, 2014. Each line repeats the date 10 times with line numbers listed before each date. The document simply repeats the same date 10 times in a listed format.
This document is a log of entries from September 5th, 2014. It contains 5 brief entries all dated September 5th, 2014 with increasing numbering from 1 to 5. The log entries do not contain any other details.
This document appears to be a series of 3 short sections, each labeled 1.3 and dated September 08, 2014. While no other contextual information is provided, the document seems to contain 3 brief pieces of information or entries from that date.