or just I if the set A is clear. Hence we have for all x in A the identity function I ( )
|
|
|
- Vanessa Lewis
- 5 years ago
- Views:
Transcription
1 3 Functions 46 SECTON E Properties of Composite Functions the end of this section ou will be able to understand what is meant b the identit function prove properties of inverse function prove properties of composition of functions E Properties of the dentit Function What do ou think the term identit means? The word identit in everda language means being the same. What does identit function mean? The function which has the same input and output. The definition of the identit function is: Definition (3.2. Let be an set. The function f : given b f ( for all in is called the identit function on the set. The identit function on a set is normall denoted b or just if the set is clear. Hence we have for all in the identit function : satisfing ( The identit function transforms back to. This can be illustrated b f Fig is in then what is 2 2. f equal to? + is in then ( + +. What is ( provided is in set equal to? The identit function transforms back to itself. The input is same as the output. The identit function has man important properties that we use. n this section we state and prove some of these properties. Proposition (3.3. Let f be an function, f :, then f f f Let be an arbitrar element in which means it is in the domain of f because we are given f :. s it in the domain of f? f f ( Fig 32 f
2 3 Functions 47 Yes because : and so f :. We have ( f ( f ( ( [ Definition of ] ecause f f f therefore f f. Since Net we show f f. s the element in the domain of f? f f f ( Fig 33 Yes because f : and therefore f :. We have ( f ( f ( [ Definition of ] ecause f f f f f therefore f f. Since Remember both and are ident ns not change b appling these functions. Hence we have our required result, f f f. it functio so their arguments and f do Prior to this section we have used lower case letters such as f, g and h to represent functions but for the identit function we use upper case to represent this function. E2 Properties of the nverse Function Proposition (3.4. Let f : be a bijective function. Then f f and f f where is the identit function on and is the identit function on. We prove f f and leave the proof of 3(e. Since f is bijective therefore the inverse function f f as a question in Eercise f eists. Wh? ecause from Section C we have (3.5 f has an inverse f is bijective Let be an arbitrar element in the set and f (. Therefore is in because f :. f f f f f ( f [ Definition of ]
3 3 Functions 48 s in the domain of Yes because f? f : that is the domain of nc f ( ( f f ( f ( [ From bove] ec ( f f ( ( f is and f ( definition of the inverse fu tion we have and therefore. the ause the identit function Since therefore we have the required result, f f. Proposition (3.5. Let the functions f : and g: Then the function g is unique and satisf both g f and f g g f. Comment. For g f we need the function f to be bijective. Wh? ecause from Section C we have (3.5 f has an inverse f is bijective First we prove th at f is bijective and then g f. Hence first we show that f is injective and surjective. How do we prove that f is injective (one to one? (3.2 we are required to show that for all, in we have f ( f ( implies Let both, f f. Then ( [ dentit Function] g f ( [ecause we are given g f g f definition of be in and ] ( [ ] g( f ( ecause f ( f ( ( g f ( [ definition of ] ( [ ecause we are given g f ] Hence therefore b (3.2 we conclude that f is injective. Net we prove f is surjective. How? Let be in (codomain of f and then show that there is an element, call it (domain of f such that f (. We have ( [ dentit Function] f g ecause we are given f g [ f g( [ Definition of ] ], in is an injection f f (3.2 implies f
4 3 Functions 49 Since we are given g: therefore g( is in. Let g( we have found an element, g(, in, such that f ( f is surjective. ecause f is both injective and surjective therefore it is bijective. (3.5 f has an inverse f is bijective. we can sa the function f has an inverse and therefore. Hence the function f. Have we completed the proof? No because we still need to show that g is unique an d g f. How do we prove this? contradiction. Suppose the are not equal, that is g f. This means there is another function besides f with the above stated properties: g f and f g The domain of both f : and g: is the set. There must be an element sa in such that g f ecause functions g f Since the function f is injective (one to one therefore ut and ( f f f g [Not Equal] ( ( [ Definition of ] ( [ ecause we are given f g ] f g f g f ( f ( ( f f ( [ Definition of ] ecause b (3.4 we have We have f ( g( f ( f ( were not equal, that is f g( f f. Hence f f. Contradiction because earlier we stated the ( we have proved our result g f. Can ou see an relationship between the above propositions (3.4 and (3.5? n (3.5 we have proven that the inverse function f which satisfies proposition (3.4 is unique: f f and f f f is the onl function with this propert for a given function f. What does this mean? Let f : and be in the domain then ( Similarl if is in then ( f f (. f f Note that g f onl if both the following conditions are satisfied: g f and f g
5 3 Functions 50 Remember bije ctive functions are crucial in the discussion of inverse functions but the are also preserved under the composition of functions as we will show in the eercises. E3 Properti es of the Composite Functions Net we prove an etremel important propert of function composition. We prove that the composition of functions is associative. Proposition (3.6 Let h:, g: C and f : C D be functions. Then We can illustrate the functions b: ( f g h f ( g h h g f Fig 34 C D The domain of both ( f g h and f ( g h is the set. Let be an arbitrar element in. Then f g h f g h Definition of lso [ ] f g( h( Definition of [ ] ( f ( g h ( f ( g h( [ Definition of ] f g( h( Definition of [ ] Since for the arbitrar element we have f g h f g h therefore we have our required result: f g h f g h SUMMRY The identit function on a set is denoted b and has the propert ( for all in. The input and output of the function sta the same. Let f : and g: be functions which satisf both g f and f g Then g f and is unique. Composition of functions is associative: f g h f g h
Section 3.1. ; X = (0, 1]. (i) f : R R R, f (x, y) = x y
![Section 3.1. ; X = (0, 1]. (i) f : R R R, f (x, y) = x y](/thumbs/83/88503328.jpg "Section 3.1. ; X = (0, 1]. (i) f : R R R, f (x, y) = x y")
Paul J. Bruillard MATH 0.970 Problem Set 6 An Introduction to Abstract Mathematics R. Bond and W. Keane Section 3.1: 3b,c,e,i, 4bd, 6, 9, 15, 16, 18c,e, 19a, 0, 1b Section 3.: 1f,i, e, 6, 1e,f,h, 13e,
Mathematics of Cryptography Part I
CHAPTER 2 Mathematics of Crptograph Part I (Solution to Practice Set) Review Questions 1. The set of integers is Z. It contains all integral numbers from negative infinit to positive infinit. The set of
6. Linear transformations. Consider the function. f : R 2 R 2 which sends (x, y) (x, y)
Consider the function 6 Linear transformations f : R 2 R 2 which sends (x, ) (, x) This is an example of a linear transformation Before we get into the definition of a linear transformation, let s investigate
Engineering Mathematics I
Engineering Mathematics I_ 017 Engineering Mathematics I 1. Introduction to Differential Equations Dr. Rami Zakaria Terminolog Differential Equation Ordinar Differential Equations Partial Differential
3.7 InveRSe FUnCTIOnS
CHAPTER functions learning ObjeCTIveS In this section, ou will: Verif inverse functions. Determine the domain and range of an inverse function, and restrict the domain of a function to make it one-to-one.
Eigenvectors and Eigenvalues 1
Ma 2015 page 1 Eigenvectors and Eigenvalues 1 In this handout, we will eplore eigenvectors and eigenvalues. We will begin with an eploration, then provide some direct eplanation and worked eamples, and
6.4 graphs OF logarithmic FUnCTIOnS
SECTION 6. graphs of logarithmic functions 9 9 learning ObjeCTIveS In this section, ou will: Identif the domain of a logarithmic function. Graph logarithmic functions. 6. graphs OF logarithmic FUnCTIOnS
Definition: div Let n, d 0. We define ndiv d as the least integer quotient obtained when n is divided by d. That is if
Section 5. Congruence Arithmetic A number of computer languages have built-in functions that compute the quotient and remainder of division. Definition: div Let n, d 0. We define ndiv d as the least integer
LESSON 35: EIGENVALUES AND EIGENVECTORS APRIL 21, (1) We might also write v as v. Both notations refer to a vector.
LESSON 5: EIGENVALUES AND EIGENVECTORS APRIL 2, 27 In this contet, a vector is a column matri E Note 2 v 2, v 4 5 6 () We might also write v as v Both notations refer to a vector (2) A vector can be man
4 Inverse function theorem
Tel Aviv Universit, 2013/14 Analsis-III,IV 53 4 Inverse function theorem 4a What is the problem................ 53 4b Simple observations before the theorem..... 54 4c The theorem.....................
Second-Order Linear Differential Equations C 2
C8 APPENDIX C Additional Topics in Differential Equations APPENDIX C. Second-Order Homogeneous Linear Equations Second-Order Linear Differential Equations Higher-Order Linear Differential Equations Application
Section 7.2: One-to-One, Onto and Inverse Functions
Section 7.2: One-to-One, Onto and Inverse Functions In this section we shall developed the elementary notions of one-to-one, onto and inverse functions, similar to that developed in a basic algebra course.
Math 214 Spring problem set (a) Consider these two first order equations. (I) dy dx = x + 1 dy
Math 4 Spring 08 problem set. (a) Consider these two first order equations. (I) d d = + d (II) d = Below are four direction fields. Match the differential equations above to their direction fields. Provide
Foundations of Databases
Foundations of Databases (Slides adapted from Thomas Eiter, Leonid Libkin and Werner Nutt) Foundations of Databases 1 Quer optimization: finding a good wa to evaluate a quer Queries are declarative, and
Roberto s Notes on Integral Calculus Chapter 3: Basics of differential equations Section 3. Separable ODE s
Roberto s Notes on Integral Calculus Chapter 3: Basics of differential equations Section 3 Separable ODE s What ou need to know alread: What an ODE is and how to solve an eponential ODE. What ou can learn
Introduction to Differential Equations
Math0 Lecture # Introduction to Differential Equations Basic definitions Definition : (What is a DE?) A differential equation (DE) is an equation that involves some of the derivatives (or differentials)
can only hit 3 points in the codomain. Hence, f is not surjective. For another example, if n = 4
.. Conditions for Injectivity and Surjectivity In this section, we discuss what we can say about linear maps T : R n R m given only m and n. We motivate this problem by looking at maps f : {,..., n} {,...,
Review Topics for MATH 1400 Elements of Calculus Table of Contents
Math 1400 - Mano Table of Contents - Review - page 1 of 2 Review Topics for MATH 1400 Elements of Calculus Table of Contents MATH 1400 Elements of Calculus is one of the Marquette Core Courses for Mathematical
1.6 CONTINUITY OF TRIGONOMETRIC, EXPONENTIAL, AND INVERSE FUNCTIONS
.6 Continuit of Trigonometric, Eponential, and Inverse Functions.6 CONTINUITY OF TRIGONOMETRIC, EXPONENTIAL, AND INVERSE FUNCTIONS In this section we will discuss the continuit properties of trigonometric
Functions. Introduction
Functions,00 P,000 00 0 70 7 80 8 0 000 00 00 Figure Standard and Poor s Inde with dividends reinvested (credit "bull": modification of work b Praitno Hadinata; credit "graph": modification of work b MeasuringWorth)
Introduction to Vector Spaces Linear Algebra, Spring 2011
Introduction to Vector Spaces Linear Algebra, Spring 2011 You probabl have heard the word vector before, perhaps in the contet of Calculus III or phsics. You probabl think of a vector like this: 5 3 or
LECTURE NOTES IN EQUIVARIANT ALGEBRAIC GEOMETRY. Spec k = (G G) G G (G G) G G G G i 1 G e
LECTURE NOTES IN EQUIVARIANT ALEBRAIC EOMETRY 8/4/5 Let k be field, not necessaril algebraicall closed. Definition: An algebraic group is a k-scheme together with morphisms (µ, i, e), k µ, i, Spec k, which
Tensor products in Riesz space theory
Tensor products in Riesz space theor Jan van Waaij Master thesis defended on Jul 16, 2013 Thesis advisors dr. O.W. van Gaans dr. M.F.E. de Jeu Mathematical Institute, Universit of Leiden CONTENTS 2 Contents
In everyday speech, a continuous. Limits and Continuity. Critical Thinking Exercises
062 Chapter Introduction to Calculus Critical Thinking Eercises Make Sense? In Eercises 74 77, determine whether each statement makes sense or does not make sense, and eplain our reasoning. 74. I evaluated
2.2 Equations of Lines
660_ch0pp07668.qd 10/16/08 4:1 PM Page 96 96 CHAPTER Linear Functions and Equations. Equations of Lines Write the point-slope and slope-intercept forms Find the intercepts of a line Write equations for
Review of Functions. Functions. Philippe B. Laval. Current Semester KSU. Philippe B. Laval (KSU) Functions Current Semester 1 / 12
Review of Functions Functions Philippe B. Laval KSU Current Semester Philippe B. Laval (KSU) Functions Current Semester 1 / 12 Introduction Students are expected to know the following concepts about functions:
Exact Differential Equations. The general solution of the equation is f x, y C. If f has continuous second partials, then M y 2 f
APPENDIX C Additional Topics in Differential Equations APPENDIX C. Eact First-Order Equations Eact Differential Equations Integrating Factors Eact Differential Equations In Chapter 6, ou studied applications
Systems of Linear Equations: Solving by Graphing
8.1 Sstems of Linear Equations: Solving b Graphing 8.1 OBJECTIVE 1. Find the solution(s) for a set of linear equations b graphing NOTE There is no other ordered pair that satisfies both equations. From
f(x) = 2x 2 + 2x - 4
4-1 Graphing Quadratic Functions What You ll Learn Scan the tet under the Now heading. List two things ou will learn about in the lesson. 1. Active Vocabular 2. New Vocabular Label each bo with the terms
Section 1.5 Formal definitions of limits
Section.5 Formal definitions of limits (3/908) Overview: The definitions of the various tpes of limits in previous sections involve phrases such as arbitraril close, sufficientl close, arbitraril large,
Particular Solutions
Particular Solutions Our eamples so far in this section have involved some constant of integration, K. We now move on to see particular solutions, where we know some boundar conditions and we substitute
The Coordinate Plane and Linear Equations Algebra 1
Name: The Coordinate Plane and Linear Equations Algebra Date: We use the Cartesian Coordinate plane to locate points in two-dimensional space. We can do this b measuring the directed distances the point
4.1 Circles. Explore Deriving the Standard-Form Equation
COMMON CORE r Locker LESSON Circles.1 Name Class Date.1 Circles Common Core Math Standards The student is epected to: COMMON CORE A-CED.A.3 Represent constraints b equations or inequalities,... and interpret
Exponential and Logarithmic Functions
Eponential and Logarithmic Functions 6 Figure Electron micrograph of E. Coli bacteria (credit: Mattosaurus, Wikimedia Commons) CHAPTER OUTLINE 6. Eponential Functions 6. Logarithmic Properties 6. Graphs
MATRIX TRANSFORMATIONS
CHAPTER 5. MATRIX TRANSFORMATIONS INSTITIÚID TEICNEOLAÍOCHTA CHEATHARLACH INSTITUTE OF TECHNOLOGY CARLOW MATRIX TRANSFORMATIONS Matri Transformations Definition Let A and B be sets. A function f : A B
1. d = 1. or Use Only in Pilot Program F Review Exercises 131
or Use Onl in Pilot Program F 0 0 Review Eercises. Limit proof Suppose f is defined for all values of near a, ecept possibl at a. Assume for an integer N 7 0, there is another integer M 7 0 such that f
Math 107: Calculus II, Spring 2015: Midterm Exam II Monday, April Give your name, TA and section number:
Math 7: Calculus II, Spring 25: Midterm Exam II Monda, April 3 25 Give our name, TA and section number: Name: TA: Section number:. There are 5 questions for a total of points. The value of each part of
MATH 1324 (Finite Mathematics or Business Math I) Lecture Notes Author / Copyright: Kevin Pinegar
MTH Finite Mathematics or usiness Math Lecture Notes uthor / opright: Kevin Pinegar MTH Module Notes: SYSTEMS OF EQUTONS & MTES. MT NVESES & POPETES OF MTES Definition: We cannot discuss the inverse of
Fixed Point Theorem and Sequences in One or Two Dimensions
Fied Point Theorem and Sequences in One or Two Dimensions Dr. Wei-Chi Yang Let us consider a recursive sequence of n+ = n + sin n and the initial value can be an real number. Then we would like to ask
1.2 Functions What is a Function? 1.2. FUNCTIONS 11
1.2. FUNCTIONS 11 1.2 Functions 1.2.1 What is a Function? In this section, we only consider functions of one variable. Loosely speaking, a function is a special relation which exists between two variables.
Trigonometry Outline
Trigonometr Outline Introduction Knowledge of the content of this outline is essential to perform well in calculus. The reader is urged to stud each of the three parts of the outline. Part I contains the
v t t t t a t v t d dt t t t t t 23.61
SECTION 4. MAXIMUM AND MINIMUM VALUES 285 The values of f at the endpoints are f 0 0 and f 2 2 6.28 Comparing these four numbers and using the Closed Interval Method, we see that the absolute minimum value
Properties of Limits
33460_003qd //04 :3 PM Page 59 SECTION 3 Evaluating Limits Analticall 59 Section 3 Evaluating Limits Analticall Evaluate a it using properties of its Develop and use a strateg for finding its Evaluate
Review of Essential Skills and Knowledge
Review of Essential Skills and Knowledge R Eponent Laws...50 R Epanding and Simplifing Polnomial Epressions...5 R 3 Factoring Polnomial Epressions...5 R Working with Rational Epressions...55 R 5 Slope
You don't have to be a mathematician to have a feel for numbers. John Forbes Nash, Jr.
Course Title: Real Analsis Course Code: MTH3 Course instructor: Dr. Atiq ur Rehman Class: MSc-II Course URL: www.mathcit.org/atiq/fa5-mth3 You don't have to be a mathematician to have a feel for numbers.
In last semester, we have seen some examples about it (See Tutorial Note #13). Try to have a look on that. Here we try to show more technique.
MATH202 Introduction to Analysis (2007 Fall and 2008 Spring) Tutorial Note #4 Part I: Cauchy Sequence Definition (Cauchy Sequence): A sequence of real number { n } is Cauchy if and only if for any ε >
Math 4242 Fall 2016 (Darij Grinberg): homework set 6 due: Mon, 21 Nov 2016 Let me first recall a definition.
Math 4242 Fall 206 homework page Math 4242 Fall 206 Darij Grinberg: homework set 6 due: Mon, 2 Nov 206 Let me first recall a definition. Definition 0.. Let V and W be two vector spaces. Let v = v, v 2,...,
FIRST- AND SECOND-ORDER IVPS The problem given in (1) is also called an nth-order initial-value problem. For example, Solve: Solve:
.2 INITIAL-VALUE PROBLEMS 3.2 INITIAL-VALUE PROBLEMS REVIEW MATERIAL Normal form of a DE Solution of a DE Famil of solutions INTRODUCTION We are often interested in problems in which we seek a solution
9.2. Cartesian Components of Vectors. Introduction. Prerequisites. Learning Outcomes
Cartesian Components of Vectors 9.2 Introduction It is useful to be able to describe vectors with reference to specific coordinate sstems, such as the Cartesian coordinate sstem. So, in this Section, we
2.1 Rates of Change and Limits AP Calculus
.1 Rates of Change and Limits AP Calculus.1 RATES OF CHANGE AND LIMITS Limits Limits are what separate Calculus from pre calculus. Using a it is also the foundational principle behind the two most important
TROPICAL SCHEME THEORY
TROPICAL SCHEME THEORY 1. Tropical ideals The idea of tropical scheme theor in general is the following. An ideal I K[ 1,..., n ] gives rise to a variet V (I) A n K a variet. With classical tropicalization
Trigonometric. equations. Topic: Periodic functions and applications. Simple trigonometric. equations. Equations using radians Further trigonometric
Trigonometric equations 6 sllabusref eferenceence Topic: Periodic functions and applications In this cha 6A 6B 6C 6D 6E chapter Simple trigonometric equations Equations using radians Further trigonometric
2.5 CONTINUITY. a x. Notice that Definition l implicitly requires three things if f is continuous at a:
SECTION.5 CONTINUITY 9.5 CONTINUITY We noticed in Section.3 that the it of a function as approaches a can often be found simpl b calculating the value of the function at a. Functions with this propert
Chapter 4 Analytic Trigonometry
Analtic Trigonometr Chapter Analtic Trigonometr Inverse Trigonometric Functions The trigonometric functions act as an operator on the variable (angle, resulting in an output value Suppose this process
LESSON 4.3 GRAPHING INEQUALITIES
LESSON.3 GRAPHING INEQUALITIES LESSON.3 GRAPHING INEQUALITIES 9 OVERVIEW Here s what ou ll learn in this lesson: Linear Inequalities a. Ordered pairs as solutions of linear inequalities b. Graphing linear
Definition Suppose S R n, V R m are subspaces. A map U : S V is linear if
.6. Restriction of Linear Maps In this section, we restrict linear maps to subspaces. We observe that the notion of linearity still makes sense for maps whose domain and codomain are subspaces of R n,
MATH 2300 review problems for Exam 3 ANSWERS
MATH 300 review problems for Eam 3 ANSWERS. Check whether the following series converge or diverge. In each case, justif our answer b either computing the sum or b b showing which convergence test ou are
Trigonometric Functions
Trigonometric Functions This section reviews radian measure and the basic trigonometric functions. C ' θ r s ' ngles ngles are measured in degrees or radians. The number of radians in the central angle
Mathematics 309 Conic sections and their applicationsn. Chapter 2. Quadric figures. ai,j x i x j + b i x i + c =0. 1. Coordinate changes
Mathematics 309 Conic sections and their applicationsn Chapter 2. Quadric figures In this chapter want to outline quickl how to decide what figure associated in 2D and 3D to quadratic equations look like.
Topic 3 Notes Jeremy Orloff
Topic 3 Notes Jerem Orloff 3 Line integrals and auch s theorem 3.1 Introduction The basic theme here is that comple line integrals will mirror much of what we ve seen for multivariable calculus line integrals.
First Order Equations
10 1 Linear and Semilinear Equations Chapter First Order Equations Contents 1 Linear and Semilinear Equations 9 Quasilinear Equations 19 3 Wave Equation 6 4 Sstems of Equations 31 1 Linear and Semilinear
4 A Catalogue of Functions and Inequalities 8. SECTION E Modulus Function
4 A Catalogue of Functions and Inequalities 8 SECTION E Modulus Function By the end of this section you will be able to visualize the geometric interpretation of the modulus function derive some inequalities
Homework 12; Due Monday, 4/9/2018
Homework 2; Due Monday, 4/9/28 Answer-Only Questions Credit based solely on the answer Question For each of the following relations, determine whether each one is symmetric, transitive, or refleive If
Functions Functions and Modeling A UTeach/TNT Course
Definition of a Function DEFINITION: Let A and B be sets. A function between A and B is a subset of A B with the property that if (a, b 1 )and(a, b 2 ) are both in the subset, then b 1 = b 2. The domain
Quick Review 4.1 (For help, go to Sections 1.2, 2.1, 3.5, and 3.6.)
Section 4. Etreme Values of Functions 93 EXPLORATION Finding Etreme Values Let f,.. Determine graphicall the etreme values of f and where the occur. Find f at these values of.. Graph f and f or NDER f,,
Symmetry Arguments and the Role They Play in Using Gauss Law
Smmetr Arguments and the Role The la in Using Gauss Law K. M. Westerberg (9/2005) Smmetr plas a ver important role in science in general, and phsics in particular. Arguments based on smmetr can often simplif
Say It With Symbols Answers
Investigation Additional Practice. p w and p ( w). a. p w () () b. p (w) w and p w. (.) m. A w Q Properties used for items will var, but all include the Distributive Propert.. Possible answer: 7 and ().
Number Plane Graphs and Coordinate Geometry
Numer Plane Graphs and Coordinate Geometr Now this is m kind of paraola! Chapter Contents :0 The paraola PS, PS, PS Investigation: The graphs of paraolas :0 Paraolas of the form = a + + c PS Fun Spot:
Systems of Linear and Quadratic Equations. Check Skills You ll Need. y x. Solve by Graphing. Solve the following system by graphing.
NY- Learning Standards for Mathematics A.A. Solve a sstem of one linear and one quadratic equation in two variables, where onl factoring is required. A.G.9 Solve sstems of linear and quadratic equations
Solutions to Homework Set 1
Solutions to Homework Set 1 1. Prove that not-q not-p implies P Q. In class we proved that A B implies not-b not-a Replacing the statement A by the statement not-q and the statement B by the statement
1.3 LIMITS AT INFINITY; END BEHAVIOR OF A FUNCTION
. Limits at Infinit; End Behavior of a Function 89. LIMITS AT INFINITY; END BEHAVIOR OF A FUNCTION Up to now we have been concerned with its that describe the behavior of a function f) as approaches some
TABLE OF CONTENTS - UNIT 1 CHARACTERISTICS OF FUNCTIONS
TABLE OF CONTENTS - UNIT CHARACTERISTICS OF FUNCTIONS TABLE OF CONTENTS - UNIT CHARACTERISTICS OF FUNCTIONS INTRODUCTION TO FUNCTIONS RELATIONS AND FUNCTIONS EXAMPLES OF FUNCTIONS 4 VIEWING RELATIONS AND
Essential Question How can you solve a nonlinear system of equations?
.5 Solving Nonlinear Sstems Essential Question Essential Question How can ou solve a nonlinear sstem of equations? Solving Nonlinear Sstems of Equations Work with a partner. Match each sstem with its graph.
16.5. Maclaurin and Taylor Series. Introduction. Prerequisites. Learning Outcomes
Maclaurin and Talor Series 6.5 Introduction In this Section we eamine how functions ma be epressed in terms of power series. This is an etremel useful wa of epressing a function since (as we shall see)
8. BOOLEAN ALGEBRAS x x
8. BOOLEAN ALGEBRAS 8.1. Definition of a Boolean Algebra There are man sstems of interest to computing scientists that have a common underling structure. It makes sense to describe such a mathematical
Chapter 6. Self-Adjusting Data Structures
Chapter 6 Self-Adjusting Data Structures Chapter 5 describes a data structure that is able to achieve an epected quer time that is proportional to the entrop of the quer distribution. The most interesting
WUCT121. Discrete Mathematics. Logic
WUCT11 Discrete Mathematics Logic 1. Logic. Predicate Logic 3. Proofs 4. Set Theor 5. Relations and Functions WUCT11 Logic 1 Section 1. Logic 1.1. Introduction. In developing a mathematical theor, assertions
Functions If (x, y 1 ), (x, y 2 ) S, then y 1 = y 2
Functions 4-3-2008 Definition. A function f from a set X to a set Y is a subset S of the product X Y such that if (, y 1 ), (, y 2 ) S, then y 1 = y 2. Instead of writing (, y) S, you usually write f()
Which of the following expressions are monomials?
9 1 Stud Guide Pages 382 387 Polnomials The epressions, 6, 5a 2, and 10cd 3 are eamples of monomials. A monomial is a number, a variable, or a product of numbers and variables. An eponents in a monomial
Math 53 Homework 4 Solutions
Math 5 Homework 4 Solutions Problem 1. (a) z = is a paraboloid with its highest point at (0,0,) and intersecting the -plane at the circle + = of radius. (or: rotate the parabola z = in the z-plane about
Using Intercept Form
8.5 Using Intercept Form Essential Question What are some of the characteristics of the graph of f () = a( p)( q)? Using Zeros to Write Functions Work with a partner. Each graph represents a function of
LESSON #48 - INTEGER EXPONENTS COMMON CORE ALGEBRA II
LESSON #8 - INTEGER EXPONENTS COMMON CORE ALGEBRA II We just finished our review of linear functions. Linear functions are those that grow b equal differences for equal intervals. In this unit we will
11.4 Polar Coordinates
11. Polar Coordinates 917 11. Polar Coordinates In Section 1.1, we introduced the Cartesian coordinates of a point in the plane as a means of assigning ordered pairs of numbers to points in the plane.
Analytic Geometry in Three Dimensions
Analtic Geometr in Three Dimensions. The Three-Dimensional Coordinate Sstem. Vectors in Space. The Cross Product of Two Vectors. Lines and Planes in Space The three-dimensional coordinate sstem is used
Introduction to Differential Equations. National Chiao Tung University Chun-Jen Tsai 9/14/2011
Introduction to Differential Equations National Chiao Tung Universit Chun-Jen Tsai 9/14/011 Differential Equations Definition: An equation containing the derivatives of one or more dependent variables,
Section 4.1 Increasing and Decreasing Functions
Section.1 Increasing and Decreasing Functions The graph of the quadratic function f 1 is a parabola. If we imagine a particle moving along this parabola from left to right, we can see that, while the -coordinates
10.3 Solving Nonlinear Systems of Equations
60 CHAPTER 0 Conic Sections Identif whether each equation, when graphed, will be a parabola, circle, ellipse, or hperbola. Then graph each equation.. - 7 + - =. = +. = + + 6. + 9 =. 9-9 = 6. 6 - = 7. 6
Section 1.2: A Catalog of Functions
Section 1.: A Catalog of Functions As we discussed in the last section, in the sciences, we often tr to find an equation which models some given phenomenon in the real world - for eample, temperature as
Solutions to the Math 1051 Sample Final Exam (from Spring 2003) Page 1
Solutions to the Math 0 Sample Final Eam (from Spring 00) Page Part : Multiple Choice Questions. Here ou work out the problems and then select the answer that matches our answer. No partial credit is given
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
MAC 110 Module Test Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem using our calculator. 1) The paired data below consist
MAT389 Fall 2016, Problem Set 2
MAT389 Fall 2016, Problem Set 2 Circles in the Riemann sphere Recall that the Riemann sphere is defined as the set Let P be the plane defined b Σ = { (a, b, c) R 3 a 2 + b 2 + c 2 = 1 } P = { (a, b, c)
INTRODUCTION TO DIOPHANTINE EQUATIONS
INTRODUCTION TO DIOPHANTINE EQUATIONS In the earl 20th centur, Thue made an important breakthrough in the stud of diophantine equations. His proof is one of the first eamples of the polnomial method. His
1.1 The Equations of Motion
1.1 The Equations of Motion In Book I, balance of forces and moments acting on an component was enforced in order to ensure that the component was in equilibrium. Here, allowance is made for stresses which
Study Guide and Intervention
6- NAME DATE PERID Stud Guide and Intervention Graphing Quadratic Functions Graph Quadratic Functions Quadratic Function A function defined b an equation of the form f () a b c, where a 0 b Graph of a
Start at the origin. Move left 3 units since the x-coordinate. Start at the origin. Since the x-coordinate is 0, the point
Answers (Lesson -) Lesson - - Stud Guide and Intervention The Coordinate Plane Identif Points In the diagram at the right, points are located in reference to two perpendicular number lines called aes.
APPENDIX D Rotation and the General Second-Degree Equation
APPENDIX D Rotation and the General Second-Degree Equation Rotation of Aes Invariants Under Rotation After rotation of the - and -aes counterclockwise through an angle, the rotated aes are denoted as the
We have examined power functions like f (x) = x 2. Interchanging x
CHAPTER 5 Eponential and Logarithmic Functions We have eamined power functions like f =. Interchanging and ields a different function f =. This new function is radicall different from a power function
CCSSM Algebra: Equations
CCSSM Algebra: Equations. Reasoning with Equations and Inequalities (A-REI) Eplain each step in solving a simple equation as following from the equalit of numbers asserted at the previous step, starting
Chapter 3. Expectations
pectations - Chapter. pectations.. Introduction In this chapter we define five mathematical epectations the mean variance standard deviation covariance and correlation coefficient. We appl these general
Practice Midterm Exam
CS/MATH320L - Applied Discrete Math Spring 2013 Instructor: Marc Pomplun Practice Midterm Exam Sample Solutions Question 1: out of points Question 2: out of points Question 3: out of points Question 4: