171S2.2q The Algebra of Functions. February 13, MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College
|
|
- Felix Barber
- 5 years ago
- Views:
Transcription
1 MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 2: More on Functions 2.1 Increasing, Decreasing, and Piecewise Functions; Applications 2.2 The Algebra of Functions 2.3 The Composition of Functions 2.4 Symmetry 2.5 Transformations 2.6 Variation and Applications Mathematica Interactive Figures are available through Tools for Success, Activities and Projects in CourseCompass. You may access these through CourseCompass or from the Important Links webpage. You must Login to MML to use this link. Feb 12 2:03 PM 2.2 The Algebra of Functions Find 1. the sum, 2. the difference, 3. the product, 4. the quotient of two functions, 5. determine the domains of the resulting functions. Find the difference quotient for a function. See the animation about the algebra of functions. It is in Course Documents of CourseCompass. Sums, Differences, Products, and Quotients of Functions If f and g are functions and x is in the domain of each function, then Example Given that f (x) = x + 2 and g(x) = 2x + 5, find each of the following. a) (f + g)(x)b) (f + g)(5) Solution: a) b) We can find ( f + g)(5) provided 5 is in the domain of each function. This is true. f(5) = = 7 g(5) = 2(5) + 5 = 15 (f + g)(5) = f(5) + g(5) = = 22 or using the function from (a) above (f + g)(x) = 3x + 7, we get (f + g)(5) = 3(5) + 7 = 22 1
2 Another Example Given that f(x) = x and g(x) = x 3, find each of the following. a) The domain of f + g, f g, fg, and f/g b) (f g)(x) c) (f/g)(x) Solution: a) The domain of f is the set of all real numbers. The domain of g is also the set of all real numbers. The domains of f + g, f g, and fg are the set of numbers in the intersection of the domains that is, the set of numbers in both domains, or all real numbers. For f/g, we must exclude 3, since g(3) = 0. Remember: f(x) = x and g(x) = x 3 b) (f g)(x) = f(x) g(x) = (x 2 + 2) (x 3) = x 2 x + 5 c) (f/g)(x) = Remember to add the stipulation that x 3, since 3 is not in the domain of (f/g)(x). Difference Quotient The ratio below is called the difference quotient, or average rate of change. Mathematica Interactive Figures are available through Tools for Success, Activities and Projects in CourseCompass. You may access these through CourseCompass or from the Important Links webpage. You must Login to MML to use this link. Another Example Example For the function f given by f (x) = 5x 1, find the difference quotient For the function f given by f (x) = x 2 + 2x 3, find the difference quotient. Solution: We first find f (x + h): Solution: We first find f (x + h): 2
3 180/4. Given f(x) = x 2 3 and g(x) = 2x + 1, find (fg)(2). 180/8. Given f(x) = x 2 3 and g(x) = 2x + 1, find. 180/5. Given f(x) = x 2 3 and g(x) = 2x + 1, find (f / g)( 1/2). 180/10. Given f(x) = x 2 3 and g(x) = 2x + 1, find (g / f)( 1/2). Sep 15 9:28 PM Sep 15 9:28 PM 180/ /24. Sep 15 9:35 PM Sep 15 9:35 PM 3
4 180/ /32. Sep 15 9:38 PM Sep 15 9:38 PM 181/36. Find the domain of F G, FG, and F / G. 181/36. Find the domain of F G, FG, and F / G. 181/38. Graph F + G. 181/40. Graph F G. Sep 15 9:39 PM Sep 15 9:39 PM 4
5 181/47. Total Cost, Revenue, and Profit. In economics, functions that involve revenue, cost, and profit are used. For example, suppose that and denote the total revenue and the total cost, respectively, of producing a new kind of tool for King Hardware Wholesalers. Then the difference P(x) = R(x) C(x) represents the total profit for producing x tools. Given R(x) = 60x 0.4x 2; C(x) = 3x + 13, find each of the following: b) R(100) C(100) P(100) 181/47. Total Cost, Revenue, and Profit. In economics, functions that involve revenue, cost, and profit are used. For example, suppose that and denote the total revenue and the total cost, respectively, of producing a new kind of tool for King Hardware Wholesalers. Then the difference P(x) = R(x) C(x) represents the total profit for producing x tools. Given R(x) = 60x 0.4x 2; C(x) = 3x + 13, find each of the following: b) R(100) C(100) P(100) 181/48. Total Cost, Revenue, and Profit. Given that R(x) = 200x x 2 and C(x) = x for a new weather radio produced by Clear Communication, find each of the following. (See Exercise 47.) b) R(175), C(175), and P(175) c) Using a graphing calculator, graph the three functions in the viewing window [0, 200, 0, 10,000]. 181/48. Total Cost, Revenue, and Profit. Given that R(x) = 200x x 2 and C(x) = x for a new weather radio produced by Clear Communication, find each of the following. (See Exercise 47.) b) R(175), C(175), and P(175) c) Using a graphing calculator, graph the three functions in the viewing window [0, 200, 0, 10,000]. 5
6 181/52. For the function f(x) = 5x + 3, construct and simplify the difference quotient 181/54. For the function f(x) = ( 1/2)x + 7, construct and simplify the difference quotient 181/60. For the function f(x) = x 2 3, construct and simplify the difference quotient 181/62. For the function f(x) = 2 x 2, construct and simplify the difference quotient 6
171S4.6q Polynomial and Rational Inequalities. April 02, Polynomial and Rational Inequalities. Polynomial Inequalities
MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 4: Polynomial and Rational Functions 4.1 Polynomial Functions and Models 4.2 Graphing Polynomial Functions 4.3 Polynomial
More information171S4.3 Polynomial Division; The Remainder and Factor Theorems. October 26, Polynomial Division; The Remainder and Factor Theorems
MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 4: Polynomial and Rational Functions 4.1 Polynomial Functions and Models 4.2 Graphing Polynomial Functions 4.3 Polynomial
More information171S4.3 Polynomial Division; The Remainder and Factor Theorems. March 24, Polynomial Division; The Remainder and Factor Theorems
MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 4: Polynomial and Rational Functions 4.1 Polynomial Functions and Models 4.2 Graphing Polynomial Functions 4.3 Polynomial
More informationCHAPTER 3: Quadratic Functions and Equations; Inequalities
171S MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 3: Quadratic Functions and Equations; Inequalities 3.1 The Complex Numbers 3.2 Quadratic Equations, Functions, Zeros,
More informationCHAPTER 3: Quadratic Functions and Equations; Inequalities
MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 3: Quadratic Functions and Equations; Inequalities 3.1 The Complex Numbers 3.2 Quadratic Equations, Functions, Zeros, and
More informationOctober 28, S4.4 Theorems about Zeros of Polynomial Functions
MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 4: Polynomial and Rational Functions 4.1 Polynomial Functions and Models 4.2 Graphing Polynomial Functions 4.3 Polynomial
More informationApril 12, S5.3q Logarithmic Functions and Graphs
MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 5: Exponential and Logarithmic Functions 5.1 Inverse Functions 5.2 Exponential Functions and Graphs 5.3 Logarithmic Functions
More informationFebruary 27, S3.4q Solving Rational Equations and Radical Equations TRUE. The solution is 5.
MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 3: Quadratic Functions and Equations; Inequalities 3.1 The Complex Numbers 3.2 Quadratic Equations, Functions, Zeros, and
More informationCHAPTER 4: Polynomial and Rational Functions
MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 4: Polynomial and Rational Functions 4.1 Polynomial Functions and Models 4.2 Graphing Polynomial Functions 4.3 Polynomial
More information171S4.4 Theorems about Zeros of Polynomial Functions. March 27, 2012
MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 4: Polynomial and Rational Functions 4.1 Polynomial Functions and Models 4.2 Graphing Polynomial Functions 4.3 Polynomial
More information171S3.4 Solving Rational Equations & Radical Equations. February 23, Some Media for this Section
MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 3: Quadratic Functions and Equations; Inequalities 3.1 The Complex Numbers 3.2 Quadratic Equations, Functions, Zeros, and
More informationCHAPTER 4: Polynomial and Rational Functions
MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 4: Polynomial and Rational Functions 4.1 Polynomial Functions and Models 4.2 Graphing Polynomial Functions 4.3 Polynomial
More informationSeptember 26, S2.5 Variation and Applications
MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 2: More on Functions 2.1 Increasing, Decreasing, and Piecewise Functions; Applications 2.2 The Algebra of Functions 2.3
More informationCHAPTER 9: Systems of Equations and Matrices
MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 9: Systems of Equations and Matrices 9.1 Systems of Equations in Two Variables 9.2 Systems of Equations in Three Variables
More informationNote: The actual exam will consist of 20 multiple choice questions and 6 show-your-work questions. Extra questions are provided for practice.
College Algebra - Unit 2 Exam - Practice Test Note: The actual exam will consist of 20 multiple choice questions and 6 show-your-work questions. Extra questions are provided for practice. MULTIPLE CHOICE.
More informationMarch 20, S4.1q Polynomial Functions and Models
MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 4: Polynomial and Rational Functions 4.1 Polynomial Functions and Models 4.2 Graphing Polynomial Functions 4.3 Polynomial
More informationCHAPTER 5: Exponential and Logarithmic Functions
MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 5: Exponential and Logarithmic Functions 5.1 Inverse Functions 5.2 Exponential Functions and Graphs 5.3 Logarithmic Functions
More informationMath 110 Midterm 1 Study Guide October 14, 2013
Name: For more practice exercises, do the study set problems in sections: 3.4 3.7, 4.1, and 4.2. 1. Find the domain of f, and express the solution in interval notation. (a) f(x) = x 6 D = (, ) or D = R
More informationOBJECTIVE Find limits of functions, if they exist, using numerical or graphical methods.
1.1 Limits: A Numerical and Graphical Approach OBJECTIVE Find limits of functions, if they exist, using numerical or graphical methods. 1.1 Limits: A Numerical and Graphical Approach DEFINITION: As x approaches
More information1 Fundamental Concepts From Algebra & Precalculus
Fundamental Concepts From Algebra & Precalculus. Review Exercises.. Simplify eac expression.. 5 7) [ 5)) ]. ) 5) 7) 9 + 8 5. 8 [ 5) 8 6)] [9 + 8 5 ]. 9 + 8 5 ) 8) + 5. 5 + [ )6)] 7) 7 + 6 5 6. 8 5 ) 6
More informationQuadratic function and equations Quadratic function/equations, supply, demand, market equilibrium
Exercises 8 Quadratic function and equations Quadratic function/equations, supply, demand, market equilibrium Objectives - know and understand the relation between a quadratic function and a quadratic
More informationDownloaded from
Question 1: Exercise 2.1 The graphs of y = p(x) are given in following figure, for some polynomials p(x). Find the number of zeroes of p(x), in each case. (i) (ii) (iii) Page 1 of 24 (iv) (v) (v) Page
More information171S4.1 Polynomial Functions and Models. March 20, 2012
MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 4: Polynomial and Rational Functions 4.1 Polynomial Functions and Models 4.2 Graphing Polynomial Functions 4.3 Polynomial
More informationMathematics for Business and Economics - I. Chapter 5. Functions (Lecture 9)
Mathematics for Business and Economics - I Chapter 5. Functions (Lecture 9) Functions The idea of a function is this: a correspondence between two sets D and R such that to each element of the first set,
More information171S3.2 Quadratic Equations, Functions, Zeros, and Models September 30, Quadratic Equations, Functions, Zeros, and Models
MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 3: Quadratic Functions and Equations; Inequalities 3.1 The Complex Numbers 3.2 Quadratic Equations, Functions, Zeros, and
More informationCHAPTER 3: Quadratic Functions and Equations; Inequalities
MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 3: Quadratic Functions and Equations; Inequalities 3.1 The Complex Numbers 3.2 Quadratic Equations, Functions, Zeros, and
More informationTest 2 Review Math 1111 College Algebra
Test 2 Review Math 1111 College Algebra 1. Begin by graphing the standard quadratic function f(x) = x 2. Then use transformations of this graph to graph the given function. g(x) = x 2 + 2 *a. b. c. d.
More informationMTH4100 Calculus I. Lecture notes for Week 2. Thomas Calculus, Sections 1.3 to 1.5. Rainer Klages
MTH4100 Calculus I Lecture notes for Week 2 Thomas Calculus, Sections 1.3 to 1.5 Rainer Klages School of Mathematical Sciences Queen Mary University of London Autumn 2009 Reading Assignment: read Thomas
More information10-1: Composite and Inverse Functions
Math 95 10-1: Composite and Inverse Functions Functions are a key component in the applications of algebra. When working with functions, we can perform many useful operations such as addition and multiplication
More information, a 1. , a 2. ,..., a n
CHAPTER Points to Remember :. Let x be a variable, n be a positive integer and a 0, a, a,..., a n be constants. Then n f ( x) a x a x... a x a, is called a polynomial in variable x. n n n 0 POLNOMIALS.
More informationCHAPTER 9: Systems of Equations and Matrices
From Section 1.4: Equations of Lines and Modeling Use a graphing calculator to model the data with a linear function. MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 9:
More informationSection 0.2 & 0.3 Worksheet. Types of Functions
MATH 1142 NAME Section 0.2 & 0.3 Worksheet Types of Functions Now that we have discussed what functions are and some of their characteristics, we will explore different types of functions. Section 0.2
More informationMAT 171. August 22, S1.4 Equations of Lines and Modeling. Section 1.4 Equations of Lines and Modeling
MAT 171 WebAdvisor: http://reg.cfcc.edu Dr. Claude Moore, CFCC Session 1 introduces the Course, CourseCompass, and Chapter 1: Graphs, Functions, and Models. This session is available in CourseCompass.
More information(MATH 1203, 1204, 1204R)
College Algebra (MATH 1203, 1204, 1204R) Departmental Review Problems For all questions that ask for an approximate answer, round to two decimal places (unless otherwise specified). The most closely related
More informationCHAPTER 2 POLYNOMIALS KEY POINTS
CHAPTER POLYNOMIALS KEY POINTS 1. Polynomials of degrees 1, and 3 are called linear, quadratic and cubic polynomials respectively.. A quadratic polynomial in x with real coefficient is of the form a x
More informationChapter 8. Exploring Polynomial Functions. Jennifer Huss
Chapter 8 Exploring Polynomial Functions Jennifer Huss 8-1 Polynomial Functions The degree of a polynomial is determined by the greatest exponent when there is only one variable (x) in the polynomial Polynomial
More informationFinal Exam Review Packet
1 Exam 1 Material Sections A.1, A.2 and A.6 were review material. There will not be specific questions focused on this material but you should know how to: Simplify functions with exponents. Factor quadratics
More informationFinal Exam Review Packet
1 Exam 1 Material Sections A.1, A.2 and A.6 were review material. There will not be specific questions focused on this material but you should know how to: Simplify functions with exponents. Factor quadratics
More informationPolynomials 6c Classifying the Zeros of a Polynomial Functions
Polynomials 6c Classifying the Zeros of a Polynomial Functions Standards: A APR.2, A APR.3, F IF.7c, N CN.9 Learning Target(s): How many zeros does a polynomial have? How can we find all the exact zeros
More informationCHAPTER 9: Systems of Equations and Matrices
MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 9: Systems of Equations and Matrices 9.1 Systems of Equations in Two Variables 9.2 Systems of Equations in Three Variables
More informationMAT College Algebra - Gateway 2: Linear Equations and. Name: Date: B
Mathematics Department MAT 10 - College Algebra - Gateway : Linear Equations and Name: Date: 9919 B This exam covers material from Sections 1.1, 1.7,.1,.3, 3.1-3.3,.1, and.. The topics covered are function
More informationTheta Functions MAΘ National Convention 2018
Theta Functions MAΘ National Convention 018 Note that choice stands for None of the above answers is correct 1. How many times do y = x and y = x 3 intersect? A) 0 B) 1 C) D) 3. Let f(x) = x + 19 and g(x)
More informationNotes: Piecewise Functions
Objective: Students will be able to write evaluate piecewise defined functions, graph piecewise defined functions, evaluate the domain and range for piecewise defined functions, and solve application problems.
More informationEquation Solving Principles. Equations and Solutions. Zeros of Linear Functions. August 30, 2012
MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 1: Graphs, Functions, and Models 1.1 Introduction to Graphing 1.2 Functions and Graphs 1.3 Linear Functions, Slope, and
More informationQuestion 1: The graphs of y = p(x) are given in following figure, for some Polynomials p(x). Find the number of zeroes of p(x), in each case.
Class X - NCERT Maths EXERCISE NO:.1 Question 1: The graphs of y = p(x) are given in following figure, for some Polynomials p(x). Find the number of zeroes of p(x), in each case. (i) (ii) (iii) (iv) (v)
More informationFUNCTIONS - PART 2. If you are not familiar with any of the material below you need to spend time studying these concepts and doing some exercises.
Introduction FUNCTIONS - PART 2 This handout is a summary of the basic concepts you should understand and be comfortable working with for the second math review module on functions. This is intended as
More informationMATH 1113 Exam 1 Review
MATH 1113 Exam 1 Review Topics Covered Section 1.1: Rectangular Coordinate System Section 1.3: Functions and Relations Section 1.4: Linear Equations in Two Variables and Linear Functions Section 1.5: Applications
More informationCHAPTER 9: Systems of Equations and Matrices
MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 9: Systems of Equations and Matrices 9.1 Systems of Equations in Two Variables 9.2 Systems of Equations in Three Variables
More informationLimit Theorems. MATH 464/506, Real Analysis. J. Robert Buchanan. Summer Department of Mathematics. J. Robert Buchanan Limit Theorems
Limit s MATH 464/506, Real Analysis J. Robert Buchanan Department of Mathematics Summer 2007 Bounded Functions Definition Let A R, let f : A R, and let c R be a cluster point of A. We say that f is bounded
More informationPreCalculus: Semester 1 Final Exam Review
Name: Class: Date: ID: A PreCalculus: Semester 1 Final Exam Review Short Answer 1. Determine whether the relation represents a function. If it is a function, state the domain and range. 9. Find the domain
More informationMAT 155. Key Concept. Density Curve
MAT 155 Dr. Claude Moore Cape Fear Community College Chapter 6 Normal Probability Distributions 6 1 Review and Preview 6 2 The Standard Normal Distribution 6 3 Applications of Normal Distributions 6 4
More informationChapter 4E - Combinations of Functions
Fry Texas A&M University!! Math 150!! Chapter 4E!! Fall 2015! 121 Chapter 4E - Combinations of Functions 1. Let f (x) = 3 x and g(x) = 3+ x a) What is the domain of f (x)? b) What is the domain of g(x)?
More informationPolynomial Review Problems
Polynomial Review Problems 1. Find polynomial function formulas that could fit each of these graphs. Remember that you will need to determine the value of the leading coefficient. The point (0,-3) is on
More informationMath 211 Business Calculus TEST 3. Question 1. Section 2.2. Second Derivative Test.
Math 211 Business Calculus TEST 3 Question 1. Section 2.2. Second Derivative Test. p. 1/?? Math 211 Business Calculus TEST 3 Question 1. Section 2.2. Second Derivative Test. Question 2. Section 2.3. Graph
More informationReview all the activities leading to Midterm 3. Review all the problems in the previous online homework sets (8+9+10).
MA109, Activity 34: Review (Sections 3.6+3.7+4.1+4.2+4.3) Date: Objective: Additional Assignments: To prepare for Midterm 3, make sure that you can solve the types of problems listed in Activities 33 and
More informationSample Math 115 Midterm Exam Spring, 2014
Sample Math 5 Midterm Exam Spring, 04 The midterm examination is on Wednesday, March at 5:45PM 7:45PM The midterm examination will be in Budig 0 Look for your instructor who will direct you where to sit
More information171S1.1 Graphs, Functions, Models August 23, 2012
MAT 171 D10 8:00 9:15 MAT 171 D11 9:30 10:45 Aug 21 7:11 AM Aug 21 7:14 AM MAT 171 D14 2:00 3:15 MAT 171 D15 3:30 4:45 Aug 21 7:14 AM Aug 21 7:14 AM 1 MAT 171 Dr. Claude Moore, CFCC CHAPTER 1: Graphs,
More informationMidterm 1 Review Problems Business Calculus
Midterm 1 Review Problems Business Calculus 1. (a) Show that the functions f and g are inverses of each other by showing that f g(x) = g f(x) given that (b) Sketch the functions and the line y = x f(x)
More information171S5.6o Applications and Models: Growth and Decay; and Compound Interest November 21, 2011
MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 5: Exponential and Logarithmic Functions 5.1 Inverse Functions 5.2 Exponential Functions and Graphs 5.3 Logarithmic Functions
More information1. Use the properties of exponents to simplify the following expression, writing your answer with only positive exponents.
Math120 - Precalculus. Final Review. Fall, 2011 Prepared by Dr. P. Babaali 1 Algebra 1. Use the properties of exponents to simplify the following expression, writing your answer with only positive exponents.
More informationA Partial List of Topics: Math Spring 2009
A Partial List of Topics: Math 112 - Spring 2009 This is a partial compilation of a majority of the topics covered this semester and may not include everything which might appear on the exam. The purpose
More informationChapter 1. Functions 1.1. Functions and Their Graphs
1.1 Functions and Their Graphs 1 Chapter 1. Functions 1.1. Functions and Their Graphs Note. We start by assuming that you are familiar with the idea of a set and the set theoretic symbol ( an element of
More informationA) (-1, -1, -2) B) No solution C) Infinite solutions D) (1, 1, 2) A) (6, 5, -3) B) No solution C) Infinite solutions D) (1, -3, -7)
Algebra st Semester Final Exam Review Multiple Choice. Write an equation that models the data displayed in the Interest-Free Loan graph that is provided. y = x + 80 y = -0x + 800 C) y = 0x 00 y = 0x +
More informationName: Algebra 1 Section 3 Homework Problem Set: Introduction to Functions
Name: Algebra 1 Section 3 Homework Problem Set: Introduction to Functions Remember: To receive full credit, you must show all of your work and circle/box your final answers. If you run out of room for
More informationMath Section Bekki George: 01/16/19. University of Houston. Bekki George (UH) Math /16/19 1 / 31
Math 1431 Section 12200 Bekki George: bekki@math.uh.edu University of Houston 01/16/19 Bekki George (UH) Math 1431 01/16/19 1 / 31 Office Hours: Mondays 1-2pm, Tuesdays 2:45-3:30pm (also available by appointment)
More informationB) Increasing on (-1, ); Decreasing on (-, -1) C) Increasing on (-, -1); Decreasing on (-1, ) D) Increasing on (-, 1); Decreasing on (1, ) 2) 2)
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine the intervals on which the function is increasing, decreasing, and constant. 1) 1) Increasing
More informationD) Increasing on (-1, ); Decreasing on (-, -1) B) Increasing on (-, -1); Decreasing on (-1, ) C) Increasing on (-, 1); Decreasing on (1, ) 2) 2)
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine the intervals on which the function is increasing, decreasing, and constant. 1) 1) Increasing
More informationSection Properties of Rational Expressions
88 Section. - Properties of Rational Expressions Recall that a rational number is any number that can be written as the ratio of two integers where the integer in the denominator cannot be. Rational Numbers:
More informationEquations and Solutions
MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 1: Graphs, Functions, and Models 1.1 Introduction to Graphing 1.2 Functions and Graphs 1.3 Linear Functions, Slope, and
More informationSection 5.1 Composite Functions
Section 5. Composite Functions Objective #: Form a Composite Function. In many cases, we can create a new function by taking the composition of two functions. For example, suppose f(x) x and g(x) x +.
More information(a) Write down the value of q and of r. (2) Write down the equation of the axis of symmetry. (1) (c) Find the value of p. (3) (Total 6 marks)
1. Let f(x) = p(x q)(x r). Part of the graph of f is shown below. The graph passes through the points ( 2, 0), (0, 4) and (4, 0). (a) Write down the value of q and of r. (b) Write down the equation of
More informationObjectives. 171S5.6_p Applications of Exponential and Logarithmic Functions. April 21, 2011
MAT 171 Precalculus Algebra Trigsted Pilot Test Dr. Claude Moore Cape Fear Community College CHAPTER 5: Exponential and Logarithmic Functions and Equations 5.1 Exponential Functions 5.2 The Natural Exponential
More informationChapter 2. Polynomial and Rational Functions. 2.6 Rational Functions and Their Graphs. Copyright 2014, 2010, 2007 Pearson Education, Inc.
Chapter Polynomial and Rational Functions.6 Rational Functions and Their Graphs Copyright 014, 010, 007 Pearson Education, Inc. 1 Objectives: Find the domains of rational functions. Use arrow notation.
More informationChapter 3 Notes, Applied Calculus, Tan
Contents 3.1 Basic Rules of Differentiation.............................. 2 3.2 The Prouct an Quotient Rules............................ 6 3.3 The Chain Rule...................................... 9 3.4
More informationSO x is a cubed root of t
7.6nth Roots 1) What do we know about x because of the following equation x 3 = t? All in one.docx SO x is a cubed root of t 2) Definition of nth root: 3) Study example 1 4) Now try the following problem
More informationKey Concept. Properties. February 23, S6.4_3 Sampling Distributions and Estimators
MAT 155 Statistical Analysis Dr. Claude Moore Cape Fear Community College Chapter 6 Normal Probability Distributions 6 1 Review and Preview 6 2 The Standard Normal Distribution 6 3 Applications of Normal
More informationSection 11.7 The Chain Rule
Section.7 The Chain Rule Composition of Functions There is another way of combining two functions to obtain a new function. For example, suppose that y = fu) = u and u = gx) = x 2 +. Since y is a function
More informationHomework 1. 3x 12, 61.P (x) = 3t 21 Section 1.2
Section 1.1 Homework 1 (34, 36) Determine whether the equation defines y as a function of x. 34. x + h 2 = 1, 36. y = 3x 1 x + 2. (40, 44) Find the following for each function: (a) f(0) (b) f(1) (c) f(
More informationSection 11.3 Rates of Change:
Section 11.3 Rates of Change: 1. Consider the following table, which describes a driver making a 168-mile trip from Cleveland to Columbus, Ohio in 3 hours. t Time (in hours) 0 0.5 1 1.5 2 2.5 3 f(t) Distance
More informationAlgebra II: Strand 5. Power, Polynomial, and Rational Functions; Topic 3. Rational Functions; Task 5.3.1
TASK 5.3.: RATIONAL FUNCTIONS AND THEIR RECIPROCALS Solutions Rational functions appear frequently in business, science, engineering, and medical applications. This activity explores some aspects of rational
More informationMath Exam Jam Concise. Contents. 1 Algebra Review 2. 2 Functions and Graphs 2. 3 Exponents and Radicals 3. 4 Quadratic Functions and Equations 4
Contents 1 Algebra Review 2 2 Functions and Graphs 2 3 Exponents and Radicals 3 4 Quadratic Functions and Equations 4 5 Exponential and Logarithmic Functions 5 6 Systems of Linear Equations 6 7 Inequalities
More informationSkills Practice Skills Practice for Lesson 10.1
Skills Practice Skills Practice for Lesson.1 Name Date Higher Order Polynomials and Factoring Roots of Polynomial Equations Problem Set Solve each polynomial equation using factoring. Then check your solution(s).
More informationCollege Algebra. George Voutsadakis 1. LSSU Math 111. Lake Superior State University. 1 Mathematics and Computer Science
College Algebra George Voutsadakis 1 1 Mathematics and Computer Science Lake Superior State University LSSU Math 111 George Voutsadakis (LSSU) College Algebra December 2014 1 / 74 Outline 1 Additional
More informationMath 110 Final Exam General Review. Edward Yu
Math 110 Final Exam General Review Edward Yu Da Game Plan Solving Limits Regular limits Indeterminate Form Approach Infinities One sided limits/discontinuity Derivatives Power Rule Product/Quotient Rule
More information8 + 6) x 2 ) y = h(x)
. a. Horizontal shift 6 left and vertical shift up. Notice B' is ( 6, ) and B is (0, 0). b. h(x) = 0.5(x + 6) + (Enter in a grapher to check.) c. Use the graph. Notice A' to see h(x) crosses the x-axis
More informationGiven the table of values, determine the equation
3.1 Properties of Quadratic Functions Recall: Standard Form f(x) = ax 2 + bx + c Factored Form f(x) = a(x r)(x s) Vertex Form f(x) = a(x h) 2 + k Given the table of values, determine the equation x y 1
More informationThe Graph of a Quadratic Function. Quadratic Functions & Models. The Graph of a Quadratic Function. The Graph of a Quadratic Function
8/1/015 The Graph of a Quadratic Function Quadratic Functions & Models Precalculus.1 The Graph of a Quadratic Function The Graph of a Quadratic Function All parabolas are symmetric with respect to a line
More informationRewriting Absolute Value Functions as Piece-wise Defined Functions
Rewriting Absolute Value Functions as Piece-wise Defined Functions Consider the absolute value function f ( x) = 2x+ 4-3. Sketch the graph of f(x) using the strategies learned in Algebra II finding the
More informationCollege Algebra. George Voutsadakis 1. LSSU Math 111. Lake Superior State University. 1 Mathematics and Computer Science
College Algebra George Voutsadakis 1 1 Mathematics and Computer Science Lake Superior State University LSSU Math 111 George Voutsadakis (LSSU) College Algebra December 2014 1 / 71 Outline 1 Higher Degree
More informationRational Functions. Elementary Functions. Algebra with mixed fractions. Algebra with mixed fractions
Rational Functions A rational function f (x) is a function which is the ratio of two polynomials, that is, Part 2, Polynomials Lecture 26a, Rational Functions f (x) = where and are polynomials Dr Ken W
More informationFunction? c. {(-1,4);(0,-4);(1,-3);(-1,5);(2,-5)} {(-2,3);(-1,3);(0,1);(1,-3);(2,-5)} a. Domain Range Domain Range
Section 3.1: Functions Definitions (pages 226 227): A relation is a correspondence between two sets. A function is a correspondence to a first set, called the domain, to a second set, called the range,
More informationLogarithms Dr. Laura J. Pyzdrowski
1 Names: (8 communication points) About this Laboratory An exponential function of the form f(x) = a x, where a is a positive real number not equal to 1, is an example of a one-to-one function. This means
More informationMAT 12 - SEC 021 PRECALCULUS SUMMER SESSION II 2014 LECTURE 3
MAT 12 - SEC 021 PRECALCULUS SUMMER SESSION II 2014 LECTURE 3 JAMIE HADDOCK 1. Agenda Functions Composition Graphs Average Rate of Change..............................................................................................................
More information8. Limit Laws. lim(f g)(x) = lim f(x) lim g(x), (x) = lim x a f(x) g lim x a g(x)
8. Limit Laws 8.1. Basic Limit Laws. If f and g are two functions and we know the it of each of them at a given point a, then we can easily compute the it at a of their sum, difference, product, constant
More informationApplications of differential calculus Relative maxima/minima, points of inflection
Exercises 15 Applications of differential calculus Relative maxima/minima, points of inflection Objectives - be able to determine the relative maxima/minima of a function. - be able to determine the points
More information6.1 Polynomial Functions
6.1 Polynomial Functions Definition. A polynomial function is any function p(x) of the form p(x) = p n x n + p n 1 x n 1 + + p 2 x 2 + p 1 x + p 0 where all of the exponents are non-negative integers and
More informationChapter 3: Inequalities, Lines and Circles, Introduction to Functions
QUIZ AND TEST INFORMATION: The material in this chapter is on Quiz 3 and Exam 2. You should complete at least one attempt of Quiz 3 before taking Exam 2. This material is also on the final exam and used
More informationWarm-Up: Sketch a graph of the function and use the table to find the lim 3x 2. lim 3x. x 5. Level 3 Finding Limits Algebraically.
Warm-Up: 1. Sketch a graph of the function and use the table to find the lim 3x 2 x 5 lim 3x 2. 2 x 1 finding limits algibraically Ch.15.1 Level 3 2 Target Agenda Purpose Evaluation TSWBAT: Find the limit
More informationExample. 171S Introduction to Graphing. January 06, Introduction to Graphing. Cartesian Coordinate System
MAT 171 Dr. Claude Moore, CFCC CHAPTER 1: Graphs, Functions, and Models 1.1 Introduction to Graphing 1.2 Functions and Graphs 1.3 Linear Functions, Slope, and Applications 1.4 Equations of Lines and Modeling
More informationStudent: Date: Instructor: kumnit nong Course: MATH 105 by Nong https://xlitemprodpearsoncmgcom/api/v1/print/math Assignment: CH test review 1 Find the transformation form of the quadratic function graphed
More information