# Functions, function notation and (IGCSE composite functions)

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 New concepts and terms: (see below for explanations) Function Function notation Composite function f(g(x)) or f o g(x) Functions What is a function? A function is a relationship for which every input has only-one unique output. Ex.A The relation G has the rule: the output is a multiple of the input Input x: Output y: G G is not a function because elements from the input have more than one output. Ex. is in relation with elements (,) (,) (,8) also, is in relation with elements (,), (,8) Ex.B The relation S has the rule: the output is double the input Input x: Output y: 5 S 8 10 S is a function because each element in the input is in relation with only one element in the output Ex.C Is the relation P a function? Input x: Output y: P 8 1

2 } } h1 Function notation Functions are generally represented by the letters f, g, h, The output y of an element from the input, x, is written f(x). or as f : x a The point (x, y) can therefore be written as (x, f(x)) Ex. The function f has the rule the output is triple the input f -1 - The equation of the function is written: y = x or f ( x) = x or f : x a x We say: f of x is equal to x Or in the function f, x is mapped onto x Notice that f ( 0) = 0, f ( 1) =, f ( ) =, etc. f ( ) = input (x) output (y) On the Cartesian plane: Ex. D. Decide if each of the following graphs represents a function or not. Why or why not?

3 Cartesian graph of a function Given the graph of a relation, this relation is a function if the shadow of a vertical line crosses the graph at most at one point at a time. (If the shadow of a vertical line ever crosses the graph more than once at any one time, the graph is not a function) This is known as the vertical line test or v.l.t.

4 Graphing functions To graph a function, (or any relationship) fill out the table of values. Let x be any reasonable input, and find the corresponding y coordinate by calculating the function rule. Ex.E. Given the following function rules (Funktionen Gleichungen),fill in the tables of values and graph: a. g( x) = x b. h : x a x c. f ( x) = 1 x x y x y x y

5 Composite Functions Composite functions: f(g(x)) means that the entire result of g(x) is the input for f g(f(x)) means that the entire result of f(x) is the input for g Ex. F. Given: f ( x) = x + 1 g( x) = x h( x) = x, find: i. f(g()) ii. f(g(x)) iii. g(h(x)) iv. h(f(-5)) v. g(f(x-1)) vi. g(h(x-)) 5

6 Practice Questions 1. Indicate if each of the following relations is a function. If not, justify your answer.. Indicate if each of the following relations is a function. If not, justify your answer.. Indicate which of the following graphs (R1, R, etc) represent functions:. Graph the following functions: a. g( x) = x b. h : x a x c. f ( x) = 1 x d. m( x) = x e. p : x a x 1 f. q( x) = 1 x + 5. Given: f ( x) = x + g( x) = x + 1 h( x) = x, find: a. f(g()) b. f(g(-)) c. g(f(0)) d. g(h()) e. f(g(x)) f. g(f(x)) g. f(h(x)) h. f(x+1) i. g(f(x+1))

7 . (LS p Q) Handelt es sich bei der Zuordnung um eine Funktion? Begründe deine Antwort. Skizziere im Falle einer Funktion einen Funktionsgraphen. a. Gefährene Strecke a Benzinverbrauch b. Benzinverbrauch a gegahrene Strecke c. Zeit a Körpergröße eines Menschen d. Körpergröße eines Menschen a Zeit e. Quadrat einer rationalen Zahl a Zahl f. Rationale Zahl a Quadrat der Zah (LS. P0 Q) 9. (LS p 0 Q) Welche Funktion gehört zur Wertetabelle? Gib eine Funktionsgleichung an. Ergänze gegebenenfalls in deinem Heft die fehlenden Werte in der Tabelle. 7

8 Answers 1a. Function, b. function, c. not a function (b leads to 1 and also to ) a. function, b. not a function (a leas to 1 and ), c. function, d. not a function ( a leads to 1, and ) a. function, b. function, c. not a function, d. not a function, e. function, f. not a function a. g q m p f h 5a. -1 b. - c. 0 d. e. -x-8 f. -x+0 g. ¾ x+ h. -x+ i. -x+18 8

### 8 Building New Functions from Old Ones

Arkansas Tech University MATH 2243: Business Calculus Dr. Marcel B. Finan 8 Building New Functions from Old Ones In this section we discuss various ways for building new functions from old ones. New functions

### Test 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.

### y+2 x 1 is in the range. We solve x as x =

Dear Students, Here are sample solutions. The most fascinating thing about mathematics is that you can solve the same problem in many different ways. The correct answer will always be the same. Be creative

### Inverse Functions. One-to-one. Horizontal line test. Onto

Inverse Functions One-to-one Suppose f : A B is a function. We call f one-to-one if every distinct pair of objects in A is assigned to a distinct pair of objects in B. In other words, each object of the

### COMPOSITION OF FUNCTIONS

COMPOSITION OF FUNCTIONS INTERMEDIATE GROUP - MAY 21, 2017 Finishing Up Last Week Problem 1. (Challenge) Consider the set Z 5 = {congruence classes of integers mod 5} (1) List the elements of Z 5. (2)

### C3 Revision Questions. (using questions from January 2006, January 2007, January 2008 and January 2009)

C3 Revision Questions (using questions from January 2006, January 2007, January 2008 and January 2009) 1 2 1. f(x) = 1 3 x 2 + 3, x 2. 2 ( x 2) (a) 2 x x 1 Show that f(x) =, x 2. 2 ( x 2) (4) (b) Show

### FUNCTIONS - 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

### Section 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

### Name: Date: Block: FUNCTIONS TEST STUDY GUIDE

Algebra STUDY GUIDE AII.6, AII.7 Functions Mrs. Grieser Name: Date: Block: Test covers: Graphing using transformations FUNCTIONS TEST STUDY GUIDE Analyzing functions, including finding domain/range in

### Materials and Handouts - Warm-Up - Answers to homework #1 - Keynote and notes template - Tic Tac Toe grids - Homework #2

Calculus Unit 1, Lesson 2: Composite Functions DATE: Objectives The students will be able to: - Evaluate composite functions using all representations Simplify composite functions Materials and Handouts

### Chapter-2 Relations and Functions. Miscellaneous

1 Chapter-2 Relations and Functions Miscellaneous Question 1: The relation f is defined by The relation g is defined by Show that f is a function and g is not a function. The relation f is defined as It

### Revision Questions. Sequences, Series, Binomial and Basic Differentiation

Revision Questions Sequences, Series, Binomial and Basic Differentiation 1 ARITHMETIC SEQUENCES BASIC QUESTIONS 1) An arithmetic sequence is defined a=5 and d=3. Write down the first 6 terms. ) An arithmetic

### Composition of Functions

Composition of Functions Lecture 34 Section 7.3 Robb T. Koether Hampden-Sydney College Mon, Mar 25, 2013 Robb T. Koether (Hampden-Sydney College) Composition of Functions Mon, Mar 25, 2013 1 / 29 1 Composition

### Identify polynomial functions

EXAMPLE 1 Identify polynomial functions Decide whether the function is a polynomial function. If so, write it in standard form and state its degree, type, and leading coefficient. a. h (x) = x 4 1 x 2

### 3 FUNCTIONS. 3.1 Definition and Basic Properties. c Dr Oksana Shatalov, Fall

c Dr Oksana Shatalov, Fall 2014 1 3 FUNCTIONS 3.1 Definition and Basic Properties DEFINITION 1. Let A and B be nonempty sets. A function f from A to B is a rule that assigns to each element in the set

### Combining functions: algebraic methods

Combining functions: algebraic metods Functions can be added, subtracted, multiplied, divided, and raised to a power, just like numbers or algebra expressions. If f(x) = x 2 and g(x) = x + 2, clearly f(x)

### Further Topics in Functions

Chapter 5 Further Topics in Functions 5. Function Composition Before we embark upon any further adventures with functions, we need to take some time to gather our thoughts and gain some perspective. Chapter

### Section 6.1: Composite Functions

Section 6.1: Composite Functions Def: Given two function f and g, the composite function, which we denote by f g and read as f composed with g, is defined by (f g)(x) = f(g(x)). In other words, the function

### ISLAMIYA ENGLISH SCHOOL ABU DHABI U. A. E.

ISLAMIYA ENGLISH SCHOOL ABU DHABI U. A. E. MATHEMATICS ASSIGNMENT-1 GRADE-A/L-II(Sci) CHAPTER.NO.1,2,3(C3) Algebraic fractions,exponential and logarithmic functions DATE:18/3/2017 NAME.------------------------------------------------------------------------------------------------

### Identifying Graphs of Functions (Vertical Line Test) Evaluating Piecewise-defined Functions Sketching the Graph of a Piecewise-defined Functions

9 Functions Concepts: The Definition of A Function Identifying Graphs of Functions (Vertical Line Test) Function Notation Piecewise-defined Functions Evaluating Piecewise-defined Functions Sketching the

### Sec\$on Summary. Definition of a Function.

Section 2.3 Sec\$on Summary Definition of a Function. Domain, Codomain Image, Preimage Injection, Surjection, Bijection Inverse Function Function Composition Graphing Functions Floor, Ceiling, Factorial

### x 4 D: (4, ); g( f (x)) = 1

Honors Math 4 Describing Functions One Giant Review Name Answer Key 1. Let f (x) = x, g(x) = 6x 3, h(x) = x 3 a. f (g(h(x))) = 2x 3 b. h( f (g(x))) = 1 3 6x 3 c. f ( f ( f (x))) = x 1 8 2. Let f (x) =

### f(x) x

2 Function 2.1 Function notation The equation f(x) = x 2 + 3 defines a function f from the set R of real numbers to itself (written f : R R). This function accepts an input value x and returns an output

### POD. A) Graph: y = 2e x +2 B) Evaluate: (e 2x e x ) 2 2e -x. e 7x 2

POD A) Graph: y = 2e x +2 B) Evaluate: (e 2x e x ) 2 2e -x e 7x 2 4.4 Evaluate Logarithms & Graph Logarithmic Functions What is a logarithm? How do you read it? What relationship exists between logs and

### Families of Functions, Taylor Polynomials, l Hopital s

Unit #6 : Rule Families of Functions, Taylor Polynomials, l Hopital s Goals: To use first and second derivative information to describe functions. To be able to find general properties of families of functions.

### More on functions. Composition

More on functions Suppose f : R! R is the function defined by f(x) =x. The letter x in the previous equation is just a placeholder. You are allowed to replace the x with any number, symbol, or combination

### Rational and Radical Functions. College Algebra

Rational and Radical Functions College Algebra Rational Function A rational function is a function that can be written as the quotient of two polynomial functions P(x) and Q(x) f x = P(x) Q(x) = a )x )

### To take the derivative of x raised to a power, you multiply in front by the exponent and subtract 1 from the exponent.

MA123, Chapter 5: Formulas for derivatives (pp. 83-102) Date: Chapter Goals: Know and be able to apply the formulas for derivatives. Understand the chain rule and be able to apply it. Know how to compute

### Analytic Geometry and Calculus I Exam 1 Practice Problems Solutions 2/19/7

Analytic Geometry and Calculus I Exam 1 Practice Problems Solutions /19/7 Question 1 Write the following as an integer: log 4 (9)+log (5) We have: log 4 (9)+log (5) = ( log 4 (9)) ( log (5)) = 5 ( log

### A simple method for solving the diophantine equation Y 2 = X 4 + ax 3 + bx 2 + cx + d

Elem. Math. 54 (1999) 32 36 0013-6018/99/010032-5 \$ 1.50+0.20/0 c Birkhäuser Verlag, Basel, 1999 Elemente der Mathematik A simple method for solving the diophantine equation Y 2 = X 4 + ax 3 + bx 2 + cx

### Section 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

### University of Toronto Mississauga

Surname: First Name: Student Number: Tutorial: University of Toronto Mississauga Mathematical and Computational Sciences MAT33Y5Y Term Test 2 Duration - 0 minutes No Aids Permitted This exam contains pages

### Continuity, Intermediate Value Theorem (2.4)

Continuity, Intermediate Value Theorem (2.4) Xiannan Li Kansas State University January 29th, 2017 Intuitive Definition: A function f(x) is continuous at a if you can draw the graph of y = f(x) without

### MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. C) ±

Final Review for Pre Calculus 009 Semester Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the equation algebraically. ) v + 5 = 7 - v

### COMPOSITION OF FUNCTIONS d INVERSES OF FUNCTIONS AND

NAME UNITS ALGEBRA II COMPOSITION OF FUNCTIONS d INVERSES OF FUNCTIONS AND LOGARITHMIC d EXPONENTIALFUNCTIONS Composition of Functions Worksheet Name -3O I. Let f (x) = 2x -, g(x) = 3x, and h(x) = x2 +.

### Exponential functions are defined and for all real numbers.

3.1 Exponential and Logistic Functions Objective SWBAT evaluate exponential expression and identify and graph exponential and logistic functions. Exponential Function Let a and b be real number constants..

### Lecture : The Chain Rule MTH 124

At this point you may wonder if these derivative rules are some sort of Sisyphean task. Fear not, by the end of the next few lectures we will complete all the differentiation rules needed for this course.

### Proving Things. 1. Suppose that all ravens are black. Which of the following statements are then true?

Proving Things 1 Logic 1. Suppose that all ravens are black. Which of the following statements are then true? (a) If X is a raven, then X is black. (b) If X is black, then X is a raven. (c) If X is not

### DIFFERENTIATION RULES

3 DIFFERENTIATION RULES DIFFERENTIATION RULES 3. The Product and Quotient Rules In this section, we will learn about: Formulas that enable us to differentiate new functions formed from old functions by

### North Carolina State University

North Carolina State University MA 141 Course Text Calculus I by Brenda Burns-Williams and Elizabeth Dempster August 7, 2014 Section1 Functions Introduction In this section, we will define the mathematical

### Investigating Limits in MATLAB

MTH229 Investigating Limits in MATLAB Project 5 Exercises NAME: SECTION: INSTRUCTOR: Exercise 1: Use the graphical approach to find the following right limit of f(x) = x x, x > 0 lim x 0 + xx What is the

### Module 6 Lecture Notes

Module 6 Lecture Notes Contents 6. An Introduction to Logarithms....................... 6. Evaluating Logarithmic Expressions.................... 4 6.3 Graphs of Logarithmic Functions......................

### Hyperreal Calculus MAT2000 Project in Mathematics. Arne Tobias Malkenes Ødegaard Supervisor: Nikolai Bjørnestøl Hansen

Hyperreal Calculus MAT2000 Project in Mathematics Arne Tobias Malkenes Ødegaard Supervisor: Nikolai Bjørnestøl Hansen Abstract This project deals with doing calculus not by using epsilons and deltas, but

### Math 421, Homework #9 Solutions

Math 41, Homework #9 Solutions (1) (a) A set E R n is said to be path connected if for any pair of points x E and y E there exists a continuous function γ : [0, 1] R n satisfying γ(0) = x, γ(1) = y, and

### Proportional Relationships (situations)

Proportional Relationships (situations) Recall: A proportion is an equality between two ratios or two rates. If the ratio of a to b is equal to the ratio of c to d, then... The following situations are

### Notes. Functions. Introduction. Notes. Notes. Definition Function. Definition. Slides by Christopher M. Bourke Instructor: Berthe Y.

Functions Slides by Christopher M. Bourke Instructor: Berthe Y. Choueiry Fall 2007 Computer Science & Engineering 235 Introduction to Discrete Mathematics Section 2.3 of Rosen cse235@cse.unl.edu Introduction

### Unit #1 - Transformation of Functions, Exponentials and Logarithms

Unit #1 - Transformation of Functions, Exponentials and Logarithms Some problems and solutions selected or adapted from Hughes-Hallett Calculus. Note: This unit, being review of pre-calculus has substantially

### ACCRS/QUALITY CORE CORRELATION DOCUMENT: ALGEBRA I

ACCRS/QUALITY CORE CORRELATION DOCUMENT: ALGEBRA I Revised March 25, 2013 Extend the properties of exponents to rational exponents. 1. [N-RN1] Explain how the definition of the meaning of rational exponents

### How much can they save? Try \$1100 in groceries for only \$40.

It s Not New, It s Recycled Composition of Functions.4 LEARNING GOALS KEY TERM In this lesson, you will: Perform the composition of two functions graphically and algebraically. Use composition of functions

### A function relate an input to output

Functions: Definition A function relate an input to output In mathematics, a function is a relation between a set of outputs and a set of output with the property that each input is related to exactly

### AP Calculus Free-Response Questions 1969-present AB

AP Calculus Free-Response Questions 1969-present AB 1969 1. Consider the following functions defined for all x: f 1 (x) = x, f (x) = xcos x, f 3 (x) = 3e x, f 4 (x) = x - x. Answer the following questions

### Polynomial functions right- and left-hand behavior (end behavior):

Lesson 2.2 Polynomial Functions For each function: a.) Graph the function on your calculator Find an appropriate window. Draw a sketch of the graph on your paper and indicate your window. b.) Identify

### De Morgan Systems on the Unit Interval

De Morgan Systems on the Unit Interval Mai Gehrke y, Carol Walker, and Elbert Walker Department of Mathematical Sciences New Mexico State University Las Cruces, NM 88003 mgehrke, hardy, elbert@nmsu.edu

GROUP ACTIONS KEITH CONRAD 1. Introduction The symmetric groups S n, alternating groups A n, and (for n 3) dihedral groups D n behave, by their very definition, as permutations on certain sets. The groups

### Functions. Chapter Continuous Functions

Chapter 3 Functions 3.1 Continuous Functions A function f is determined by the domain of f: dom(f) R, the set on which f is defined, and the rule specifying the value f(x) of f at each x dom(f). If f is

### physicsandmathstutor.com Paper Reference Core Mathematics C3 Advanced Level Monday 23 January 2006 Afternoon Time: 1 hour 30 minutes

Centre No. Candidate No. Paper Reference(s) 6665/01 Edexcel GCE Core Mathematics C3 Advanced Level Monday 23 January 2006 Afternoon Time: 1 hour 30 minutes Materials required for examination Mathematical

ADDITIONAL MATHEMATICS GCE NORMAL ACADEMIC LEVEL (016) (Syllabus 4044) CONTENTS Page INTRODUCTION AIMS ASSESSMENT OBJECTIVES SCHEME OF ASSESSMENT 3 USE OF CALCULATORS 3 SUBJECT CONTENT 4 MATHEMATICAL FORMULAE

### Advanced Calculus I Chapter 2 & 3 Homework Solutions October 30, Prove that f has a limit at 2 and x + 2 find it. f(x) = 2x2 + 3x 2 x + 2

Advanced Calculus I Chapter 2 & 3 Homework Solutions October 30, 2009 2. Define f : ( 2, 0) R by f(x) = 2x2 + 3x 2. Prove that f has a limit at 2 and x + 2 find it. Note that when x 2 we have f(x) = 2x2

### 4.1 Real-valued functions of a real variable

Chapter 4 Functions When introducing relations from a set A to a set B we drew an analogy with co-ordinates in the x-y plane. Instead of coming from R, the first component of an ordered pair comes from

### Elementay Math Models EMM Worksheet: Logistic Growth

Elementay Math Models EMM Worksheet: Logistic Growth This worksheet discusses 5 sample models that will give you some practice working with the ideas of logistic growth. The basic framework for each model

### Explain the mathematical processes of the function, and then reverse the process to explain the inverse.

Lesson 8: Inverse Functions Outline Inverse Function Objectives: I can determine whether a function is one-to-one when represented numerically, graphically, or algebraically. I can determine the inverse

### MATHEMATICS HIGHER SECONDARY FIRST YEAR VOLUME II REVISED BASED ON THE RECOMMENDATIONS OF THE TEXT BOOK DEVELOPMENT COMMITTEE

MATHEMATICS HIGHER SECONDARY FIRST YEAR VOLUME II REVISED BASED ON THE RECOMMENDATIONS OF THE TEXT BOOK DEVELOPMENT COMMITTEE Untouchability is a sin Untouchability is a crime Untouchability is inhuman

### Inverse Functions. Say Thanks to the Authors Click (No sign in required)

Inverse Functions Say Thanks to the Authors Click http://www.ck12.org/saythanks (No sign in required) To access a customizable version of this book, as well as other interactive content, visit www.ck12.org

### 1. Algebra and Functions

1. Algebra and Functions 1.1.1 Equations and Inequalities 1.1.2 The Quadratic Formula 1.1.3 Exponentials and Logarithms 1.2 Introduction to Functions 1.3 Domain and Range 1.4.1 Graphing Functions 1.4.2

11.2 The Quadratic Formula Solving Quadratic Equations Using the Quadratic Formula. By solving the general quadratic equation ax 2 + bx + c = 0 using the method of completing the square, one can derive

### Math 12 Final Exam Review 1

Math 12 Final Exam Review 1 Part One Calculators are NOT PERMITTED for this part of the exam. 1. a) The sine of angle θ is 1 What are the 2 possible values of θ in the domain 0 θ 2π? 2 b) Draw these angles

### Solutionbank Edexcel AS and A Level Modular Mathematics

Page of Exercise A, Question The curve C, with equation y = x ln x, x > 0, has a stationary point P. Find, in terms of e, the coordinates of P. (7) y = x ln x, x > 0 Differentiate as a product: = x + x

### Function Junction: Homework Examples from ACE

Function Junction: Homework Examples from ACE Investigation 1: The Families of Functions, ACE #5, #10 Investigation 2: Arithmetic and Geometric Sequences, ACE #4, #17 Investigation 3: Transforming Graphs,

### Chapter Product Rule and Quotient Rule for Derivatives

Chapter 3.3 - Product Rule and Quotient Rule for Derivatives Theorem 3.6: The Product Rule If f(x) and g(x) are differentiable at any x then Example: The Product Rule. Find the derivatives: Example: The

### 1. 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.

### Functions - Algebra of Functions

10.2 Functions - Algebra of Functions Several functions can work together in one larger function. There are 5 common operations that can be performed on functions. The four basic operations on functions

### College 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

### Characteristics of Polynomials and their Graphs

Odd Degree Even Unit 5 Higher Order Polynomials Name: Polynomial Vocabulary: Polynomial Characteristics of Polynomials and their Graphs of the polynomial - highest power, determines the total number of

### Solution: It could be discontinuous, or have a vertical tangent like y = x 1/3, or have a corner like y = x.

1. Name three different reasons that a function can fail to be differentiable at a point. Give an example for each reason, and explain why your examples are valid. It could be discontinuous, or have a

### Common Core Algebra II. MRS21 Course Overview (Tentative)

Common Core Algebra II MRS21 Course Overview (Tentative) Unit #1 Total: 6 days Algebraic Expressions and Operations on Polynomials Lesson #1: Classifying Polynomials and Evaluating Expressions Lesson #2:

### Georgia Department of Education Common Core Georgia Performance Standards Framework CCGPS Advanced Algebra Unit 2

Polynomials Patterns Task 1. To get an idea of what polynomial functions look like, we can graph the first through fifth degree polynomials with leading coefficients of 1. For each polynomial function,

### Chapter 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)?

### AP Calculus Summer Homework

Class: Date: AP Calculus Summer Homework Show your work. Place a circle around your final answer. 1. Use the properties of logarithms to find the exact value of the expression. Do not use a calculator.

### Chapter 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

### dt 2 The Order of a differential equation is the order of the highest derivative that occurs in the equation. Example The differential equation

Lecture 18 : Direction Fields and Euler s Method A Differential Equation is an equation relating an unknown function and one or more of its derivatives. Examples Population growth : dp dp = kp, or = kp

### Bayesian D-optimal Design

Bayesian D-optimal Design Susanne Zaglauer, Michael Deflorian Abstract D-optimal and model based experimental designs are often criticised because of their dependency to the statistical model and the lac

### College Algebra Notes

Metropolitan Community College Contents Introduction 2 Unit 1 3 Rational Expressions........................................... 3 Quadratic Equations........................................... 9 Polynomial,

### Tommo So Vowel Harmony: The Math

Tommo So Vowel Harmony: The Math Laura McPherson and Bruce Hayes January 205. Goal of this document I. BACKGROUND AND STARTING POINT Explain and justify all of the math in our article. Intended audience:

### Written Homework # 1 Solution

Math 516 Fall 2006 Radford Written Homework # 1 Solution 10/11/06 Remark: Most of the proofs on this problem set are just a few steps from definitions and were intended to be a good warm up for the course.

### CS100: DISCRETE STRUCTURES

1 CS100: DISCRETE STRUCTURES Computer Science Department Lecture 2: Functions, Sequences, and Sums Ch2.3, Ch2.4 2.3 Function introduction : 2 v Function: task, subroutine, procedure, method, mapping, v

### Composition of Functions

Math 120 Intermediate Algebra Sec 9.1: Composite and Inverse Functions Composition of Functions The composite function f g, the composition of f and g, is defined as (f g)(x) = f(g(x)). Recall that a function

### Find the slope of the curve at the given point P and an equation of the tangent line at P. 1) y = x2 + 11x - 15, P(1, -3)

Final Exam Review AP Calculus AB Find the slope of the curve at the given point P and an equation of the tangent line at P. 1) y = x2 + 11x - 15, P(1, -3) Use the graph to evaluate the limit. 2) lim x

### ON FUNCTIONS WHOSE GRAPH IS A HAMEL BASIS II

To the memory of my Mother ON FUNCTIONS WHOSE GRAPH IS A HAMEL BASIS II KRZYSZTOF P LOTKA Abstract. We say that a function h: R R is a Hamel function (h HF) if h, considered as a subset of R 2, is a Hamel

### Math 16A, Summer 2009 Exam #2 Name: Solutions. Problem Total Score / 120. (x 2 2x + 1) + (e x + x)(2x 2)

Math 16A, Summer 2009 Exam #2 Name: Solutions Each Problem is worth 10 points. You must show work to get credit. Problem 1 2 3 4 5 6 7 8 9 10 11 12 Total Score / 120 Problem 1. Compute the derivatives

### Again it does not matter if the number is negative or positive. On a number line, 6 is 6 units to the right of 0. So the absolute value is 6.

Name Working with Absolute Value - Step-by-Step Lesson Lesson 1 Absolute Value Problem: 1. Find the absolute value. 6 = Explanation: Absolute values focus on the size or magnitude of a number. They do

### ALGEBRA I CCR MATH STANDARDS

RELATIONSHIPS BETWEEN QUANTITIES AND REASONING WITH EQUATIONS M.A1HS.1 M.A1HS.2 M.A1HS.3 M.A1HS.4 M.A1HS.5 M.A1HS.6 M.A1HS.7 M.A1HS.8 M.A1HS.9 M.A1HS.10 Reason quantitatively and use units to solve problems.

ADDITIONAL MATHEMATICS GCE Ordinary Level (06) (Syllabus 4047) CONTENTS Page INTRODUCTION AIMS ASSESSMENT OBJECTIVES SCHEME OF ASSESSMENT 3 USE OF CALCULATORS 3 SUBJECT CONTENT 4 MATHEMATICAL FORMULAE

### Homework #5 Solutions

Homework #5 Solutions p 83, #16. In order to find a chain a 1 a 2 a n of subgroups of Z 240 with n as large as possible, we start at the top with a n = 1 so that a n = Z 240. In general, given a i we will

### The Plane of Complex Numbers

The Plane of Complex Numbers In this chapter we ll introduce the complex numbers as a plane of numbers. Each complex number will be identified by a number on a real axis and a number on an imaginary axis.

### Tennessee s State Mathematics Standards - Algebra I

Domain Cluster Standards Scope and Clarifications Number and Quantity Quantities The Real (N Q) Number System (N-RN) Use properties of rational and irrational numbers Reason quantitatively and use units

### Matrix Multiplication

228 hapter Three Maps etween Spaces IV2 Matrix Multiplication After representing addition and scalar multiplication of linear maps in the prior subsection, the natural next operation to consider is function

### WHAT IS AN SMT SOLVER? Jaeheon Yi - April 17, 2008

WHAT IS AN SMT SOLVER? Jaeheon Yi - April 17, 2008 WHAT I LL TALK ABOUT Propositional Logic Terminology, Satisfiability, Decision Procedure First-Order Logic Terminology, Background Theories Satisfiability