Copyright © 2014, 2010, 2006 Pearson Education, Inc. 1 Chapter 1 Introduction to Functions and Graphs Book cover will go here.

Slides:



Advertisements
Similar presentations
Functions and Their Representations
Advertisements

MAT 105 SP09 Functions and Graphs
Chapter 2: Functions and Graphs
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Chapter 10 Graphing Equations and Inequalities.
2.3) Functions, Rules, Tables and Graphs
Function: Definition A function is a correspondence from a first set, called the domain, to a second set, called the range, such that each element in the.
Introduction to Functions
Learning Objectives for Section 2.1 Functions
Function A function is a relation in which, for each distinct value of the first component of the ordered pair, there is exactly one value of the second.
Chapter 3 Section 6 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley.
Chapter 1 Functions and Graphs Copyright © 2014, 2010, 2007 Pearson Education, Inc Basics of Functions and Their Graphs.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Relations and Functions Another Foundational Concept Copyright © 2014 – Curt Hill.
SECTION 1.1 FUNCTIONS FUNCTIONS. DEFINITION OF A RELATION A relation is a correspondence between two sets. If x and y are two elements in these sets and.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall. 1.2 Basics of Functions and Their Graphs.
Introduction to Functions
Slide 1-1 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley.
1. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Graphing Linear Equations and Inequalities CHAPTER 4.1The Rectangular.
TH EDITION LIAL HORNSBY SCHNEIDER COLLEGE ALGEBRA.
Copyright © 2013, 2009, 2005 Pearson Education, Inc. 1 2 Graphs and Functions Copyright © 2013, 2009, 2005 Pearson Education, Inc.
Chapter 1 A Beginning Library of Elementary Functions
Chapter 2.2 Functions. Relations and Functions Recall from Section 2.1 how we described one quantity in terms of another. The letter grade you receive.
Copyright © 2011 Pearson Education, Inc. Functions Section 2.1 Functions and Graphs.
C ollege A lgebra Functions and Graphs (Chapter1) L:8 1 Instructor: Eng. Ahmed abo absa University of Palestine IT-College.
Copyright © 2007 Pearson Education, Inc. Slide 1-1.
Section 2.1 Functions. 1. Relations A relation is any set of ordered pairs Definition DOMAINRANGE independent variable dependent variable.
Functions: Definitions and Notation 1.3 – 1.4 P (text) Pages (pdf)
Formalizing Relations and Functions
1.3 and 1.4 January 6, p #2-12 even, even 2) rational, real 4) natural, integer, rational, real 6) Rational, real 8) integer, rational,
Homework Questions? Welcome back to Precalculus. Review from Section 1.1 Summary of Equations of Lines.
Copyright © 2010 Pearson Education, Inc. All rights reserved Sec
Functions Section 1.4. Relation The value of one variable is related to the value of a second variable A correspondence between two sets If x and y are.
Relations Relation: a set of ordered pairs Domain: the set of x-coordinates, independent Range: the set of y-coordinates, dependent When writing the domain.
Copyright © 2013, 2009, 2005 Pearson Education, Inc. 1 4 Inverse, Exponential, and Logarithmic Functions Copyright © 2013, 2009, 2005 Pearson Education,
Functions and Their Representations
HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Section 9.5.
Chapter 2 Linear Functions and Models. Ch 2.1 Functions and Their Representations A function is a set of ordered pairs (x, y), where each x-value corresponds.
Sections 7.1, 7.2 Sections 7.1, 7.2 Functions and Domain.
 Analyze and graph relations.  Find functional values. 1) ordered pair 2) Cartesian Coordinate 3) plane 4) quadrant 5) relation 6) domain 7) range 8)
Section 1.2 Functions and Graphs. Relation A relation is a correspondence between the first set, called the domain, and a second set, called the range,
1 Copyright © Cengage Learning. All rights reserved. 3 Functions and Graphs 3.4 Definition of function.
Copyright © 2015, 2011, 2007 Pearson Education, Inc. 1 1 Chapter 3 Equations and Inequalities in Two Variables; Functions.
Copyright © 2014, 2010, and 2006 Pearson Education, Inc. 1 Chapter 7 Functions and Graphs.
I CAN DETERMINE WHETHER A RELATION IS A FUNCTION AND I CAN FIND DOMAIN AND RANGE AND USE FUNCTION NOTATION. 4.6 Formalizing Relations and Functions.
Copyright © Cengage Learning. All rights reserved. Fundamentals.
Chapter 2 Functions and Linear Equations. Functions vs. Relations A "relation" is just a relationship between sets of information. A “function” is a well-behaved.
Copyright © Cengage Learning. All rights reserved. Graphs; Equations of Lines; Functions; Variation 3.
Review Chapter 1 Functions and Their Graphs. Lines in the Plane Section 1-1.
Algebra 2 Foundations, pg 64  Students will be able to graph relations and identify functions. Focus Question What are relations and when is a relation.
Section 7.6 Functions Math in Our World. Learning Objectives  Identify functions.  Write functions in function notation.  Evaluate functions.  Find.
Topic 4 Functions Graphs’ key features: Domain and Range Intercepts
Introduction Functions have many characteristics, such as domain, range, asymptotes, zeros, and intercepts. These functions can be compared even when given.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
1.2 Functions and Graphs Determine whether a correspondence or a relation is a function. Find function values, or outputs, using a formula or a graph.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Functions Introduction.
Splash Screen.
Graphs, Linear Equations, and Functions
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Functions and Their Representations
UNDERSTANDING FUNCTIONS
3 Chapter Chapter 2 Graphing.
Presentation transcript:

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 1 Chapter 1 Introduction to Functions and Graphs Book cover will go here

2 Copyright © 2014, 2010, 2006 Pearson Education, Inc. Functions and Their Representations ♦ Learn function notation ♦ Represent a function four different ways ♦ Define a function formally ♦ Identify the domain and range of a function ♦ Use calculators to represent functions (optional) ♦ Identify functions ♦ Represent functions with diagrams and equations 1.3

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 3 Basic Concepts The following table lists the approximate distance y in miles between a person and a bolt of lightning when there is a time lapse of x seconds between seeing the lightning and hearing the thunder. The value of y can be found by dividing the corresponding value of x by 5. x (seconds) y (miles) 12345

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 4 Basic Concepts This table establishes a special type of relationship between two sets of numbers, where each valid input x in seconds determines exactly one output y in miles. The table represents or defines a function f, where function f computes the distance between an observer and a lightning bolt. x (seconds) y (miles) 12345

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 5 Basic Concepts The distance y depends on the time x, and so y is called the dependent variable and x is called the independent variable. The notation y = f(x) is used to emphasize that f is a function (not multiplication). It is read “y equals f of x” and denotes that function f with input x produces output y. That is, f (Input) = Output

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 6 Function Notation The notation y = f (x) is called function notation. The input is x, the output is y, and the name of the function is f. Output Name y = f (x) Input

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 7 Function Notation The variable y is called the dependent variable, and the variable x is called the independent variable. The expression f(20) = 4 is read “f of 20 equals 4” and indicates that f outputs 4 when the input is 20. A function computes exactly one output for each valid input. The letters f, g, and h are often used to denote names of functions.

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 8 Domain and Range of a Function The set of all meaningful inputs x is called the DOMAIN of the function. The set of corresponding outputs y is called the RANGE of the function. A function f that computes the height after t seconds of a ball thrown into the air, has a domain that might include all the times while the ball is in flight, and the range would include all heights attained by the ball.

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 9 Representation of Functions Functions can be represented by Verbal descriptions Tables Symbols Graphs

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 10 Verbal Representation (Words) In the lightning example, “Divide x seconds by 5 to obtain y miles.” OR “f calculates the number of miles from a lightning bolt when the delay between thunder and lightning is x seconds.”

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 11 Numerical Representation (Table of Values) Here is a table of the lightning example using different input-output pairs (the same relationship still exists): x (seconds) y (miles) Since it is inconvenient or impossible to list all possible inputs x, we refer to this type of table as a partial numerical representation.

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 12 Symbolic Representation (Formula) In the lightning example, Similarly, if a function g computes the square of a number x, then

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 13 Graphical Representation (Graph) A graph visually pairs and x-input with a y-output. Using the lightning data:

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 14 Graphical Representation (Graph) The scatterplot suggests a line for the graph of f.

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 15 Formal Definition of a Function A function is a relation in which each element of the domain corresponds to exactly one element in the range. The ordered pairs for a function can be either finite or infinite.

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 16 Let a function f be represented symbolically by (a) Evaluate f(2), f(1), and f(a + 1) (b) Find the domain of f. Solution (a) Example: Evaluating a function and determining its domain

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 17 (b)The expression for f is not defined when the denominator x – 1 = 0, that is, when x = 1. So the domain of f is all real numbers except for 1. Example: Evaluating a function and determining its domain

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 18 Set-Builder Notation The expression {x | x ≠ 1} is written in set- builder notation and represents the set of all real numbers x such that x does not equal 1. Another example is {y | 1 < y < 5}, which represents the set of all real numbers y such that y is greater than 1 and less than 5.

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 19 A function g is given by g(x) = x 2 – 2x, and its graph is shown. (a)Find the domain and range of g. (b)Use g(x) to evaluate g(–1). (c)Use the graph of g to evaluate g(–1). Example: Evaluating a function symbolically and graphically

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 20 (a)The domain for g(x) = x 2 – 2x, is all real numbers. (b) g(–1) = (–1) 2 – 2(–1) = = 3 (c) Find x = –1 on the x-axis. Move upward to the graph of g. Move across (to the right) to the y-axis. Read the y-value: g(–1) = 3. Example: Evaluating a function symbolically and graphically

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 21 A graph of (a) Evaluate f(1) (b) Find the domain and range of f. is shown. Example: Find the domain and range graphically

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 22 (a) Start by finding 1 on the x-axis. Move up and down on the grid. Note that we do not intersect the graph of f. Thus f(1) is undefined. (b) Arrow indicates x and y increase without reaching a maximum. Domain is in green: D = {x | x ≥ 2} Range is in red: R = {y | y ≥ 0} Example: Find the domain and range graphically

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 23 Identifying Functions: Vertical Line Test If every vertical line intersects a graph at no more than one point, then the graph represents a function. Note: If a vertical line intersects a graph more than once, then the graph does not represents a function.

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 24 Use the vertical line test to determine if the graph represents a function. (b) Solution (a) (a) Yes(b) No Example: Identifying a function graphically

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 25 Functions Represented by Diagrams and Equations There are two other ways that we can represent, or define, a function: Diagram Equation

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 26 Diagrammatic Representation (Diagram) Function Sometimes referred to as mapping; 1 is the image of 5; 5 is the preimage of 1.

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 27 Diagrammatic Representation (Diagram) Not a function

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 28 Functions Defined by Equations The equation x + y = 1 defines the function f given by f(x) = 1 – x where y = f(x). Notice that for each input x, there is exactly one y output determined by y = 1 – x.

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 29 Determine if y is a function of x. a) x = y 2 (b) y = x 2 – 2 Solution (a) If we let x = 4, then y could be either 2 or –2. So, y is not a function of x. The graph shows it fails the vertical line test. Example: Identifying a function

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 30 (b) y = x 2 – 2 Each x-value determines exactly one y-value, so y is a function of x. The graph shows it passes the vertical line test. Example: Identifying a function