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Chapter 3 Equations and Inequalities in Two Variables; Functions Copyright © 2015, 2011, 2007 Pearson Education, Inc. 1 CHAPTER 3 Equations and Inequalities in Two Variables; Functions 3.2 The Slope of a Line 3.3 The Equation of a Line 3.5 Introduction to Functions and Function Notation Copyright © 2015, 2011, 2007 Pearson Education, Inc. 2 3.5 Introduction to Functions and Function Notation 1. Identify the domain and range of a relation and determine whether a relation is a function. 2. Find the value of a function. Copyright © 2015, 2011, 2007 Pearson Education, Inc. 3 Relation: A set of ordered pairs. Domain: The set containing initial values of a relation; its input values; the first coordinates in ordered pairs. Range: The set containing all values that are paired to domain values in a relation; its output values; the second coordinates in ordered pairs. Copyright © 2015, 2011, 2007 Pearson Education, Inc. 4 Function: A relation in which each value in the domain is assigned to exactly one value in the range. Domain 0 1 2 3 4 Range 2 4 6 8 10 Each element in the domain has a single arrow pointing to an element in the range. Copyright © 2015, 2011, 2007 Pearson Education, Inc. 5 Every function is a relation, but not every relation is a function. If any value in the domain is assigned to more than one value in the range, then the relation is not a function. Domain Range 0 2 1 4 2 6 10 12 not a function Copyright © 2015, 2011, 2007 Pearson Education, Inc. 6 Example Identify the domain and range of the relation, then determine if it is a function. Birthdate Family member March 1 Donna April 17 Dennis Sept. 3 Catherine October 9 Denise Nancy Domain: {March April 17,because Sept 3, an Oct 9} The relation is not 1, a function element the domain, Sept. 3, is assigned toNancy} Range: in {Donna, Dennis, Catherine, Denise, two names in the range. Copyright © 2015, 2011, 2007 Pearson Education, Inc. 7 Determining the Domain and Range of a Graph The domain is a set containing the first coordinate (x-coordinate) of every point on the graph. The range is a set containing the second coordinate (y-coordinate) of every point on the graph. Copyright © 2015, 2011, 2007 Pearson Education, Inc. 8 Vertical Line Test To determine whether a graphical relation is a function, draw or imagine vertical lines through each value in the domain. If each vertical line intersects the graph at only one point, the relation is a function. If any vertical line intersects the graph more than once, the relation is not a function. Copyright © 2015, 2011, 2007 Pearson Education, Inc. 9 Example For each graph, identify the domain and range. Then state whether each relation is a function. a. b. Domain: {x|x 1} Not a function Range: all real numbers Domain: all real numbers Function Range: {y 1} Copyright © 2015, 2011, 2007 Pearson Education, Inc. 10 When written as an equation, the notation for a function is a modification of an equation in two variables. y = 3x + 4 could be written as f(x) = 3x + 4 f(x) is read as “a function in terms of x” or “f of x” Copyright © 2015, 2011, 2007 Pearson Education, Inc. 11 Finding the Value of a Function Given a function f(x), to find f(a), where a is a real number in the domain of f, replace x in the function with a and then evaluate or simplify. Copyright © 2015, 2011, 2007 Pearson Education, Inc. 12 Example For the function f(x) = 3x – 5, find the following. a. f(2) b. f(4) c. f(a) Solution b. f(4) = 3x – 5 c. f(a) = 3x – 5 a. f(2) = 3x – 5 = 3(a) – 5 = 3(4) – 5 = 3(2) – 5 =6–5 = 3a – 5 = 12 – 5 =1 = 17 Copyright © 2015, 2011, 2007 Pearson Education, Inc. 13 Example Use the graph to find the indicated value of the function. a. f(0) b. f(2) c. f(8) 10 Y 8 Solution a. When x = 0, y = 0, so f(0) = 0. 6 4 2 -10 -8 -6 b. When x = 2, y = 2, so f(2) = 2. -4 -2 0 -2 X 2 4 6 8 10 -4 -6 c. When x = 8, y = 4, so f(8) = 4. Copyright © 2015, 2011, 2007 Pearson Education, Inc. -8 -10 14