Copyright © 2015, 2011, 2007 Pearson Education, Inc. 1 1 Chapter 3 Equations and Inequalities in Two Variables; Functions

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 2 Equations and Inequalities in Two Variables; Functions 3.2The Slope of a Line 3.3The Equation of a Line 3.5Introduction to Functions and Function Notation CHAPTER 3

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 3 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. 3.5

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 4 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. 5 Function: A relation in which each value in the domain is assigned to exactly one value in the range. Domain Range Each element in the domain has a single arrow pointing to an element in the range.

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 6 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 not a function

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 7 Example Identify the domain and range of the relation, then determine if it is a function. BirthdateFamily member March 1Donna April 17Dennis Sept. 3Catherine October 9Denise Nancy The relation is not a function because an element in the domain, Sept. 3, is assigned to two names in the range. Domain: {March 1, April 17, Sept 3, Oct 9} Range: {Donna, Dennis, Catherine, Denise, Nancy}

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 8 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. 9 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. 10 Example For each graph, identify the domain and range. Then state whether each relation is a function. a.b. Domain: {x|x  1} Range: all real numbers Not a function Domain: all real numbers Range: {y   1} Function

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 11 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. 12 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. 13 Example For the function f(x) = 3x – 5, find the following. a. f(2)b. f(  4)c. f(a) Solution a. f(2) = 3x – 5 = 3(2) – 5 = 6 – 5 = 1 b. f(  4) = 3x – 5 = 3(  4) – 5 =  12 – 5 =  17 c. f(a) = 3x – 5 = 3(a) – 5 = 3a – 5

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 14 Example Use the graph to find the indicated value of the function. a. f(0)b. f(2)c. f(8) Solution a. When x = 0, y = 0, so f(0) = 0. b. When x = 2, y = 2, so f(2) = 2. c. When x = 8, y = 4, so f(8) = 4.

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 15 We create the graph of a function the same way that we create the graph of an equation in two variables. Slope-intercept form: y = mx + b Linear function: f(x) = mx + b

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 16 Example Consider f(x) = 2x + 1. Give the domain and range. Solution We could make a table of values or use the fact that the slope is 2 and the y-intercept is 1. Domain: Range: xf(x)f(x) f(x) = 2x + 1