Chapter 1: Linear Functions, Equations, and Inequalities 1.1 Real Numbers and the Rectangular Coordinate System 1.2 Introduction to Relations and Functions 1.3 Linear Functions 1.4 Equations of Lines and Linear Models 1.5 Linear Equations and Inequalities 1.6 Applications of Linear Functions 1.1 Real Numbers and the Coordinate System 1.2 Introduction to Relations and Functions 1.3 Linear Functions 1.4 Equations of Lines and Inequalities 1.5 Linear Equations and Inequalities 1.6 Applications of Linear Functions

1.2 Introduction to Relations and Functions Two Types of Notation: 1. Set Builder Notation {x | x > –2} is read “The set of all x such that x is greater than –2” Interval Notation (–2,) represents the set of all numbers greater than –2 Note that a left parenthesis “(“ indicates that –2 is not included. A parenthesis is always next to the infinity symbol .

1.2 Interval Notation Example of Set-Builder Corresponding Corresponding Type of Interval Notation Interval Notation Graph

1.2 Relation, Domain, and Range A relation is a set of ordered pairs. If we denote the ordered pairs of a relation by (x,y), the set of all x-values is called the domain, and the set of all y-values is called the range.

1.2 Example of a Relation Let F be a relation where F = {(1, 2),(–2, 5),(3, –1 )}. The Domain = {1, –2, 3} and the Range = {2, 5, – 1}. The graph of F looks like the following:

1.2 Graph of a Relation A graph of a line or curve in the xy-plane represents a relation. Let F represent a relation consisting of all ordered pairs having the form (x,2x), where x is a real number. Example: (-2,-4),(-1,-2),(0,0),(1,2),(2,4) (-2,-4)

1.2 Diagram of a Relation Relation F can be illustrated with a diagram. An arrow from 1 to 2 indicates that the ordered pair (1,2) belongs to F. F -2 5 1 2 3 -1

1.2 Domain and Range from a Graph

1.2 Definition of a Function A function is a relation in which each element in the domain corresponds to exactly one element in the range. If x represents any element in the domain, then x is called the independent variable. If y represents any element in the range, then y is called the dependent variable.

1.2 Definition of a Function Examples Indicate whether the following relations are functions. {(1,1),(1,2),(1,3),(2,4)} 2. Yes, since each element in the domain corresponds to exactly one element in the range. x -4 -3 -2 -1 1 y 2

1.2 Vertical Line Test for Functions If every vertical line intersects a graph in no more than one point, then the graph is the graph of a function. This is the graph of a function. This is not the graph of a function.

1.2 Definition of a Function Function (Alternative Definition) A function is a correspondence in which each element x from a set called the domain is paired with one and only one element y from a set called the range.