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Chapter 1: Linear Functions, Equations, and Inequalities

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

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Chapter 1: Linear Functions, Equations, and Inequalities

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  1. 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

  2. 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 “(“ indicates that –2 is not included and a parenthesis is always next to the infinity symbol .

  3. 1.2 Interval Notation Example of Set-Builder Corresponding Corresponding Type of Interval Notation Interval Notation Graph

  4. 1.2 Definitions: Relation, Domain, and Range • A set of ordered pairs is called a relation. • 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.

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

  6. 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 -1 3

  7. 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)

  8. Range Range Domain Domain 1.2 Domain and Range from a Graph

  9. 1.2 Definition of a Function • 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. • 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.

  10. 1.2 Test for Functions Vertical Line Test – If every vertical line intersects a graph in no more than one point, the graph is the graph of a function. This graph is of a function. This graph is not of a function.

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