Functions: Ordered Pairs, Relations, and Domain-Range, Slides of Algebra

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§2.1 Intro to

Functions

Review §

 Any QUESTIONS About

• §1.6 → Exponent Rules & Properties

 Any QUESTIONS About HomeWork

• §1.6 → HW-

1.6 MTH 55

Ordered Pair Dependency

• Frequently, the numerical values of the variable y can be determined by assigning appropriate values to the variable x. For this reason, y is sometimes referred to as the dependent variable and x as the independent variable. - i.e., if we KNOW x , we can CALCULATE y

Mathematical RELATION

• Any set of ordered pairs is called a

relation. The set of all first components

is called the domain of the relation, and

the set of all SECOND components is

called the RANGE of the relation

Example  Domain & Range

• Find the Domain and Range for the relation: - { (Titanic, $600.8), (Star Wars IV, $461.0), (Shrek 2, $441.2), (E.T., $435.1), (Star Wars I, $431.1), (Spider-Man, $403.7)}
• SOLUTION
  • The RANGE is the set of all second components, or {$600.8, $461.0, $441.2, $435.1, $431.1, $403.7)}.

FUNCTION Defined

• A function which “takes” a set X to a

set Y is a relation in which each element of X corresponds to ONE, and ONLY ONE, element of Y.

Example  Is Relation a Fcn?

• Determine whether the relations that follow are functions. The domain of each relation is the family consisting of Malcolm (father), Maria (mother), Ellen (daughter), and Duane (son).

1. For the relation defined by the following diagram, the range consists of the ages of the four family members, and each family member corresponds to that family member’s age.

Example  Is Relation a Fcn?

Example  Is Relation a Fcn?

2. For the relation defined by the diagram on the next slide, the range consists of the family’s home phone number, the office phone numbers for both Malcolm and Maria, and the cell phone number for Maria. Each family member corresponds to all phone numbers at which that family member can be reached.

Example  Is Relation a Fcn?

Function Notation

• Typically use single letters such as f , F , g , G , h , H , and so on as the name of a function.
• For each x in the domain of f , there corresponds a unique y in its range. The number y is denoted by f ( x ) read as “ f of x ” or “ f at x ”.
• We call f ( x ) the value of f at the number x and say that f assigns the f ( x ) value to y. - Since the value of y depends on the given value of x , y is called the dependent variable and x is called the independent variable.

Function Forms

• Functions can be described by:
  • A Table
  • A Graph

y

x

Evaluating a Function

• Let g be the function defined by the equation  y = g ( x ) = x^2 – 6 x + 8
• Evaluate each function value:

a. g (^) ( ) 3 b.^ g^ (− 2 ) c. g

^

d. g a ( + (^2) ) e.^ g x (^^ +^ h )

 SOLUTION

a. g (^) ( ) 3 = 32 − 6 3( ) + 8 = − 1

Evaluating a Function

• Evaluate fcn  y = g ( x ) = x^2 – 6 x + 8

b. g (^) (− 2 ) c. g

^

d. g a ( + (^2) ) e.^ g x (^^ +^ h )

 SOLUTION

b. g (^) (− 2 ) = (^) (− 2 )^2 − (^6) (− 2 ) + 8 = 24

c. g

^

^

2 − 6

^