Relation A relation is a correspondence between two sets where each element in the first set, called...

Relation

A relation is a correspondence

between two sets where each element

in the first set, called the domain, corresponds to at least one element in the

second set, called the range.

Relation

Person Blood type

ORDERED PAIR

Michael A (Michael, A)

Tania A (Tania, A)

Dylan AB (Dylan, AB)

Trevor 0 (Trevor, O)

Megan 0 (Megan, O)

Relation

Person Blood type

ORDERED PAIR

Michael A (Michael, A)

Tania A (Tania, A)

Dylan AB (Dylan, AB)

Trevor 0 (Trevor, O)

Megan 0 (Megan, O)

Relation

Person Blood type

ORDERED PAIR

Michael A (Michael, A)

Tania A (Tania, A)

Dylan AB (Dylan, AB)

Trevor 0 (Trevor, O)

Megan 0 (Megan, O)

The domain is the set of all the first components.

{Michael, Tania, Dylan, Trevor, Megan}

The range is the set of all the second components.

{A, AB, O}

Relation

Person Blood type

ORDERED PAIR

Michael A (Michael, A)

Tania A (Tania, A)

Dylan AB (Dylan, AB)

Trevor 0 (Trevor, O)

Megan 0 (Megan, O)

The domain is the set of all the first components.

{Michael, Tania, Dylan, Trevor, Megan}

The range is the set of all the second components.

{A, AB, O}

Function

Function

• A function is a correspondence between two sets where each element in the first set, called the domain, corresponds to exactly one element in the second set, called the range

Function

• A function is a correspondence between two sets where each element in the first set, called the domain, corresponds to exactly one element in the second set, called the range

• Note that the definition of a function is more restrictive than the definition of a relation.

Function

Time of day Competition

1:00 P.M. Football

2:00 P.M. Volleyball

7:00 P.M. Soccer

7:00 P.M. Basketball

Functions Defined by Equations

Functions Defined by Equations

y = x2 − 3x

Functions Defined by Equations

y = x2 − 3x

x y = x2 − 3x y

Functions Defined by Equations

y = x2 − 3x

x y = x2 − 3x y

 1

Functions Defined by Equations

y = x2 − 3x

x y = x2 − 3x y

 1 y = (1)2 − 3(1)

Functions Defined by Equations

y = x2 − 3x

x y = x2 − 3x y

 1 y = (1)2 − 3(1) −2

Functions Defined by Equations

y = x2 − 3x

x y = x2 − 3x y

 1 y = (1)2 − 3(1) −2

5

Functions Defined by Equations

y = x2 − 3x

x y = x2 − 3x y

 1 y = (1)2 − 3(1) −2

5 y = (5)2 − 3(5)

Functions Defined by Equations

y = x2 − 3x

x y = x2 − 3x y

 1 y = (1)2 − 3(1) −2

5 y = (5)2 − 3(5) 10

Functions Defined by Equations

y = x2 − 3x

x y = x2 − 3x y

 1 y = (1)2 − 3(1) −2

5 y = (5)2 − 3(5) 10

1.2

Functions Defined by Equations

y = x2 − 3x

x y = x2 − 3x y

 1 y = (1)2 − 3(1) −2

5 y = (5)2 − 3(5) 10

1.2 y = (1.2)2 − 3(1.2)

Functions Defined by Equations

y = x2 − 3x

x y = x2 − 3x y

 1 y = (1)2 − 3(1) −2

5 y = (5)2 − 3(5) 10

1.2 y = (1.2)2 − 3(1.2) −2.16

Functions Defined by Equations

y = x2 − 3x

x y = x2 − 3x y

•  Since the variable y depends on what value of x is selected, we denote y as

the dependent variable. (output)

Functions Defined by Equations

y = x2 − 3x

x y = x2 − 3x y

•  Since the variable y depends on what value of x is selected, we denote y as

the dependent variable. (output)

• The variable x can be any number in the domain; therefore, we denote x as the independent variable. (input)

Function Notation

Function Notation

• The notation y = f(x) denotes that the variable y is function of x.

Function Notation

• The notation y = f(x) denotes that the variable y is function of x.

INPUT FUNCTION OUTPUT EQUATION

x f f (x) f (x) = 2x + 5

Function Notation

• A Linear function is a function defined by an equation that can be written in the form

f(x) = mx + b , or y = mx + b

where m is the slope of the line graph and

(0, b) is the y - intercept

Function Notation

• A Linear function is a function defined by an equation that can be written in the form

f(x) = mx + b , or y = mx + b

where m is the slope of the line graph and

(0, b) is the y - intercept

Ex. y = -3x + 8 f(x) = 5x – 4

The Graph of the Function

• The graph of the function is the graph of the ordered pairs (x, f(x)), that define the function.

Use the given graphs to evaluate the function.

Find f (0), f (1). f (2) , 4f (3),

Find x such that f (x) = 10, f (x) = 2

Find x such that f (x) = 10, f (x) = 2

Use the given graphs to evaluate the function.

T(−5)

T(−2)

T(4)

Vertical Line Test

• Given the graph of an equation, if any vertical line that can be drawn intersects the graph at no more than one point, the

equation defines a function of x.

This test is called the vertical line test.

Vertical Line Test

Evaluating the Difference Quotient

Evaluating the Difference Quotient

( ) ( )f x h f x

h

Evaluating the Difference Quotient

For the function f (x) = x2 − x, find

( ) ( )f x h f x

h

Falling Objects: Firecrackers.

• A firecracker is launched straight up, and its height is a function of time,

h(t) = −16t2 + 128t, where h is the height in feet and t is the time in seconds with

t = 0 corresponding to the instant it launches. What is the height 4 seconds after launch?