Relation A relation is a correspondence between two

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 ORDERED type PAIR Michael A (Michael, A) Tania A (Tania, A) Dylan AB (Dylan, AB) Trevor 0 (Trevor, O) Megan 0 (Megan, O)

Relation Person Blood ORDERED type 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 = x 2 − 3 x

Functions Defined by Equations y = x 2 − 3 x x y = x 2 − 3 x y

Functions Defined by Equations y = x 2 − 3 x x 1 y = x 2 − 3 x y

Functions Defined by Equations y = x 2 − 3 x x y = x 2 − 3 x 1 y = (1)2 − 3(1) y

Functions Defined by Equations y = x 2 − 3 x x y = x 2 − 3 x 1 y = (1)2 − 3(1) y − 2

Functions Defined by Equations y = x 2 − 3 x x y = x 2 − 3 x 1 y = (1)2 − 3(1) 5 y − 2

Functions Defined by Equations y = x 2 − 3 x x y = x 2 − 3 x 1 y = (1)2 − 3(1) 5 y = (5)2 − 3(5) y − 2

Functions Defined by Equations y = x 2 − 3 x x y = x 2 − 3 x 1 y = (1)2 − 3(1) 5 y = (5)2 − 3(5) y − 2 10

Functions Defined by Equations y = x 2 − 3 x x y = x 2 − 3 x 1 y = (1)2 − 3(1) 5 y = (5)2 − 3(5) 1. 2 y − 2 10

Functions Defined by Equations y = x 2 − 3 x x y = x 2 − 3 x 1 y = (1)2 − 3(1) 5 y = (5)2 − 3(5) 1. 2 y = (1. 2)2 − 3(1. 2) y − 2 10

Functions Defined by Equations y = x 2 − 3 x x y = x 2 − 3 x 1 y = (1)2 − 3(1) 5 y = (5)2 − 3(5) 1. 2 y = (1. 2)2 − 3(1. 2) y − 2 10 − 2. 16

Functions Defined by Equations y = x 2 − 3 x x y = x 2 − 3 x 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 = x 2 − 3 x x y = x 2 − 3 x 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 x FUNCTION f OUTPUT f (x) EQUATION f (x) = 2 x + 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 = -3 x + 8 f(x) = 5 x – 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) , 4 f (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 For the function f (x) = x 2 − x, find

Falling Objects: Firecrackers. • A firecracker is launched straight up, and its height is a function of time, h(t) = − 16 t 2 + 128 t, 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?