Unit 1: Functions Unit 1: Functions Sec1: Relations, Sec1: Relations,

Functions, Domain and Functions, Domain and RangeRangeAlgebra IIAlgebra II

Goals/ObjectivesGoals/Objectives

Essential Question: Essential Question: • How do you solve, graph, and apply How do you solve, graph, and apply

linear functions to predict and solve real linear functions to predict and solve real world problems?world problems?

Objectives:Objectives:• Students will identify functions and Students will identify functions and

relations.relations.• Students will identify the domain and Students will identify the domain and

range of a function or relation.range of a function or relation.• Students will graph linear equations.Students will graph linear equations.

RelationsRelations A A relationrelation is a mapping, or is a mapping, or

pairing, of input values with output pairing, of input values with output values.values.

A relation is a A relation is a functionfunction if there is if there is exactly one output for each input.exactly one output for each input.

It is It is NOTNOT a function if at least one a function if at least one input has more than one outputinput has more than one output

Domain and Range

The domain is the horizontal width of the graph ( all x-values).

The range is the vertical height of the graph ( all y-values).

Input Output

-3 3

1 -2

4 1

4

Identify the Domain and Range. Then tell if the relation is a function for the points (-3, 3), (1, -2), (1, 1) and (4, 4).

Domain = {-3, 1,4}Range = {3,-2,1,4}

Function?No: input 1 is mapped onto Both -2 & 1

Notice the set notation!!!

Identify the Domain and Range. Then Identify the Domain and Range. Then tell if the relation is a function for the tell if the relation is a function for the

points (-3, 3), (1, 1), (3, 1) and (4, -2).points (-3, 3), (1, 1), (3, 1) and (4, -2).Input Output

-3 3

1 1

3 -2

4

Domain = {-3, 1,3,4}Range = {3,1,-2}

Function?Yes: each input is mappedonto exactly one output

Vertical Line TestVertical Line Test You can use the vertical line test to You can use the vertical line test to

visually determine if a relation is a visually determine if a relation is a function.function.

Slide any vertical line (pencil) across the Slide any vertical line (pencil) across the graph to see if any two points lie on the graph to see if any two points lie on the same vertical line.same vertical line.

If there are If there are notnot two points on the same two points on the same vertical line then the relation vertical line then the relation is a is a functionfunction..

If there If there areare two points on the same two points on the same vertical line then the relation is vertical line then the relation is NOT a NOT a functionfunction

Graphing RelationsGraphing Relations Graph the relations in the Graph the relations in the previous examples on 2 previous examples on 2 separate graphs.separate graphs.

Plot the pointsPlot the points Do the Do the Vertical Line Test Vertical Line Test to to determine if the relation is a determine if the relation is a function.function.

(-3,3)(4,4)

(1,1)

(1,-2)

Use the vertical line test to visually check if the relation is a function.

Function?No, Two points are on The same vertical line.

(-3,3)

(4,-2)

(1,1) (3,1)

Use the vertical line test to visually check if the relation is a function.

Function?Yes, no two points are on the same vertical line

Graphing and Evaluating FunctionsGraphing and Evaluating Functions

Many functions can be represented Many functions can be represented by an by an equationequation in 2 variables: in 2 variables:

y=x²-7 and y = 2x + 1y=x²-7 and y = 2x + 1

A. Is the graph a function?A. Is the graph a function? B. What is the domain of the B. What is the domain of the

function?function? C. What is the range of the C. What is the range of the

function?function?

AssignmentHandout

Relation and Functions