Chapter 7 Section 6

Functions

Learning Objective

Find the Domain and range of a relation.

Recognize functions

Evaluate functions

Graph linear functions

Key Vocabulary

Relations

Domain

Range

Function

Vertical Line Test

Function Notation

Linear Functions

Find the Domain and Range of a Function

Functions are a special type of relation and are common from algebra through calculus

Relation is any set of ordered pairs and can have elements other than numbers.

(x, y) ordered pairs

x and y are the components of the ordered pair

Every graph will represent a relation (candy, sales) (class, students) (birth year, life expectancy) (year, unemployment)

Find the Domain and Range of a Function

(Domain, Range)

Domain (x) of the relation is the first component of the ordered pair

Range (y) of the relation is the second component of the ordered pair

Think of a functions as a set of rules or directions that get you from one place (domain) to another (range)

Find the Domain and Range of a Function

This example is a function because each person is assigned to exactly one seat.

Sarah

Brad

Phillip

Amanda

Seat 1

Seat 2

Seat 3

Seat 4

Domain Range

(Sarah, Seat 1)

(Brad, Seat 2)

(Phillip, Seat 3)

(Amanda, Seat 4)

Relation

Functions are ordered pairs where the first component (domain, x) corresponds with exactly one second component (range, y).

First component is thought of as the input

Second component is thought of as the output.

For a function the output depends on the input.

Garbage In = Garbage Out

Find the Domain and Range of a Function

Do the following figures represent functions?

Yes is a functionBecause the first components (domain) 1,2,& 3 only have one second component (range).As ordered pairs all the x’s are different.

1

2

3

A

B

(1, A)

(2, A)

(3, B)

Domain (x) Range (y) Ordered Pair

Do the following figures represent functions?

No, not a function

Because the first component, domain (x), has two second components, range (y). 1 is associated with A and B

As ordered pairs two have the same x. NOT a function

1

2

A

B

C

(1, A)

(1, B)

(2, C)

Domain Ordered PairRange

Find the Domain and Range of a Function

Functions are useful because they allow us to predict the output when the input is known.

Example

Assume each candy bars cost $0.50

Write a function to determine the cost, c, when, n, candy bars are purchased.

The function is c = 0.65n

The cost (domain) depends on the number of candy bars purchased (range).

One 0.50(1) 0.50

Two 0.50(2) 1.00

n 0.50(n) 0.50n

-3

2

4

Determine which sets are functions?

-6

2

Domain Range

Example{ (2,-3), (-6, 2), (2, 4)}

No, not a function

Because the first component, Domain (x) has two second components. 2 is assigned to -3 and 4

Domain: {-6, 2}

Range: {-3, 2, 4}

Determine which sets are functions?

-2

1

2

Domain Range

Example{ (1, 3), (2, 4), (-2, 3)}

3

4

Yes is a functionBecause the first components (1,2,& -2) only have one second component.

Domain: {-2, 1, 2}

Range: {3, 4}

Vertical Line Test

If a vertical line can be drawn through any part of a graph and the vertical line intersects another part of the graph, then each value of x does not correspond to exactly one value of y and the graph does not represent a function.

If a vertical line cannot be drawn to intersect the graph at more than one point, each value of x corresponds to exactly one value of y and the graph represents a function.

x

y

Example: Points on the Graph.

This is a FUNCTION becauseno vertical line Intersects more than one point

Vertical Line Test

Example:

Is not a function

because one vertical line

Intersects 3 points

(1, 2) (1, -1) (1, -3)

x

y

Vertical Line Test

Example:

FunctionNot a Function

Vertical Line Test

Example:

Not a functionFunction

Evaluate Functions

Graphs that we see every day represents function. Income, Sales Births, Year

Function notation is f(x) reads as “f of x”

Graph y = x + 2 is a function

y depends on x

Therefore, y = f(x)

y is a function of the variable x

y = f(x) = x + 2

f(x) = x + 2

Evaluate Functions

Example:Let f(x) = x2 – x + 2Find f(-1)

f(-1) = (-1)2 – (-1) + 2f(-1) = 1 + 1 + 2f(-1) = 4

When x = -1 f(x) or y = 4

NOTE:f(-1) is a shorthand for “Evaluate the expression that the function equals for x = -1”

Evaluate Functions

Example:Let f(x) = x2 – x + 2Find f(3)

f(3) = (3)2 – 3 + 2f(3) = 9 – 3 + 2f(3) = 8

When x = 3 f(x) or y = 8

NOTE:f(3) is a shorthand for “Evaluate the expression that the function equals for x = 3”

Evaluate Functions

Example:Let f(x) = x2 – x + 2If x = 4 find the value of y

y = f(x) = x2 – x + 2y = f(4) = (4)2 – 4 + 2y = f(4) = 16 – 4 + 2y = f(4) = 14

When x = 4 f(x) or y = 14

NOTE:f(4) is a shorthand for “Evaluate the expression that the function equals for x = 4”

Graph Linear Functions

Remember that the graph of y = ax + b is a straight line that is a function

y = mx + b and f(x) = mx + b

are linear functions

Graph Linear Functions

Example:Graph f(x) = -2x + 6

Let x = 3 (3, 0)f(3) = -2(3) + 6f(3) = -6 + 6f(3) = 0

Let x = 2 (2, 2)f(2) = -2(2) + 6f(2) = -4 + 6f(2) = 2

Let x = 1 (1, 4)f(1) = -2(1) + 6f(1) = -2 + 6f(1) = 4 (3,0)

(2,2)

(1,4)

Two way to graph 1.Using the Slope = -2 and y-intercept (0, 6)

down 2 right 12. Plotting the points

(0,6)

Graph Linear FunctionsExample: Ice Skating Rink, the weekly profits (p) on an ice skating

rink is a function of the number of skaters per week (n). The function approximating the profit is p = f(n) = 8n – 600, where 0 ≤ n ≤ 400.

What is the profit for 300 skaters?Graph f(n) = 8x - 600

Let n = 300f(300) = 8(300) - 600f(300) = 2400 - 600f(300) = 1800

Let n = 200f(200) = 8(200) - 600f(200) = 1600 - 600f(200) = 1000

Let n = 100f(100) = 8(100) - 600f(100) = 800 - 600f(100) = 200

3000

2500

2000

1500

1000

500

100 200 300 400 500 600

Remember

Think of a function as a set of rules or directions that get you from one place (domain) to another (range)

Parentheses usually means multiply however, remember that function notation does not mean multiply

Use the definition of a function to determine if an equation or a set of ordered pairs is a function.

The vertical line test let you check if a graph is a function.

What you do to one side of an equation , you do to the other side. When you put the value for x in the function notation on the left side of equal sign, you use the same value for x in the function definition on the right side

HOMEWORK 7.6

Page 476:

# 35, 39, 45, 47, 57