1.7 An Introduction to Functions

GOAL 1Identify a function and make an input-output table for a function.

GOAL 2Write an equation for a real-life function, such as the relationship between water pressure and depth.

What you should learn

Functions

• A relationship where one thing depends upon another is called a function.

• A function is a rule that establishes a relationship between two quantities called the input and output.

• In a function each input has exactly one output. More than one input can have the same output

Vocabulary

Domain: collection of all input values.

Range: collection of all output values.

Again: Cannot have the same input (domain) more than once or it is NOT a function.

Input n

Output

1

1

2

3

1

6

Input n

Output

1

1

2

3

3

6

NO YES

Identifying Functions

• The key to identifying functions is the rule that each input has exactly one output.

• If an input has more than one output…then the data is not a function

• Often times you will be given a table or a list of ordered pairs be asked to identify if the data is a function.

• Let’s look at some examples…

Identifying Functions

• Look at the table to the right…notice that each input has exactly one output…

• Therefore, this set of data is considered a function

Input Output

5 3

6 4

7 5

8 6

Identifying Functions

• Look at the table to the right…notice that the input of 9 has two different outputs (5 and 4 respectively)

• Therefore, this set of data is not considered to be a function

Input Output

9 5

9 4

8 3

7 2

Identifying Functions

• Look at the table to the right…notice that the input of 1 and 2 have the same output of 3

• In this instance this is considered a function because each input has exactly one output…it’s ok to have different inputs with the same output

Input Output

1 3

2 3

3 4

4 4

Your Turn – Identifying a Function

• Does the table represent a function? Explain

Input Output

1 3

1 4

2 5

3 6

Input Output

1 1

2 3

3 6

4 10

Input Output

1 3

2 6

3 11

4 18

Input Output

5 9

4 8

3 9

2 7

1.

2.

3.

4.

Example 1

Input 0 1 2 2.5 3 4

Output 3 1 2.3 2 1 2

For the input 1, what is the output?

For the input 3, what is the output?

What is the domain of the function?

What is the range of the function?

Vertical Line Test

Function

Extra Example 1

The profit on the school play is $4 per ticket minus $280, the expense to build the set. There are 300 seats in the theater. The profit for n tickets sold is

p = 4n – 280 for 70 ≤ n ≤ 300.

a. Make an input-output table.

b. Is this a function?

c. Describe the domain and range.

EXAMPLE 2

n 70 71 72 73 … 300

p

n 70 71 72 73 … 300

p 0 4 8 12 … 920

Yes; none of the inputs are repeated.

Domain: 70, 71, 72, 73,… , 300

Range: 0, 4, 8, 12,… ,920

You bicycle 4 mi and decide to ride for 2.5 more hours at 6 mi/hr. The distance you have traveled d after t hours is given by d = 4 + 6t, where 0 ≤ t ≤ 2.5.

a. Make an input-output table. Calculate d for each half-hour (t = 0, 0.5, 1, 1.5, 2, 2.5).

b. Draw a line graph.

Example 2

t 0 0.5 1 1.5 2 2.5

d

t 0 0.5 1 1.5 2 2.5

d 4 7 10 13 16 19

Example 2 (cont.)

t 0 0.5 1 1.5 2 2.5

d 4 7 10 13 16 19

Bicycle Distance

0

5

10

15

20

0 0.5 1 1.5 2 2.5

Time (hours)

Dis

tan

ce (

mile

s)

• Input-Output Table

• Description in Words

• Equation

• Graph

4 WAYS TO DESCRIBE A FUNCTION

Checkpoint

A plane is at 2000 ft. It climbs at a rate of 1000 ft/min for 4 min. The altitude h for t minutes is given by

h = 2000 + 1000t for 0 ≤ t ≤ 4.

1.Make a table (use 0, 1, 2, 3, and 4 minutes).

2.Draw a line graph.

3.Describe the domain and range.

Checkpoint (cont.)

t 0 1 2 3 4

d 2000 3000 4000 5000 6000

Plane Altitude

0

2000

4000

6000

8000

0 1 2 3 4

Time (minutes)

Hei

gh

t (f

eet)

Checkpoint (cont.)Plane Altitude

0

2000

4000

6000

8000

0 1 2 3 4

Time (minutes)

Hei

gh

t (f

eet)

Domain: all numbers between and including 0 and 4

Range: all numbers between and including 2000 and 6000

t 0 1 2 3 4

d 2000 3000 4000 5000 6000

All numbers are included because time is continuous. This is what is shown by connecting the data points with a line. Even numbers such as 1.73 minutes or 2148.4 ft are included as the plane climbs.

GOAL 2 WRITING EQUATIONS FOR FUNCTIONS

1.7 An Introduction to Functions

Use the problem solving strategy to:

•Write a verbal model

•Assign labels

•Write an algebraic model

EXAMPLE 3

Example 3

An internet service provider charges $9.00 for the first 10 hours and $0.95 per hour for any hours above 10 hours. Represent the cost c as a function of the number of hours (over 10) h.

a.Write an equation.

b.Create an input-output table for hours 10-14.

c. Make a line graph.

Example 3 (cont.)

VERBAL MODEL

LABELS

ALGEBRAIC MODEL

Cost • Number of hours

= Connection fee

+ Rateper

hour

c $9 $0.95 h

c = $9 + $0.95h

h 10 11 12 13 14

c 9 9.95 10.90 11.85 12.80

h 10 11 12 13 14

c

Example 3 (cont.)

h 10 11 12 13 14

c 9 9.95 10.90 11.85 12.80

Internet Cost

0

2

4

6

8

10

12

14

10 11 12 13 14

Time (hours)

Am

ou

nt

($)

Checkpoint

The temperature at 6:00 a.m. was 62°F and rose 3°F every hour until 9:00 a.m. Represent the temperature T as a function of the number of hours h after 6:00 a.m.

1.Write an equation.

2.Make an input-output table, using a one-half hour interval.

3.Make a line graph.

Checkpoint (cont.)

a. T = 62 + 3h

b.h 0 0.5 1 1.5 2 2.5 3

T 62 63.5 65 66.5 68 69.5 71

c.

Temperature Change

55

60

65

70

75

0 0.5 1 1.5 2 2.5 3

Time (hours)

Tem

per

atu

re (

F)

Homework

• Pg. 49-50

Numbers 1-8, 14, 18, 22, 25