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

To represent real-life relationships between two quantities such as time and altitude for a rising hot-air balloon.

Why you should learn it

GOAL 1 INPUT-OUTPUT TABLES

EXAMPLE 1

1.6 Tables and Graphs

VOCABULARY

•function

•input/domain

•output/range

•input-output table

In a function, each input has exactly one output. Another way to put it is no number in the input can be repeated.

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.

Extra 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

Extra 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

By the end of the lesson you should be able to move comfortably among all four representations. You will then have a variety of ways to model real-life situations.

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 from Section 1.5 to:

•Write a verbal model

•Assign labels

•Write an algebraic model

EXAMPLE 3

Extra Example 3An 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.

Extra 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

Extra 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)

QUESTIONS?