Transcript function. - Lowellville Local Schools

1.7 An Introduction to Functions
What you should learn
GOAL
GOAL
1
Identify a function and make an input-output
table for a function.
2
Write an equation for a real-life function,
such as the relationship between water
pressure and depth.
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
NO
2
3
1
6
Input n
Output
1
1
2
3
YES
3
6
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
3.
1.
Input
1
2
3
4
Output
1
3
6
10
2.
Input
Output
1
1
2
3
3
4
5
6
Input
Output
1
2
3
4
3
6
11
18
4.
Input
Output
5
4
3
2
9
8
9
7
Example 1
Input
0
Output 3
1
1
2
2.3
2.5
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?
3
1
4
2
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.
n
70
71
72
73
… 300
p
0
4
8
12
… 920
b. Is this a function? Yes; none of the inputs are repeated.
c. Describe the domain and range.
Domain: 70, 71, 72, 73,… , 300
Range: 0, 4, 8, 12,… ,920
EXAMPLE 2
Example 2
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).
t
0
0.5
1
1.5
2
2.5
d
4
7
10
13
16
19
b. Draw a line graph.
Example 2 (cont.)
t
0
0.5
1
1.5
2
2.5
d
4
7
10
13
16
19
Bicycle Distance
Distance (miles)
20
15
10
5
0
0
0.5
1
1.5
Time (hours)
2
2.5
4 WAYS TO DESCRIBE A FUNCTION
• Input-Output Table
• Description in Words
• Equation
• Graph
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
d
1
2
3
4
2000 3000 4000 5000
Plane Altitude
Height (feet)
8000
6000
4000
2000
0
0
1
2
3
Time (minutes)
4
6000
Checkpoint (cont.)
0
1
2
3
4
d 2000 3000 4000 5000 6000
Domain: all numbers between
and including 0 and 4
Range: all numbers between
and including 2000 and 6000
8000
Height (feet)
t
Plane Altitude
6000
4000
2000
0
0
1
2
3
Time (minutes)
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.
4
1.7 An Introduction to Functions
GOAL
2
WRITING EQUATIONS FOR 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
Cost
LABELS
c
=
$9
ALGEBRAIC
MODEL
h
c
Connection
+
fee
Rate
per
hour
• Number
of hours
$0.95
c = $9 + $0.95h
10
9
11
9.95
12
13
14
10.90 11.85 12.80
h
Example 3 (cont.)
h
c
10
9
11
9.95
12
13
14
10.90 11.85 12.80
Internet Cost
14
Amount ($)
12
10
8
6
4
2
0
10
11
12
Time (hours)
13
14
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
Temperature (F)
Temperature Change
c.
75
70
65
60
55
0
0.5
1
1.5
Time (hours)
2
2.5
3
Homework
• Pg. 49-50
Numbers 1-8, 14, 18, 22, 25