functions. objectives you will represent functions as rules and as tables you will represent...
TRANSCRIPT
Objectives
You will represent functions as rules and as tablesYou will represent functions as graphs
Function
A function consists of:
• A set called the domain containing numbers called inputs and a set called the range containing numbers called outputs.
• A pairing of inputs with outputs such that each input is paired with exactly one output.
Dependent Variable
The dependent variable is the output variable because its value depends on the value of the input variable.
Identify the domain and range of a functionEXAMPLE 1
The input-output table shows the cost of various amounts of regular unleaded gas from the same pump. Identify the domain and range of the function.
ANSWER
19.99, 23.99, 25.99, and 33.98.
The domain is the set of inputs: 10, 12, 13, and 17.The range is the set of outputs:
10Input gallons
Output dollars
12 13 17
19.99 23.99 25.99 33.98
GUIDED PRACTICE for Example 1
1. Identify the domain and range of the function.
Input 0 1 2 4Output 5 2 2 1
domain: 0, 1, 2, and 4
range: 1, 2, and 5
ANSWER
Reminder: Function
A function consists of:
• A set called the domain containing numbers called inputs and a set called the range containing numbers called outputs.
• A pairing of inputs with outputs such that each input is paired with exactly one output.
Tell whether the pairing is a function.
Identify a functionEXAMPLE 2
a.
The pairing is not a function because the input 0 is paired with both 2 and 3.
b.
Identify a functionEXAMPLE 2
Output Input
2 1
0 0
4 8
6 12
The pairing is a function because each input is pairedwith exactly one output.
GUIDED PRACTICE for Example 2
Tell whether the pairing is a function.
1221Output
12963Input2.
ANSWER
function
GUIDED PRACTICE for Example 2
Tell whether the pairing is a function.
3210Output
7422Input3.
ANSWER
not a function
Functions as a Rule
A function may be represented using a rule that relates one variable to another. The input variable is called the independent variable. The output variable is called the dependent variable because its value depends on the value of the input variable.
Functions:
Verbal Rule:
The output is 3 more than the input.
Equation:
Input, x 0 1 2 3 4
Output, y 3 4 5 6 7
Table
The range of the function is 0, 4, 10, 14, and 16.
Make a table for a functionEXAMPLE 3
The domain of the function y = 2x is 0, 2, 5, 7, and 8. Make a table for the function, then identify the range of the function.
x
y
2 5 70 8
= 2x 2 2 = 4 2 5 =10 2 7 =14 2 8 =162 0 = 0
SOLUTION
Write a function ruleEXAMPLE 4
Write a rule for the function. Input
Output 2
1
6 123
4
8
60 10
SOLUTION
and let y or dependent variable. Notice that eachbe the output,
Let x be the input, or independent variable,
output is 2 more than the corresponding input. So, a rule for the function is y = x + 2.
Write a function rule for a real-world situationEXAMPLE 5
Concert Tickets
You are buying concert tickets that cost $15 each. You can buy up to 6 tickets. Write the amount (in dollars) you spend as a function of the number of tickets you buy. Identify the independent and dependent variables. Then identify the domain and the range of the function.
Write a function rule for a real-world situationEXAMPLE 5
SOLUTION
So, the function rule is A = 15n. The amount spent depends on the number of tickets bought, so n is the independent variable and A is the dependent variable.
Write a verbal model. Then write a function rule. Let n represent the number of tickets purchased and A represent the amount spent (in dollars).
A 15 n=
Amount spent
(dollars)
Cost per ticket
(dollars/ticket)
Tickets purchased
(tickets)= •
Write a function rule for a real-world situationEXAMPLE 5
Because you can buy up to 6 tickets, the domain of the function is 0, 1, 2, 3, 4, 5, and 6. Make a table to identify the range.
Amount (dollars), A
0 1 32 654
0 15 4530 7560 90
Number of tickets, n
The range of the function is 0, 15, 30, 45, 60, 75, and 90.
GUIDED PRACTICE for Examples 3,4 and 5
4. Make a table for the function y = x – 5 with domain 10, 12, 15, 18, and 29. Then identify the range of the function.
range: 5, 7, 10, 13 and 24.
ANSWER
GUIDED PRACTICE for Examples 3,4 and 5
5. Write a rule for the function. Identify the domain and the range.
Pay (dollars)
1 2 43
8 16 3224
Time (hours)
y = 8x; domain: 1, 2, 3, and 4;range: 8, 16, 24, and 32.
ANSWER
Representing Functions as Graphs
Table Ordered Pairs
Input, x Output, y
1 2
2 3
4 5
(1, 2)(2, 3)(4, 5)
Graph a function
EXAMPLE 1
STEP 1 Make an input-output table.
SOLUTION
Graph the function y = x with domain 0, 2, 4, 6, and 8. 12
x 0 2 4 6 8
y 0 1 2 3 4
GUIDED PRACTICE for Example 1
1. Graph the function y = 2x – 1 with domain 1, 2, 3, 4, and 5.
ANSWER
Graph a function
EXAMPLE 2
The table shows the average score s on the mathematics section of the Scholastic Aptitude Test (SAT) in the United States from 1997 to 2003 as a function of the time t in years since 1997. In the table, 0 corresponds to the year 1997, 1 corresponds to 1998, and so on. Graph the function.
519516514514511512511Average score, s
6543210Years since 1997, t
SAT Scores
SOLUTION
STEP 1 Choose a scale. The scale should allow you to plot all the points on a graph that is a reasonable size.
EXAMPLE 2 Graph a function
The s-values range from 511 to 519, so label the s-axis from 510 to 520 in increments of 2 units.
The t-values range from 0 to 6, so label the t-axis from 0 to 6 in increments of 1 unit.
EXAMPLE 2GUIDED PRACTICE for Example 2
WHAT IF? In Example 2, suppose that you use a scale on the s-axis from 0 to 520 in increments of 1 unit. Describe the appearance of the graph.
2.
The graph would be very large with all the points near the top of the graph.
ANSWER
Write a function rule for a graph
EXAMPLE 3
Write a rule for the function represented by the graph.Identify the domain and the range of the function.
SOLUTION
STEP 1 Make a table for the graph.
x 1 2 3 4 5
y 2 3 4 5 6
Write a function rule for a graph
EXAMPLE 3
STEP 2 Find a relationship between the inputs and the outputs. Notice from the table that each output value is 1 more than the corresponding input value.
STEP 3
Write a function rule that describes the relationship:y = x + 1.
A rule for the function is y = x + 1. The domain of the function is 1, 2, 3, 4, and 5. The range is 2, 3, 4, 5, and 6.
ANSWER
GUIDED PRACTICE for Example 3
Write a rule for the function represented by the graph. Identify the domain and the range of the function.
3.
ANSWER y = 5 – x; domain: 0, 1, 2, 3, and 4, range: 1, 2, 3, 4, and 5
GUIDED PRACTICE for Example 3
Write a rule for the function represented by the graph. Identify the domain and the range of the function.
ANSWER y = 5x + 5; domain: 1, 2, 3, and 4, range: 10, 15, 20, and 25
4.
Analyze a graph
EXAMPLE 4
The graph shows guitar sales (in millions of dollars) for a chain of music stores for the period 1999–2005. Identify the independent variable and the dependent variable. Describe how sales changed over the period and how you would expect sales in 2006 to compare to sales in 2005.
Guitar Sales
Analyze a graph
EXAMPLE 4
SOLUTION
The independent variable is the number of years since1999. The dependent variable is the sales (in millions of dollars). The graph shows that sales were increasing. If the trend continued, sales would be greater in 2006 than in 2005.
EXAMPLE 4GUIDED PRACTICE for Example 4
Yes; the graph seems to increase about $0.2 million every two years.
ANSWER
Based on the graph in Example 4, is $1.4 million a reasonable prediction of the chain’s sales for 2006? Explain.
5. REASONING