Functions

Objectives You will represent functions as rules and as tables You will represent functions as graphs

Vocabulary Function Domain Range Independent Variable Dependent Variable

Function

Function

Function

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.

Domain The domain is the set of numbers containing all inputs for a function.

Range The range is the set of numbers containing all outputs for a function.

Independent Variable The independent variable is the input variable.

Dependent Variable The dependent variable is the output variable because its value depends on the value of the input variable.

The domain is the set of inputs: 10, 12, 13, EXAMPLE 1 Identify the domain and range of a function 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. 10 Input gallons Output dollars 12 13 17 19.99 23.99 25.99 33.98 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:

1 2 4 5 GUIDED PRACTICE for Example 1 Input 1. Identify the domain and range of the function. Input 1 2 4 Output 5 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.

a. EXAMPLE 2 Identify a function Tell whether the pairing is a function. a. The pairing is not a function because the input 0 is paired with both 2 and 3.

EXAMPLE 2 Identify a function b. Output Input 2 1 4 8 6 12 The pairing is a function because each input is paired with exactly one output.

GUIDED PRACTICE for Example 2 Tell whether the pairing is a function. 1 2 Output 12 9 6 3 Input 2. ANSWER function

GUIDED PRACTICE for Example 2 Tell whether the pairing is a function. 3 2 1 Output 7 4 Input 3. 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: 𝑦=𝑥+3 Table Input, x 1 2 3 4 Output, y 5 6 7

EXAMPLE 3 Make a table for a function 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. SOLUTION x y 2 5 7 8 = 2x 4 10 14 16 The range of the function is 0, 4, 10, 14, and 16.

rule for the function is y EXAMPLE 4 Write a function rule Write a rule for the function. Input Output 2 1 6 12 3 4 8 10 SOLUTION and let y or dependent variable. Notice that each be 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.

EXAMPLE 5 Write a function rule for a real-world situation 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 situation EXAMPLE 5 Write a function rule for a real-world situation SOLUTION 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) • 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.

EXAMPLE 5 Write a function rule for a real-world situation 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 1 3 2 6 5 4 15 45 30 75 60 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 4 3 8 16 32 24 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 3 4 5 (1, 2) (2, 3) (4, 5)

EXAMPLE 1 Graph a function Graph the function y = x with domain 0, 2, 4, 6, and 8. 1 2 SOLUTION STEP 1 Make an input-output table. x 2 4 6 8 y 1 3

EXAMPLE 1 Graph a function STEP 2 Plot a point for each ordered pair (x, y).

GUIDED PRACTICE for Example 1 1. Graph the function y = 2x – 1 with domain 1, 2, 3, 4, and 5. ANSWER

EXAMPLE 2 Graph a function SAT Scores 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. 519 516 514 511 512 Average score, s 6 5 4 3 2 1 Years since 1997, t

EXAMPLE 2 Graph a function SOLUTION STEP 1 Choose a scale. The scale should allow you to plot all the points on a graph that is a reasonable size. The t-values range from 0 to 6, so label the t-axis from 0 to 6 in increments of 1 unit. The s-values range from 511 to 519, so label the s-axis from 510 to 520 in increments of 2 units.

EXAMPLE 2 Graph a function STEP 2 Plot the points.

EXAMPLE 2 GUIDED 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

EXAMPLE 3 Write a function rule for a graph 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 6

EXAMPLE 3 Write a function rule for a graph 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. ANSWER 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.

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. 4. ANSWER y = 5x + 5; domain: 1, 2, 3, and 4, range: 10, 15, 20, and 25

EXAMPLE 4 Analyze a graph Guitar Sales 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.

EXAMPLE 4 Analyze a graph SOLUTION The independent variable is the number of years since 1999. 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 4 GUIDED PRACTICE for Example 4 Based on the graph in Example 4, is $1.4 million a reasonable prediction of the chain’s sales for 2006? Explain. 5. REASONING Yes; the graph seems to increase about $0.2 million every two years. ANSWER

Ways to Represent a Function Verbal Rule: The output is 1 less than twice the input. Equation: 𝒚=𝟐𝒙−𝟏 Table: Graph x y 1 2 3 5 4 7