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Functions

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:

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Functions

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  1. Functions

  2. Objectives • You will represent functions as rules and as tables • You will represent functions as graphs

  3. Vocabulary • Function • Domain • Range • Independent Variable • Dependent Variable

  4. Function

  5. Function

  6. Function

  7. 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.

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

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

  10. Independent Variable The independent variable is the input variable.

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

  12. 10 Input gallons 12 13 17 Output dollars 19.99 23.99 25.99 33.98 ANSWER The domain is the set of inputs: 10, 12, 13, and17.The range is the set of outputs: 19.99, 23.99, 25.99,and 33.98. 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.

  13. ANSWER domain:0, 1, 2, and 4range:1, 2, and 5 for Example 1 GUIDED PRACTICE 1. Identify the domain and range of the function.

  14. 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.

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

  16. Output Input 0 0 1 2 8 4 6 12 EXAMPLE 2 Identify a function b. The pairing is a function because each input is pairedwith exactly one output.

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

  18. 3. Input 2 2 4 7 Output 0 1 2 3 ANSWER not a function for Example 2 GUIDED PRACTICE Tell whether the pairing is a function.

  19. 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.

  20. Functions: Verbal Rule: The output is 3 more than the input. Equation: Table

  21. x 0 2 5 7 8 y 7 8 = 2x 2 14 2 16 2 0 2 2 10 0 2 5 4 = = = = = 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 The range of the function is 0, 4, 10, 14, and 16.

  22. Input 0 4 6 10 1 6 Output 8 12 2 3 and let y Let x be the input, or independent variable, be the output, or dependent variable. Notice that each output is 2 more than the corresponding input. So, a rule for the function isy 2. x = + EXAMPLE 4 Write a function rule Write a rule for the function. SOLUTION

  23. 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.

  24. Amount spent(dollars) Tickets purchased (tickets) Cost per ticket (dollars/ticket) • = 15 n A = EXAMPLE 5 Write a function rule for a real-world situation SOLUTION Write a verbal model. Then write a function rule. Let nrepresent the number of tickets purchased and Arepresent the amount spent (in dollars). So, the function rule is A =15n. The amount spent depends on the number of tickets bought, so nis the independent variable and Ais the dependent variable.

  25. Number of tickets, n 0 1 2 3 4 5 6 60 75 90 Amount (dollars), A 15 30 45 0 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. The range of the function is 0, 15, 30, 45, 60, 75, and 90.

  26. ANSWER range: 5, 7, 10, 13 and 24. for Examples 3,4 and 5 GUIDED PRACTICE 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.

  27. Time (hours) 1 2 3 4 Pay (dollars) 8 16 24 32 ANSWER y = 8x; domain: 1, 2, 3, and 4; range: 8, 16, 24, and 32. for Examples 3,4 and 5 GUIDED PRACTICE 5. Write a rule for the function. Identify the domain and the range.

  28. Representing Functions as Graphs Table Ordered Pairs (1, 2) (2, 3) (4, 5)

  29. Graph the functiony = xwith domain 0, 2, 4, 6, and 8. 1 2 EXAMPLE 1 Graph a function SOLUTION STEP 1 Make an input-output table.

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

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

  32. Years since1997,t 0 1 2 3 4 5 6 Average score, s 511 512 511 514 514 516 519 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.

  33. EXAMPLE 2 Graph a function SOLUTION STEP1 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 from511to519, so label the s-axis from510 to520in increments of2units.

  34. EXAMPLE 2 Graph a function STEP2 Plot the points.

  35. 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. ANSWER The graph would be very large with all the points near the top of the graph. EXAMPLE 2 for Example 2 GUIDED PRACTICE

  36. 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.

  37. 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.

  38. 3. y = 5 – x;domain:0, 1, 2, 3,and4,range: 1, 2, 3, 4,and5 ANSWER for Example 3 GUIDED PRACTICE Write a rule for the function represented by the graph. Identify the domain and the range of the function.

  39. ANSWER y = 5x + 5; domain:1, 2, 3,and4, range: 10, 15, 20, and25 for Example 3 GUIDED PRACTICE Write a rule for the function represented by the graph. Identify the domain and the range of the function. 4.

  40. 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.

  41. 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.

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

  43. Ways to Represent a Function Verbal Rule: The output is 1 less than twice the input. Equation: Graph Table:

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