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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. Why you should learn it To represent real-life relationships between two quantities such as time and altitude for a rising hot-air balloon. 1.6 Tables and Graphs GOAL 1 INPUT-OUTPUT TABLES VOCABULARY •function •input/domain •output/range In a function, each input has exactly one output. Another way to put it is no number in the input can be repeated. •input-output table EXAMPLE 1 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 Extra 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. Extra 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 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 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 from Section 1.5 to: •Write a verbal model •Assign labels •Write an algebraic model EXAMPLE 3 Extra 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. Extra 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 + $0.95h 10 9 11 9.95 12 13 14 10.90 11.85 12.80 h Extra Example 3 (cont.) h c 10 9 11 9.95 12 13 14 10.90 11.85 12.80 Internet Cost 14 Amount ($) 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 QUESTIONS?