1. 1.7 An Introduction to Functions GOAL 1 Identify a function and make an input-output table for a function. GOAL 2 Write an equation for a real-life function, such as the relationship between water pressure and depth. What you should learn

2. 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

3. 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 2 3 1 6 Input n Output 1 1 2 3 3 6 NOYES

4. 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…

5. 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 InputOutput 53 64 75 86

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 InputOutput 95 94 83 72

7. 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 InputOutput 13 23 34 44

8. Your Turn – Identifying a Function Does the table represent a function? Explain InputOutput 13 14 25 36 InputOutput 11 23 36 410 InputOutput 13 26 311 418 InputOutput 59 48 39 27 1. 2. 3. 4.

9. Example 1 Input0122.534 Output312.3212 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?

10. Vertical Line Test Function

11. 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. b. Is this a function? c. Describe the domain and range. EXAMPLE 2 n70717273…300 p n70717273…300 p04812…920 Yes; none of the inputs are repeated. Domain: 70, 71, 72, 73,…, 300 Range: 0, 4, 8, 12,…,920

12. 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). b. Draw a line graph. Example 2 t00.511.522.5 d t00.511.522.5 d4710131619

13. Example 2 (cont.) t00.511.522.5 d4710131619

14. Input-Output Table Description in Words Equation Graph 4 WAYS TO DESCRIBE A FUNCTION

15. 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.

16. Checkpoint (cont.) t01234 d20003000400050006000

17. Checkpoint (cont.) Domain: all numbers between and including 0 and 4 Range: all numbers between and including 2000 and 6000 t01234 d20003000400050006000 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.

18. GOAL 2 WRITING EQUATIONS FOR FUNCTIONS 1.7 An Introduction to Functions Use the problem solving strategy to: Write a verbal model Assign labels Write an algebraic model EXAMPLE 3

19. 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.

20. Example 3 (cont.) VERBAL MODEL LABELS ALGEBRAIC MODEL Cost Number of hours = Connection fee + Rate per hour c$9$0.95h c = $9 + $0.95h h1011121314 c99.9510.9011.8512.80 h1011121314 c

21. Example 3 (cont.) h1011121314 c99.9510.9011.8512.80

22. 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.

23. Checkpoint (cont.) a. T = 62 + 3h b. h00.511.522.53 T6263.56566.56869.571 c.

24. Homework Pg. 49-50 Numbers 1-8, 14, 18, 22, 25