Relations and Functions

Transcription

1 Lesson 5.1 Objectives Identify the domain and range of a relation. Write a rule for a sequence of numbers. Determine if a relation is a function. Relations and Functions You can estimate the distance of an approaching thunderstorm by counting the seconds between a flash of lightning and the resulting sound of thunder. This works because sound travels about 1 3 of a kilometer every second. Consequently, if 3 seconds go by between the flash and the sound, the storm is 1 kilometer away, and so on. Relations Sets of ordered pairs, tables, and graphs are all useful tools used to represent relationships between two variables. A set of ordered pairs is called a relation. Example 1 Writing Relations Use ordered pairs to represent the data for an approaching thunderstorm. Then write an equation to represent the relation. Because there are three seconds between the flash and sound when the storm is 1 kilometer away, the first ordered pair is (3, 1). R {(3, 1), (6, 2), (9, 3), (12, 4),...} The braces { } mean the set of. You can list the relation in a table or draw its graph on a coordinate plane. Time in Seconds Distance in Kilometers : : Distance (km) (12, 4) (9, 3) (6, 2) (3, 1) Time (s) There is a pattern in the ordered pairs describing the thunderstorm data. Represent time with the independent variable t and the distance with the dependent variable re d. The equation d 1 3 t represents the relationship. be This equation is a shortcut way of saying that the distance sound travels depends on time. Each value of a relation has a domain and a range. The domain is the set of all first components of the relation. The range is the set of all second components of the relation. 272 Chapter 5 Nonlinear Functions

2 Example 2 Domain and Range of Relations Identify the domain and range of each relation. a. Weight of First-Class Letter (in ounces) Cost to Mail Letter 1 $ $ $ $1.31 b. c. ( 4, 11), ( 2, 7), (0, 3), (2, 1), (4, 5) If a domain or range value is used more than once in the relation, you only need to list the value once. a. domain {1, 2, 3, 4} range {$0.41, $0.58, $0.75, $1.31} b. domain { 4, 2, 0, 2, 5} range { 3, 1, 2, 4, 5} c. domain { 4, 2, 0, 2, 4} range { 11, 7, 3, 1, 5} Sequences Examine the following sequence of numbers: 88, 96, 104, 112, 120,... The numbers in this sequence record the speed of a car on a test track each time it passes the timing line. The first speed measured is 88 feet per second and the car is steadily increasing its speed. Do you see a pattern? The difference between any two numbers is 8. You can order the numbers in the sequence with the set of whole numbers and list them in a table. Lap Number Speed Difference in Speed Relations and Functions 273

3 If you graph the ordered pairs, you can see that the graph is a line with slope 8 and y-intercept 88. The equation representing the relationship between the ordered pairs is y 8x (4, 120) 110 (3, 112) 100 (2, 104) Speed 90 (0, 88) (1, 96) Lap Number What is the domain of the function? the range? all positive numbers; all positive numbers greater than or equal to 88 Ongoing Assessment What is the speed of the car after 5 laps? By how much does the car increase its speed each lap? 128 feet per second; 8 feet per second Activity Sequences and Equations Use the sequence 3, 7, 11, 15, 19, to complete the following: 1 Use whole numbers to write the sequence as a set of ordered pairs beginning with (1, 3) and ending with (5, 19). {(1, 3), (2, 7), (3, 11), (4, 15), (5, 19)} 2 Use the ordered pairs to make a table. see margin 3 Graph the set of ordered pairs. see margin 4 Find the slope and y-intercept of the graph. 4; 1 5 Find an equation that expresses the relationship between the numbers in the sequence. y 4x 1 6 Extend the sequence to three more numbers. 23, 27, Chapter 5 Nonlinear Functions

4 Example 3 Charges for Service Calls A computer repair company has a schedule of charges for making service calls. The company charges a flat rate of $40 for each call plus $20 per hour on the job. Show the schedule of charges as a table, a graph, and an equation. Write the domain and the range. Time in Hours Charge in Dollars Service Charge ($) Time (hours) The equation c 20t 40 represents the relationship between the time worked (t) and the charge for the work (c). The domain is positive numbers. The range is positive numbers greater than or equal to 40. Function Notation A function is a relation with one additional condition. Function A function is a set of ordered pairs (x, y) such that for any value of x, there is exactly one value of y. For example, the equation y 2x 1 describes the rule of a function. To obtain each range value y, multiply the domain value x by 2 and then add 1. In this case, x is the independent variable and y is the dependent variable. Because a unique value for y is determined for each value of x, y is a function of x. A relation is a function if for each value of the independent variable x, there exists exactly one value of the dependent variable y. In function notation, you write f(x) 2x 1. The symbol f(x) is read f of x or f is a function of x. In function notation, an input x determines the output f(x) according to the rule 2x 1. Usually the letters f, g, and h are used to represent functions. 5.1 Relations and Functions 275

5 Example 4 Functions Examine the ordered pairs in part c of Example 2. Is the relation a function? List the ordered pairs. ( 4, 11), ( 2, 7), (0, 3), (2, 1), (4, 5) Each domain value is associated with exactly one range value. Therefore, the relation is a function. The Pencil or Vertical-Line Test When a relation is shown on a Cartesian coordinate system, you can use a technique called the pencil test or vertical-line test to determine if the relation is a function. With this test, move a pencil (oriented as a vertical line) from left to right across the graph of the relation. As the pencil moves, check to see how many points of the relation it intersects at any one time. If the pencil crosses the graph of the relation one point at a time, the relation is a function. If the pencil intersects more than one point at a time, the relation is not a function. The graph of the relation below on the left is a function. The graph of the relation below on the right is not a function because a pencil or vertical line intersects points M and N at the same time. Ongoing Assessment Determine whether the relations in part a and part b of Example 2 represent functions. part a is a function; part b is not a function 276 Chapter 5 Nonlinear Functions

6 Lesson Assessment Think and Discuss 1. Give examples of mathematical relations that are not functions. Explain why they are relations but not functions. 2. How many ways can you describe a function? Give examples. 3. Explain why a sequence of numbers might be considered a function. 4. Explain what the notation g(x) 3x 5 means. Practice and Problem Solving State the domain and range of each relation. Then determine if the relation is a function. see margin 5. ( 6, 5), ( 3, 6), (0, 7), (3, 8), (6, 9), (9, 10) 6. (16, 4), (9, 3), (4, 2), (0, 0), (4, 2), (9, 3), (16, 4) 7. Number of Cards Purchased Amount of Discount 1 15% 2 15% 3 20% 4 20% 5 25% 6 25% 8. Hours Worked Amount Earned 5 $ $ $ $ $ $ Relations and Functions 277

7 For each sequence, write a rule that gives the relationship between the numbers and find the next three numbers in the sequence , 4, 5, , 2, 4, , 2, 5, 8 y x 2; 7, 8, 9. y 2x 2; 8, 10, 12. y 3x 4; 11, 14, , 11, 16, , 1, 3, , 3 2, 2, 5 2 y 5x 1; 26, 31, 36 y 2x 3; 7, 9, 11.. y 1 x 2 1 ; 2 3, 7, 2 4 Determine if each relation is a function. If the relation is a function, look for a pattern and write an equation for y as a function of x. For each function, find the slope and zeros. 17. x yes; y 11x 4; y slope 11; zeros: x y x yes; y 3x 2; slope 3; zeros: 2 3 no y For each situation, make a table, find and graph the equation, and give the slope and y-intercept. 20. Roberto started walking from a point 3 miles from his house and walked away from his house at a constant speed of 4 miles per hour. What is Roberto s distance (d) from home as a function of time (t)? see margin 21. In Carlette s appliance repair business, she charges $15 for making a house call. She also charges $8 per hour. What are Carlette s total charges (c) as a function of time (t)? see margin 22. A savings account has $500. Each week, $15 is added to the account. How much is in the account (y) after x weeks? see margin Mixed Review 23. Write in scientific notation Write 100,000,000 as a power of ten Solve each equation y A s 26. 3(r 1) m A store pays $650 for a stereo system. It charges its customers $1,050 for the system. What is the profit from the sale of the system as a percent of the store s cost? 61.5% 278 Chapter 5 Nonlinear Functions