SECT

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4.3 4.3

4.3

OBJECTIVES 1. Determine whether a table of values represents a function 2. Use the vertical line test to identify the graph of a function 3. Identify the domain of a function

Identifying Functions In Section 4.2, we used a function machine as a model that enabled us to put in a value for x and get out a value that is a function of x. These two values, x and f(x), have a relationship that is usually expressed as an ordered pair. A similar type of relationship is used in every field in which mathematics is applied. ■

The physicist looks for the relationship that uses a planet’s mass to predict its gravitational pull.

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The economist looks for the relationship that uses the tax rate to predict the employment rate.

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The business marketer looks for the relationship that uses an item’s price to predict the number that will be sold.

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The college board looks for the relationship between tuition costs and the number of students enrolled at the college.

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The biologist looks for the relationship that uses temperature to predict a body of water’s nutrient level.

In each of these examples, a researcher matches an item from the given set (the domain) with an item from the related set (the range). Each pairing becomes an ordered pair. In Section 4.1, we looked at the concept of a relation, which is a set of ordered pairs. In the preceding list, we mentioned the relationship between a planet’s mass and its gravitational pull. This relationship is an example of a function. There cannot be two different gravitational pulls associated with a single planet. If you know a planet’s mass, you can find its gravitational pull. Every set of ordered pairs defines a relation, but not every set of ordered pairs defines a function. A function is a special kind of relation.

A function is a set of ordered pairs (a relation) in which no two first coordinates are equal.

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Section 4.3

Example 1

Identifying Functions

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297

Identifying a Function For each table of values below, decide whether the relation is a function. (a)

x

y

2 1 1 2

1 1 3 3

(b)

x

y

5 1 1 2

2 3 6 9

(c)

x

y

3 1 0 2

1 0 2 4

Part a represents a function. No element of the domain (x) is matched with two different elements of the range (y). Part b is not a function because 1 is matched with two different range elements, 3 and 6. Part c is a function. ✓ CHECK YOURSELF 1 ■ For each table of values below, decide whether the relation is a function. (a)

x

y

3 1 1 3

0 1 2 3

(b)

x

y

2 1 1 2

2 2 3 3

(c)

x

y

2 1 0 0

0 1 2 3

We defined a function in terms of ordered pairs. A set of ordered pairs can be specified in several ways; here are the most common.

1. We can present the ordered pairs in a list or table, as in Example 1. 2. We can give a rule or equation that will generate the ordered pairs. 3. We can use a graph to indicate the ordered pairs. The graph can show distinct ordered pairs, or it can show all the ordered pairs on a line or curve.

Vertical Line Test Let’s look at a graph of the ordered pairs from Example 1 to introduce the vertical line test, which is a graphic test for identifying a function.

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A Beginning Look at Functions (a)

(b)

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2

8 6 4 2 2

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8 6 4 2 2

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(c)

2

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y 8 6 4 2 8 6 4 2 2

2

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4 6 8

Notice that in the graphs of relations a and c, there is no vertical line that can pass through two different points of the graph. In relation b, a vertical line can pass through the two points that represent the ordered pairs (1, 3) and (1, 6). This leads to the following definition.

Vertical Line Test If no vertical line can pass through two or more points in the graph of a relation, then the relation is a function.

Example 2

Identifying a Function For each set of ordered pairs, plot the related points on the provided axes. Then use the vertical line test to determine which of the sets is a function.

Section 4.3

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Identifying Functions

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(a) {(0, 1), (2, 3), (2, 6), (4, 2), (6, 3)} y 8 6 4 2 8 6 4 2 2

2

4

6

8

x

4 6 8

Since a vertical line can be drawn through the points (2, 3) and (2, 6), the relation does not pass the vertical line test. This is not a function. (b) {(1, 1), (2, 0), (3, 3), (4, 3), (5, 3)} y 8 6 4 2 8 6 4 2 2

2

4

6

8

x

4 6 8

This is a function. Although a horizontal line can be drawn through several points, no vertical line passes through more than one point. ✓ CHECK YOURSELF 2 ■ For each set of ordered pairs, plot the related points. Then use the vertical line test to determine which of the sets is a function. (a) {(2, 4), (1, 4), (0, 4), (1, 3), (5, 5)} (b) {(3, 1), (1, 3), (1, 3), (1, 3)}

The vertical line test can be used to determine whether a graph is the graph of a function.

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Example 3

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A Beginning Look at Functions

Identifying a Function Which of the following graphs represents the graph of a function? (a)

(b)

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8 6 4 2 2

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(c)

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y 8 6 4 2 8 6 4 2 2

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4 6 8

Part a is not a function, part b is a function, and part c is a function. ✓ CHECK YOURSELF 3 ■ Which of the following graphs represents the graph of a function? (a)

(b)

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Section 4.3 (c)

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Identifying Functions

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y 8 6 4 2 8 6 4 2 2

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4 6 8

Example 4

Identifying a Function Which of the following graphs represents the graph of a function? (a)

(b)

y

x

Curves, like the number line, are made up of a continuous set of points.

(c)

y

x

y

x

Part a is not a function; it does not pass the vertical line test. Part b is a function because it passes the vertical line test. Part c is not a function.

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A Beginning Look at Functions ✓ CHECK YOURSELF 4 ■ Which of the following graphs represents the graph of a function? (a)

(b)

y

x

(c)

y

x

y

x

We used the term function several times in this chapter. We identified functions, looked at a function machine, used function notation, and found the domain and range for a function. But how does all this relate to the equations in two variables that we studied before this chapter? When is y the same as f(x)? Anytime we solved a linear equation for y, such as y  3x  2 y was a function of x. The x is considered the independent variable, and the y is considered the dependent variable. This means that y changes because x has changed. Let’s look at some examples of variables that are related and determine which is the dependent variable.

Section 4.3

Example 5

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Identifying Functions

303

Identifying the Dependent Variable From each pair, identify which variable is dependent on the other. (a) The age of a car and its resale value. The value depends on the age, so we would assign the age of the car the independent variable (x) and the value the dependent variable (y). (b) The amount of interest earned in a bank account and the amount of time the money has been in the bank.

If you think about it, you will see that time will be the independent variable in most ordered pairs. Most everything depends on time rather than the reverse.

The interest depends on the time, so interest is the dependent variable (y) and time is the independent variable (x). (c) The number of cigarettes one has smoked and the chance of dying from a smoking-related disease. The number of cigarettes is the independent variable (x), and the chance of dying from a smoking-related disease is the dependent variable (y). ✓ CHECK YOURSELF 5 ■ From each pair, identify which variable is dependent on the other. (a) The number of credits taken and the amount of tuition paid. (b) The temperature of a cup of coffee and the length of time since it was poured.

✓ CHECK YOURSELF ANSWERS ■ 1.

(a) Is a function; (b) is a function; (c) is not a function.

2.

(a) Is a function; (b) is not a function.

3.

(a) Is a function; (b) is not a function; (c) is a function.

4.

(a) Is not a function; (b) is a function; (c) is not a function.

5.

(a) Tuition is dependent on credits taken; (b) the temperature is dependent on the time since the coffee was poured.

E xercises 1. Function 2. Function 3. Function 4. Function 5. Not a function

4.3

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In Exercises 1 to 8, determine which of the relations are also functions. 1. {(1, 6), (2, 8), (3, 9)}

2. {(2, 3), (3, 4), (5, 9)}

3. {(1, 4), (2, 5), (3, 7)}

4. {(2, 1), (3, 4), (4, 6)}

5. {(1, 3), (1, 2), (1, 1)}

6. {(2, 4), (2, 5), (3, 6)}

7. {(1, 1), (2, 1), (2, 3)}

8. {(2, 1), (3, 4), (3, 1)}

6. Not a function 7. Not a function 8. Not a function

In Exercises 9 to 14, decide whether the relation is a function in each table of values. 9.

9. Function 10. Function 11. Not a function 12. Not a function 13. Function 14. Function 15. Function

12.

x

y

3 2 5 7

1 4 3 4

x

y

1 3 1 2

5 6 5 9

10.

13.

x

y

2 1 5 2

3 4 6 1

x

y

1 3 6 9

2 6 2 4

11.

14.

x

y

2 4 2 6

3 2 5 3

x

y

4 2 7 3

6 3 1 6

16. Function

In Exercises 15 to 20, for each set of ordered pairs, plot the related points. Then use the vertical line test to determine which sets are functions. 15. {(3, 1), (1, 2), (2, 3), (1, 4)}

16. {(2, 2), (1, 1), (3, 3), (4, 5) y

y 8

8

6

6

4

4 2

2 8 6 4 2 2

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8 6 4 2 2

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Section 4.3 17. Function

17. {(1, 1), (2, 2), (3, 4), (5, 6)}

18. Function

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18. {(1, 4), (1, 5), (0, 2), (2, 3)}

y

19. Not a function 20. Not a function 21. Function

y

8

8

6

6

4

4

2

22. Not a function

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Identifying Functions

2

8 6 4 2 2

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8 6 4 2 2

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19. {(1, 2), (1, 3), (2, 1), (3, 1)}

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20. {(1, 1), (3, 4), (1, 2), (5, 3)}

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2 8 6 4 2 2

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2 2

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8 6 4 2 2

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For Exercises 21 to 28, use the vertical line test to determine whether the graphs represent a function. 21.

22.

y

x

y

x

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23. Not a function

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A Beginning Look at Functions 23.

24.

y

y

24. Not a function 25. Function 26. Function 27. Function

x

x

28. Function

25.

26.

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27.

x

28.

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Section 4.3

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Identifying Functions

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29. Consider the following graph. y

(a) Identify the domain and range of the relation whose graph is given.

29. (a) D: 2  x  2; R: 1  y  2 29. (b) Yes 29. (c) Answers will vary 30. Independent: length of call; dependent: amount of bill

(b) Does this graph represent a function? Explain your answer. x

(c) How do you use the graph to determine the domain and range of the relation it represents?

31. Independent: size of tank; dependent: cost 32. Independent: time in air; dependent: height of ball 33. Independent: length of time; dependent: amount of penalty 34. Independent: number of credits; dependent: time to graduate 35. Independent: length of winter; dependent: amount of snowfall

In Exercises 30 to 35, from each pair, identify which variable is dependent and which is independent. 30. The amount of a phone bill and the length of the call. 31. The cost of filling a car’s gas tank and the size of the tank. 32. The height of a ball thrown in the air and the time in the air. 33. The amount of penalty on an unpaid tax bill and the length of the time unpaid. 34. The length of time needed to graduate from college and the number of credits taken per semester. 35. The amount of snowfall in Boston and the length of the winter. 36. Are all relations functions? Are all functions relations? Explain your answer.

36. Not every relation is a function, but every function is a relation.

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A Beginning Look at Functions 37. The following table shows the average hourly earnings for blue-collar workers from 1947 to 1993. These figures are given in “real” wages, which means that the purchasing power of the money is given rather than the actual dollar amount. In other words, the amount earned for 1947 is not the actual amount listed here; in fact, it was much lower. The amount you see here is the amount in dollars that 1947 earnings could buy in 1947 compared to what 1993 wages could buy in 1993.

Year

Average Hourly Earnings (in 1993 dollars)

1947 1967 1973 1979 1982 1989 1991 1993

$ 6.75 10.67 12.06 12.03 11.61 11.26 10.95 10.83

Make a Cartesian coordinate graph of this data, using the year as the domain and the hourly earnings as the range. You will have to decide how to set up the axes so that the data all fit on the graph nicely. (Hint: Do not start the year at 0!) In complete sentences, answer the following questions: What are the trends that you notice from reading the table? What additional information does the graph show? Is this relation a function? Why or why not?