What you should learn

To identify the domain, range, and inverse of a relation

To show relations as sets of ordered pairs, tables, mappings, and graphs.

NCTM Curriculm Standards 2, 3, 6-10

 

In doing this the teacher wants to make sure that the following words are incorporated into the introductory lesson:

Domain

Range

Mapping

Inverse

Introduction: To begin this lesson we will start with the following example. (CONNECTION: BIOLOGY)

There are 1.4 million classifid speccies of microorganisms, invertebrates, plants, fish, birds, reptiles, amphibians, and mammals. Recently, 930 species worldwide were listed as endangered. A species becomes endangered when its numbers are so low that the species is in danger of extinction.

In the United States, the manatee, an aquatic mammal, is considered to be endangered. In the table below, you will find the number of manatees that have been found dead since 1981.

Manatee Mortalit In Florida
Year	1981	1982	1983	1984	1985	1986	1987	1988	1989
Number of Manatees	116	114	81	128	119	122	114	133	168
Year	1990	1991	1992	1993	1994	1995	1996	1997	1998
Number of Manatees	206	174	163	145	193	201	415	242	231

The manatee data could be represented by a set of ordered pairs, as shown in the list below. Each first coordinate is the year, and the second coordinate is the number of manatees. Each ordered pair can then be graphed.

These points that were ploted above and in the chart, can be written as an ordered pair. For example the first one is (1981, 116). This can be done with all the points given.

Now remember that a relation is a set of ordered pairs, like the one shown above. The set of the first coordinates in the ordered pairs is called the domain. The domin usually contains the x-coordinates while the y-coordinates are usually contained with the range.

For the example above we can list the the domain and range as a set of values:

Domain = {1981, 1982, 1983, 1984, 1985, 1986, 1987, 1988, 1989, 1990, 1991, 1992, 1993, 1994, 1995, 1996, 1997, 1998}

Range = {116, 114, 81, 128, 119, 122, 114, 133, 168, 206, 174, 163, 145, 193, 201, 415, 242, 231}

In addition to ordered pairs, tables, and graphs, a relation can be represented by a mapping. A mapping illustrates how each element of the domain is paired with an element in the range. For example, the relation {(3, 3), (-1, 4), (0, -4)} can be modeled in each of the following ways:

ORDERED PAIR	TABLE	GRAPH	MAPPING
(3, 3) (-1, 4) (0, -4)	x  y  3  3  -1  4  0  -4

Exercise 1: Represent the relation shown in the graph below as

A. A set of ordered pairs

B. A table

C. A mapping

D. Then determine the domain and range

Exercise 2: To protect endangered species, there is a limit to the number of animals that can be exported to other countries. The table below shows yearly quotas for the number of wild crocodiles that could be exported from Indonesia.

Maximum Exportation of Crocodiles from Indonesia
Year	1986	1987	1988	1989	1990	1991	1992	1993	1994
Number of Crocodiles	2000	2000	4000	4000	3000	3000	2700	1500	1500

 

A. Determine the domain and range of the relation

B. Graph the data

C. What conclusions might you make from the graph of the data?

More Definitions: The inverse of any relation is obtained by switching the coordinates in each ordered pair.

Notice that the domain of the relation becomes the range of the inverse, and vice versa.

Exercise 3: Express the relation shown in the mapping as a set of ordered pairs. Write the inverse of the relation and draw a mapping to model the inverse.

Activity: Modeling Matematics: Relations and Inverses

MATERIALS: Graph paper, ruler, and colored pencils

STEP 1: Graph the relation {(4, 5), (-3, 6), (-5, -3), (4, -7), (5, 0), (0, -3)} on a coordinate plane using one color of pencil. Connect the points in order using the same color pencil.

STEP 2: Use a different color pencil to graph the inverse of the relation, connecting its points in order.

STEP 3: Fold the graph paper through the origin so that the positive y-axis lies on top of the positive x-axis. Hold the paper up to a light so that you can see all of the points you graphed.

 

Questions:

1. What do you notice about the location of the points you graphed when you looked at the folded paper?

2. Unfold the paper. Describe the pattern formed by the lines connecting the points in the relation and its inverse.

3. What do you think are the ordered pairs that represent the points on the folded line? Describe these in terms of x and y.

4. How could you graph the inverse of a function without writing the ordered pairs first?

 

Further Application: See page 269 in the student text. Working on the Investigation (Go Fish).

Closing Activity: Check for understanding by using this as a quick review before class is over. It should take about the last five to ten minutes. I would use it for my students as their 'ticket out the door'. Click Here.

Homework: The homework to be assigned for tonight would be: 17-41 odd, 42, 43, 45, 46-53

Alternative Homework: Enriched: 16-40 even, 42-53

Extra Practice: Students book page 767 Lesson 5-2

Extra Practice Worksheet: Click Here.

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