Relations and Functions

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1 CHAPTER 5 Relations and Functions What You ll Learn Describe a relation in words and represent it using a set of ordered pairs, an arrow diagram, a table of values, and a graph. Find the domain and range of a relation and decide if the relation is a function. Interpret graphs that represent situations. Find the rate of change and the intercepts of the graph of a linear function. Why It s Important Relations and functions are used by: advertising specialists, to analyze trends in sales bank tellers, to convert Canadian dollars to foreign currency nurses, to relate a patient s mass to the drug dosage required Key Words relation arrow diagram domain range function dependent variable independent variable function notation rate of change linear function horizontal intercept vertical intercept Copyright 2011 Pearson Canada Inc. 249

2 5.1 Representing Relations FOCUS Represent relations in different ways. A set is a collection of objects. An element of a set is one object in the set. A relation associates the elements of one set with the elements of another set. We can represent a relation in different ways. Example 1 Representing a Relation Given as a Table Athletes can be associated with their sports. Athlete Jennifer Botterill Jennifer Jones Jeremy Wotherspoon Jonathan Cheechoo Sport Ice hockey Curling Speed skating Ice hockey The columns represent the sets in the relation. The heading of each column describes each set. a) Describe this relation in words. b) Represent this relation as a set of ordered pairs. c) Represent this relation as an arrow diagram. Solution a) The relation shows the association takes part in from a set of athletes to a set of sports. For example, Jennifer Botterill takes part in ice hockey. b) The ordered pairs are: The athletes are the 1st {(Jennifer Botterill, ice hockey), (Jennifer Jones, curling), elements in the ordered pairs (Jeremy Wotherspoon, speed skating), and the sports are the 2nd (Jonathan Cheechoo, ice hockey)} elements. c) The athletes are written in the 1st set of the arrow diagram. The sports are written in the 2nd set; each sport is written only once. takes part in Jennifer Botterill Jennifer Jones Jeremy Wotherspoon Jonathan Cheechoo curling ice hockey speed skating Each arrow associates an element in the 1st set with an element in the 2nd set. The order is important; a relation has direction from the 1st set to the 2nd set Copyright 2011 Pearson Canada Inc.

3 Check 1. Tourist attractions can be associated with the provinces or territories they are in. Tourist Attraction Butchart Gardens Icefields Parkway Royal Canadian Mint Stanley Park Province British Columbia Alberta Manitoba British Columbia a) Describe this relation in words. The relation shows the association from a set of to a set of. For example, b) Represent this relation as a set of ordered pairs. The ordered pairs are: c) Represent this relation as an arrow diagram. Butchart Gardens Icefields Parkway Royal Canadian Mint Stanley Park Alberta British Columbia Manitoba We usually list the elements of each set in alphabetical or numerical order. When the elements of one or both sets are numbers, the relation can be represented as a bar graph. Example 2 Representing a Relation Given as a Bar Graph Consider the relation represented by this graph. Represent the relation: 70 a) as a table b) as an arrow diagram 60 Maximum Speeds of Different Animals Maximum speed (km/h) Bison Cougar Coyote Grizzly bear White-tailed deer Animal Copyright 2011 Pearson Canada Inc. 251

4 Solution a) In the table, write the types of animals in the 1st column and estimates of the maximum speeds in kilometres per hour in the 2nd column. Animal Maximum Speed (km/h) Bison 35 Cougar 55 Coyote 69 Grizzly bear 48 White-tailed deer 48 b) The association is: has a maximum speed (km/h) of In the arrow diagram, write the types of animals in the 1st set and the maximum speeds in kilometres per hour in the 2nd set. has a maximum speed (km/h) of bison cougar coyote grizzly bear white-tailed deer Check 1. Consider the relation represented by this graph. Canada s Medal Count at the Summer Olympics Number of medals Year Copyright 2011 Pearson Canada Inc.

5 Represent the relation: a) as a table b) as an arrow diagram Year Number of Medals The association is: Practice 1. Consider the relation represented by this table. Capital City Yellowknife Iqaluit Whitehorse Victoria Edmonton Regina Winnipeg Province or Territory Northwest Territories Nunavut Yukon British Columbia Alberta Saskatchewan Manitoba a) Complete this sentence. The relation associates a set of with a set of b) Circle the best choice of association for the relation above. i) is the province of ii) is the capital city of iii) is a village in iv) is a territory in c) Circle the ordered pair that belongs to the relation above. i) (Edmonton, Saskatchewan) ii) (Nunavut, Iqaluit) iii) (Calgary, Alberta) iv) (Winnipeg, Manitoba) Copyright 2011 Pearson Canada Inc. 253

6 d) Draw an arrow diagram to represent the relation. Edmonton Alberta 2. Properties in the board game Monopoly can be associated with their colours. Property Atlantic Avenue Baltic Avenue Boardwalk Marvin Gardens Pacific Avenue Colour Yellow Purple Dark Blue Yellow Green a) Describe this relation in words. The relation shows the association from a set of to a set of For example, b) Represent this relation as a set of ordered pairs. The ordered pairs are: c) Represent this relation as an arrow diagram Copyright 2011 Pearson Canada Inc.

7 3. Consider the relation represented by this graph. Number of Vowels in a Word Number of vowels Aquarium Audio Banana Evacuate Oodles Word a) Represent the relation as a table. Word Number of Vowels b) Describe the relation in words. The relation shows the association c) List 2 ordered pairs that belong to the relation. Two ordered pairs are: 4. Here are some Canadian prime ministers and the year they died. (Pierre Trudeau, 2000); (John Diefenbaker, 1979); (Lester B. Pearson, 1972); (Robert Borden, 1937); (John A. Macdonald, 1891) Use these data to represent each association below in the stated way. a) died in as a table Copyright 2011 Pearson Canada Inc. 255

8 b) is the year of death of as an arrow diagram c) Does it matter if you reverse the ovals in the arrow diagram? Explain. 5. a) Create a table of values to represent the association is the number of legs on for different types of animals. b) Represent the relation in two different ways Copyright 2011 Pearson Canada Inc.

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