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Section 1: Introduction to Functions

Elementary and Intermediate Algebra: Graphs & Models (3rd Edition)

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Section 2.1 Introduction to Functions 73 Version: Fall 2007 2.1 Introduction to Functions Our development of the function concept is a modern one, but quite quick, particularly in light of the fact that today’s definition took over 300 years to reach its present state. We begin with the definition of a relation. Relations x y 5 5 (2 , 4) (4 , 2) Figure 1. We use the notation (2 , 4) to denote what is called an ordered pair . If you think of the positions taken by the ordered pairs (4 , 2) and (2 , 4) in the coordinate plane (see Figure 1 ), then it is immediately apparent why order is important. The ordered pair (4 , 2) is simply not the same as the ordered pair (2 , 4) . The first element of an ordered pair is called its abscissa . The second element of an ordered pair is called its ordinate . Thus, for example, the abscissa of (4 , 2) is 4, while the ordinate of (4 , 2) is 2. Definition 1. A collection of ordered pairs is called a relation . For example, the collection of ordered pairs R = (0 , 1) , (0 , 2) , (3 , 4) (2) is a relation. Definition 3. The domain of a relation is the collection of all abscissas of each ordered pair. Thus, the domain of the relation R in ( 2 ) is Domain = { 0 , 3 } . Note that we list each abscissa only once. Definition 4. The range of a relation is the collection of all ordinates of each ordered pair. Thus, the range of the relation R in ( 2 ) is Range = { 1 , 2 , 4 } . Let’s look at an example. Copyrighted material. See: http://msenux.redwoods.edu/IntAlgText/ 1
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74 Chapter 2 Functions Version: Fall 2007 l⚏ Example 5. Consider the relation T defined by T = (1 , 2) , (3 , 2) , (4 , 5) . (6) What are the domain and range of this relation? The domain is the collection of abscissas of each ordered pair. Hence, the domain of T is Domain = { 1 , 3 , 4 } . The range is the collection of ordinates of each ordered pair. Hence, the range of T is Range = { 2 , 5 } . Note that we list each ordinate in the range only once. In Example 5 , the relation is described by listing the ordered pairs. This is not the only way that one can describe a relation. For example, a graph certainly represents a collection of ordered pairs. l⚏ Example 7. Consider the graph of the relation S shown in Figure 2 . x y 5 5 Figure 2. The graph of a relation. What are the domain and range of the relation S ? There are five ordered pairs (points) plotted in Figure 2 . They are S = (1 , 2) , (2 , 1) , (2 , 4) , (3 , 3) , (4 , 4) . Therefore, the relation S has Domain = { 1 , 2 , 3 , 4 } and Range = { 1 , 2 , 3 , 4 } . In the case of the range, note how we’ve sorted the ordinates of each ordered pair in ascending order, taking care not to list any ordinate more than once.
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Section 2.1 Introduction to Functions 75 Version: Fall 2007 Functions A function is a very special type of relation. We begin with a formal definition. Definition 8. A relation is a function if and only if each object in its domain is paired with one and only one object in its range.
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