Section 1: Introduction to Functions

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

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• davidvictor
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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: 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.
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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• Set Theory, Prime number, Euler, Johann Carl Friedrich Gauss

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