6.10 Functions and Their Graphs 387 Nikita wishes to download songs from iTunes and each song costs $1.29. Thus, one song costs 1 $1.29, × or $1.29. Two songs cost 2 $1.29, × or $2.58. Three songs cost 3 $1.29, × or $3.87, and so on. The relationship between two quantities—in this case, the number of songs downloaded and the cost of the songs—will be discussed in this section. Why This Is Important We will introduce the concept of function, an extremely important mathematical concept. Examples of functions in this section involve weekly salary for sales representatives, estimating the height of a spacecraft above the surface of the moon, estimating the age of fossils, estimating the growth of bacteria, and estimating the population of earth in the future. Knowledge of functions is a crucial aspect of mathematical understanding and is useful in many areas of modern society. Functions and Their Graphs SECTION 6.10 LEARNING GOAL Upon completion of this section, you will be able to: 7 Understand relations and functions. 7 Understand and graph linear functions. 7 Understand and graph quadratic functions. 7 Understand and graph exponential functions. 7 Solve problems involving natural exponential functions. Relations and Functions We begin with a discussion of relations. A relation is any set of ordered pairs. Every graph consists of a set of points, and each point corresponds to an ordered pair. Therefore, every graph represents a relation. The following table indicates the relation between the number of songs purchased and the cost of the songs discussed in the section opening paragraph. Number of Songs Cost ($) 0 0.00 1 1.29 2 2.58 3 3.87 10 12.90 In general, the cost for purchasing n songs will be $1.29 times the number of songs, or n 1.29 dollars. We can represent the cost, c, of n songs by the equation = c n 1.29 . Since the value of c depends on the value of n, we refer to c as the dependent variable and n as the independent variable. Note that for each value of the independent variable, n, there is one and only one value of the dependent variable, c. An equation such as = c n 1.29 is called a function. In the equation = c n 1.29 , the value of c depends on the value of n, so we say that “c is a function of n.” Iofoto/Shutterstock
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