8.4. If we let x denote the number of gallons pumped, then the price y in dollars can $ $1.70 $ $1.70 $ $1.70 $ $1.


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1 8.4 An Introduction to Functions: Linear Functions, Applications, and Models We often describe one quantit in terms of another; for eample, the growth of a plant is related to the amount of light it receives, the demand for a product is related to the price of the product, the cost of a trip is related to the distance traveled, and so on. To represent these corresponding quantities, we can use ordered pairs. For eample, suppose that it is time to fill up our car s tank with gasoline. At our local station, 89octane gas is selling for $1.7 per gallon. Eperience has taught ou that the final price ou pa is determined b the number of gallons ou bu multiplied b the price per gallon (in this case, $1.7). As ou pump the gas, two sets of numbers spin b: the number of gallons pumped and the price for that number of gallons. The table below uses ordered pairs to illustrate this situation. Number of Gallons Pumped Price for This Number of Gallons $. $1.7 $1.7 1$1.7 $3.4 2$1.7 $5.1 3$1.7 4 $6.8 4$1.7 If we let denote the number of gallons pumped, then the price in dollars can be found b the linear equation 1.7. Theoreticall, there are infinitel man
2 8.4 An Introduction to Functions: Linear Functions, Applications, and Models 423 ordered pairs, that satisf this equation, but in this application we are limited to nonnegative values for, since we cannot have a negative number of gallons. There also is a practical maimum value for in this situation, which varies from one car to another. What determines this maimum value? In this eample, the total price depends on the amount of gasoline pumped. For this reason, price is called the dependent variable, and the number of gallons is called the independent variable. Generalizing, if the value of the variable depends on the value of the variable, then is the dependent variable and the independent variable. Relations and Functions Independent variable ~ b b ~ Dependent variable, Since related quantities can be written using ordered pairs, the concept of relation can be defined as follows. Relation A relation is a set of ordered pairs. For eample, the sets F 1, 2, 2, 5, 3, 1 and G 4, 1, 2, 1, 2, both are relations. A special kind of relation, called a function, is ver important in mathematics and its applications. Function A function is a relation in which for each value of the first component of the ordered pairs there is eactl one value of the second component. Of the two eamples of a relation just given, onl set F is a function, because for each value, there is eactl one value. In set G, the last two ordered pairs have the same value paired with two different values, so G is a relation, but not a function. F 1,2, 2,5, (3, 1 a ~ ~~~ a ~ ~~ a Different values G 4, 1, 2,1, (2, a ~ ~~~ a Same values Function Not a function In a function, there is eactl one value of the dependent variable, the second component, for each value of the independent variable, the first component. This is what makes functions so important in applications. Another wa to think of a functional relationship is to think of the independent variable as an input and the dependent variable as an output. A calculator is an inputoutput machine, for eample. To find 8 2, we must input 8, press the squaring ke, and see that the output is 64. Inputs and outputs also can be determined from a graph or a table.
3 424 CHAPTER 8 Graphs, Functions, and Sstems of Equations and Inequalities A third wa to describe a function is to give a rule that tells how to determine the dependent variable for a specific value of the independent variable. Suppose the rule is given in words as the dependent variable is twice the independent variable. As an equation, this can be written 2. a a Dependent Independent variable variable EXAMPLE 1 Determine the independent and dependent variables for each of the following functions. Give an eample of an ordered pair belonging to the function. (a) The 2 Summer Olmpics medal winners in men s basketball were gold, United States, silver, France, bronze, Lithuania. (Source: espn.go.com/ol/summer/) Input: independent variable Output: dependent variable The independent variable (the first component in each ordered pair) is the tpe of medal; the dependent variable (the second component) is the recipient. An of the three ordered pairs could be given as an eample. (b) A calculator that finds square roots The independent variable (the input) is a nonnegative real number, since the square root of a negative number is not a real number. The dependent variable is the nonnegative square root. For eample, 81, 9 belongs to this function. (c) The graph in Figure 2, which shows the relationship between the number of gallons of water in a small swimming pool and time in hours GALLONS OF WATER IN A POOL AT TIME t Gallons g Hours t FIGURE 2
4 8.4 An Introduction to Functions: Linear Functions, Applications, and Models 425 U.S. Petroleum Imports Year Imports Source: U.S. Energ Department. The independent variable is time, in hours, and the dependent variable is the number of the gallons of water in the pool. One ordered pair is 25, 3. (d) The table of petroleum imports in millions of barrels per da for selected ears The independent variable is the ear; the dependent variable is the number of millions of barrels of petroleum. An eample of an ordered pair is 1994, 9.. (e) 3 4 The independent variable is, and the dependent variable is. One ordered pair is 13, 5. Domain and Range Domain and Range In a relation, the set of all values of the independent variable () is the domain; the set of all values of the dependent variable () is the range. EXAMPLE 2 Give the domain and range of each function in Eample 1. (a) The domain is the set of tpes of medals, {gold, silver, bronze}, and the range is the set of winning countries, {United States, France, Lithuania}. (b) Here, the domain is restricted to nonnegative numbers:,. The range also is,. (c) The domain is all possible values of t, the time in hours, which is the interval, 1. The range is the number of gallons at time t, the interval, 3. (d) The domain is the set of ears, 1994, 1995, 1996, 1997, 1998; the range is the set of imports (in millions of barrels per da), 9., 8.83, 9.4, 1.16, (e) In the defining equation (or rule), 3 4, can be an real number, so the domain is is a real number or,. Since ever real number can be produced b some value of, the range also is the set is a real number, or,. Graphs of Relations The graph of a relation is the graph of its ordered pairs. The graph gives a picture of the relation, which can be used to determine its domain and range. EXAMPLE 3 Give the domain and range of each relation. (a) ( 1, 1) (, 1) (1, 2) (4, 3) The domain is the set of values, 1,, 1, 4. The range is the set of values, 3, 1, 1, 2.
5 426 CHAPTER 8 Graphs, Functions, and Sstems of Equations and Inequalities (b) Domain Range The values of the points on the graph include all numbers between 4 and 4, inclusive. The  values include all numbers between 6 and 6, inclusive. Using interval notation, the domain is 4, 4; the range is 6, 6. (c) The arrowheads indicate that the line etends indefinitel left and right, as well as up and down. Therefore, both the domain and the range are the set of all real numbers, written,. (d) 3 2 The arrowheads indicate that the graph etends indefinitel left and right, as well as upward. The domain is,. Because there is a least  value, 3, the range includes all real numbers greater than or equal to 3, written 3,. Relations often are defined b equations, such as 2 3 and 2. It is sometimes necessar to determine the domain of a relation from its equation. In this book, the following agreement on the domain of a relation is assumed. Agreement on Domain The domain of a relation is assumed to be all real numbers that produce real numbers when substituted for the independent variable. To illustrate this agreement, since an real number can be used as a replacement for in 2 3, the domain of this function is the set of real numbers. As another eample, the function defined b 1 has all real numbers ecept as domain, since is undefined if. In general, the domain of a function defined b an algebraic epression is all real numbers, ecept those numbers that lead to division b or an even root of a negative number. Identifing Functions Most of the relations we have seen in the eamples are functions that is, each value corresponds to eactl one value. Now we look at was to determine whether a given relation, defined algebraicall, is a function.
6 8.4 An Introduction to Functions: Linear Functions, Applications, and Models 427 In a function each value of leads to onl one value of, so an vertical line drawn through the graph of a function must intersect the graph in at most one point. This is the vertical line test for a function. Vertical Line Test If a vertical line intersects the graph of a relation in more than one point, then the relation is not a function. For eample, the graph shown in Figure 21(a) is not the graph of a function, since a vertical line can intersect the graph in more than one point, while the graph in Figure 21(b) does represent a function. Not a function the same value corresponds to four different values Function each value corresponds to onl one value (a) FIGURE 21 (b) The vertical line test is a simple method for identifing a function defined b a graph. It is more difficult to decide whether a relation defined b an equation is a function. The net eample gives some hints that ma help EXAMPLE 4 Decide whether each of the following defines a function and give the domain. (a) 2 1 Here, for an choice of in the domain, there is eactl one corresponding value for (the radical is a nonnegative number), so this equation defines a function. Since the radicand cannot be negative, The graph of 2 1 supports the result in Eample 4(a). The domain is 1, 2. The domain is 1,
7 428 CHAPTER 8 Graphs, Functions, and Sstems of Equations and Inequalities (b) (c) 2 The ordered pairs 16, 4 and 16, 4 both satisf this equation. Since one value of, 16, corresponds to two values of, 4 and 4, this equation does not define a function. Solving 2 for gives or, which shows that two values of correspond to each positive value of. Because is equal to the square of, the values of must alwas be nonnegative. The domain of the relation is,. 1 B definition, is a function of if ever value of leads to eactl one value of. In this eample, a particular value of, sa 1, corresponds to man values of. The ordered pairs 1,, 1, 1, 1, 2, 1, 3, and so on, all satisf the inequalit. For this reason, the inequalit does not define a function. An number can be used for, so the domain is the set of real numbers,. (d) 5 1 Given an value of in the domain, we find b subtracting 1, then dividing the result into 5. This process produces eactl one value of for each value in the domain, so this equation defines a function. The domain includes all real numbers ecept those that make the denominator. We find these numbers b setting the denominator equal to and solving for. 1 1 Thus, the domain includes all real numbers ecept 1. In interval notation this is written as,1 1,. In summar, three variations of the definition of function are given here. Variations of the Definition of Function 1. A function is a relation in which for each value of the first component of the ordered pairs there is eactl one value of the second component. 2. A function is a set of ordered pairs in which no first component is repeated. 3. A function is a rule or correspondence that assigns eactl one range value to each domain value. Function Notation When a function f is defined with a rule or an equation using and for the independent and dependent variables, we sa is a function of to emphasize that depends on. We use the notation f,
8 8.4 An Introduction to Functions: Linear Functions, Applications, and Models 429 called function notation, to epress this and read f as f of. (In this notation the parentheses do not indicate multiplication.) The letter f stands for function. For eample, if 2 7, we write f() 2 7. Note that f is just another name for the dependent variable. For eample, if f 9 5, and 2, then we find, or f 2, b replacing with 2. f The statement if 2, then 13 is abbreviated with function notation as f Read f2 as f of 2 or f at 2. Also, f 9 5 5, and f These ideas and the smbols used to represent them can be eplained as follows. Name of the function Defining epression f 9 5 Value of the function Name of the independent variable For Y 1 9X 5, function notation capabilit of the TI83 Plus supports the discussion here. EXAMPLE 5 Let f Find the following. (a) f2 f f Replace with (b) f (c) f Replace with Linear Functions function. An important tpe of elementar function is the linear Linear Function A function that can be written in the form f() m b for real numbers m and b is a linear function.
9 43 CHAPTER 8 Graphs, Functions, and Sstems of Equations and Inequalities f() = Notice that the form f m b defining a linear function is the same as that of the slopeintercept form of the equation of a line, first seen in the previous section. We know that the graph of f m b will be a line with slope m and intercept, b. 1 (a) 3 f() = 3 (b) FIGURE EXAMPLE 6 Graph each linear function. (a) f 2 3 To graph the function, locate the intercept,, 3. From this point, use the slope to go down 2 and right 1. This second point is used to obtain the graph in Figure 22(a). (b) f 3 From the previous section, we know that the graph of 3 is a horizontal line. Therefore, the graph of f 3 is a horizontal line with intercept, 3 as shown in Figure 22(b). The function defined in Eample 6(b) and graphed in Figure 22(b) is an eample of a constant function. A constant function is a linear function of the form f b, where b is a real number. The domain of an linear function is,. The range of a nonconstant linear function (like in Eample 6(a)) is also,, while the range of the constant function f b is b. Cost and Revenue Models A compan s cost of producing a product and the revenue from selling the product can be epressed as linear functions. The idea of breakeven analsis then can be eplained using the graphs of these functions. When cost is greater than revenue earned, the compan loses mone; when cost is less than revenue, the compan makes mone; and when cost equals revenue, the compan breaks even. Compare with Eample 6(a) and Figure 22(a). 1 Compare with Eample 6(b) and Figure 22(b). 6 1 EXAMPLE 7 Peripheral Visions, Inc., produces studio qualit audiotapes of live concerts. The compan places an ad in a trade newsletter. The cost of the ad is $1. Each tape costs $2 to produce, and the compan charges $24 per tape. (a) Epress the cost C as a function of, the number of tapes produced. The fied cost is $1, and for each tape produced, the variable cost is $2. Therefore, the cost C can be epressed as a function of, the number of tapes produced: C 2 1 (C in dollars). (b) Epress the revenue R as a function of, the number of tapes sold. Since each tape sells for $24, the revenue R is given b R 24 (R in dollars). (c) For what value of does revenue equal cost? The compan will just break even (no profit and no loss) as long as revenue just equals cost, or R C. This is true whenever R C Substitute for R and C. If 25 tapes are produced and sold, the compan will break even.
10 8.4 An Introduction to Functions: Linear Functions, Applications, and Models = R() = = C() = The breakeven point is 25, 6, as indicated at the bottom of the screen. The calculator can find the point of intersection of the graphs. (d) Graph C 2 1 and R 24 on the same coordinate sstem, and interpret the graph. Figure 23 shows the graphs of the two functions. At the breakeven point, we see that when 25 tapes are produced and sold, both the cost and the revenue are $6. If fewer than 25 tapes are produced and sold (that is, when 25), the compan loses mone. When more than 25 tapes are produced and sold (that is, when 25), there is a profit. Cost (25, 6) Loss Profit R() = Tapes Produced and Sold FIGURE 23 C() = 2 + 1
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