297 2.8 Function Operations and Composition 2.8 Function Operations and Composition ■ Arithmetic Operations on Functions ■ The Difference Quotient ■ Composition of Functions and Domain Arithmetic Operations on Functions Figure 92 shows the situation for a company that manufactures DVDs. The two lines are the graphs of the linear functions for revenue R1x2 = 168x and cost C1x2 = 118x + 800, where x is the number of DVDs produced and sold, and x, R1x2, and C1x2 are given in thousands. When 30,000 (that is, 30 thousand) DVDs are produced and sold, profit is found as follows. P1x2 = R1x2 - C1x2 Profit function P1302 = R1302 - C1302 Let x = 30 . P1302 = 5040 - 4340 R1302 = 1681302; C1302 = 1181302 + 800 P1302 = 700 Subtract. Thus, the profit from the sale of 30,000 DVDs is $700,000. The profit function is found by subtracting the cost function from the revenue function. New functions can also be formed using other operations. Operations on Functions and Domains Given two functions ƒ and g, then for all values of x for which both ƒ1x2 and g1x2 are defined, the functions ƒ + g, ƒ - g, ƒg, and ƒ g are defined as follows. 1 ƒ +g2 1x2 =ƒ1x2 +g1x2 Sum function 1ƒ −g2 1x2 =ƒ1x2 −g1x2 Difference function 1ƒg2 1x2 =ƒ1x2 # g1x2 Product function a ƒ g b 1 x2 = ƒ1x2 g1x2 , g1x2 30 Quotient function The domains of ƒ +g, ƒ −g, and ƒg include all real numbers in the intersection of the domains of ƒ and g, while the domain of ƒ g includes those real numbers in the intersection of the domains of f and g for which g1x2 ≠0. 10 20 30 40 2000 4000 6000 x y P(30) C(30) C(x) R(30) R(x) Dollars (in thousands) DVDs (in thousands) DVD Production Figure 92 NOTE The condition g1x2 ≠0 in the definition of the quotient function means that the domain of A ƒ gB1x2 is restricted to all values of x for which g1x2 is not 0. The condition does not mean that g1x2 is a function that is never 0.
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