1. Slide 2.7- 1 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

2. OBJECTIVES Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Combining Functions; Composite Functions Learn basic operations on functions. Learn to form composite functions. Learn to find the domain of a composite function. Learn to decompose a function Learn to apply composition to a practical problem. SECTION 2.7 1 2 3 4 5

3. Slide 2.7- 3 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley SUM, DIFFERENCE, PRODUCT, AND QUOTIENT OF FUNCTIONS Let f and g be two functions. The sum f + g, the difference f – g, the product fg, and the quotient are functions whose domains consist of those values of x that are common to the domains of f and g. These functions are defined as follows:

4. Slide 2.7- 4 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley SUM, DIFFERENCE, PRODUCT, AND QUOTIENT OF FUNCTIONS (iv)Quotient (i)Sum (ii)Difference (iii)Product

5. Slide 2.7- 5 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 1 Combining Functions Let Find each of the following functions. Solution

6. Slide 2.7- 6 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 1 Combining Functions Solution continued

7. Slide 2.7- 7 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 1 Combining Functions Solution continued

8. Slide 2.7- 8 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 1 Combining Functions Solution continued Since f and g are polynomials, the domain of f and g is the set of all real numbers, or, in interval notation (–∞, ∞). The domain for must exclude x = 2. Its domain is (–∞, 2) U (2, ∞). The domain for f +g, f – g, and fg is (–∞, ∞).

9. Slide 2.7- 9 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley COMPOSITION OF FUNCTIONS If f and g are two functions, the composition of function f with function g is written as and is defined by the equation where the domain of values x in the domain of g for which g(x) is in the domain of f. consists of those

10. Slide 2.7- 10 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley COMPOSITION OF FUNCTIONS

11. Slide 2.7- 11 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 2 Evaluating a Composite Function Let Find each of the following. Solution

12. Slide 2.7- 12 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 2 Evaluating a Composite Function Solution continued

13. Slide 2.7- 13 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 3 Finding Composite Functions Let Find each composite function. Solution

14. Slide 2.7- 14 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 3 Finding Composite Functions Solution continued

15. Slide 2.7- 15 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 4 Finding the Domain of a Composite Function Solution

16. Slide 2.7- 16 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 4 Finding the Domain of a Composite Function Solution continued Domain is (–∞, 0) U (0, ∞). Domain is (–∞, –1) U (–1, ∞).

17. Slide 2.7- 17 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 5 Decomposing a Function Show that each of the following provides a decomposition of H(x).

18. Slide 2.7- 18 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 5 Decomposing a Function Solution

19. Slide 2.7- 19 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 6 Calculating the Area of an Oil Spill from a Tanker Oil is spilled from a tanker into the Pacific Ocean. Suppose the area of the oil spill is a perfect circle. (In practice, this does not happen, because of the winds and tides and the location of the coastline.) Suppose that the radius of the oil slick is increasing (because oil continues to spill) at the rate of 2 miles per hour. a.Express the area of the oil slick as a function of time. b.Calculate the area covered by the oil slick in 6 hours.

20. Slide 2.7- 20 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 6 Calculating the Area of an Oil Spill from a Tanker Solution The area of the oil slick is a function its radius. The radius is a function time: increasing 2 mph a. The area is a composite function b. Substitute t = 6. The area of the oil slick is 144π square miles.

21. Slide 2.7- 21 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 7 Applying Composition to Sales A car dealer offers an 8% discount off the manufacturer’s suggested retail price (MSRP) of x dollars for any new car on his lot. At the same time, the manufacturer offers a $4000 rebate for each purchase of a car. a.Write a function f (x) that represents the price after the rebate. b.Write a function g(x) that represents the price after the dealer’s discount. c.Write the function What do they represent?

22. Slide 2.7- 22 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 7 Applying Composition to Sales d.Calculate Interpret this expression. Solution a.The MSRP is x dollars, rebate is $4000, so f (x) = x – 4000 represents the price of the car after the rebate. b.The dealer’s discount is 8% of x, or 0.08x, so g(x) = x – 0.08x = 0.92x represents the price of the car after the dealer’s discount.

23. Slide 2.7- 23 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 7 Applying Composition to Sales Solution continued represents the price when the dealer’s discount is is applied first. represents the price when the manufacturer’s rebate is applied first.

24. Slide 2.7- 24 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 7 Applying Composition to Sales Solution continued This equation shows that it will cost 320 dollars more for any car, regardless of its price, if you apply the rebate first and then the discount.