1. 1 © 2010 Pearson Education, Inc. All rights reserved © 2010 Pearson Education, Inc. All rights reserved Chapter 2 Graphs and Functions

2. OBJECTIVES © 2010 Pearson Education, Inc. All rights reserved 2 Combining Functions; Composite Functions SECTION 2.8 1 2 3 4 Learn basic operations on functions. Form composite functions. Find the domain of a composite function. Decompose a function. Apply composition to practical problems. 5

3. 3 © 2010 Pearson Education, Inc. All rights reserved 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 (except for where all values x where g(x) = 0 must be excluded). These functions are defined as follows:

4. 4 © 2010 Pearson Education, Inc. All rights reserved SUM, DIFFERENCE, PRODUCT, AND QUOTIENT OF FUNCTIONS (iv)Quotient (i)Sum (ii)Difference (iii)Product

5. 5 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 1 Combining Functions Let Find each of the following functions.

6. 6 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 1 Combining Functions Solution

7. 7 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 1 Combining Functions Solution continued

8. 8 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 1 Combining Functions Solution continued

9. 9 © 2010 Pearson Education, Inc. All rights reserved 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. 10 © 2010 Pearson Education, Inc. All rights reserved COMPOSITION OF FUNCTIONS

11. 11 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 2 Evaluating a Composite Function Let Find each of the following. Solution

12. 12 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 2 Evaluating a Composite Function Solution continued

13. 13 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 3 Finding Composite Functions Let Find each composite function. Solution

14. 14 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 3 Finding Composite Functions Solution continued

15. 15 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 4 Finding the Domain of a Composite Function Solution Domain is (–∞, 0) U (0, ∞).

16. 16 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 4 Finding the Domain of a Composite Function Solution continued Domain is (–∞, –1) U (–1, ∞).

17. 17 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 5 Decomposing a Function Solution Step 1 Define g(x) as any expression in the defining equation for H. Let g(x) = 2x 2 + 1.

18. 18 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 5 Decomposing a Function Solution continued Step 2 Replace the letter H with f and replace the expression chosen for g(x) with x. becomes Step 3 Now we have:

19. 19 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 6 Calculating the Area of an Oil Spill from a Tanker Oil is spilled from a tanker into the Pacific Ocean and the area of the oil spill is a perfect circle. The radius of this oil slick increases 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 six hours.

20. 20 © 2010 Pearson Education, Inc. All rights reserved 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. 21 © 2010 Pearson Education, Inc. All rights reserved 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.

22. 22 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 7 Applying Composition to Sales a.Write a function r (x) that represents the price after the rebate. b.Write a function d(x) that represents the price after the dealer’s discount. c.Write the functions What do they represent? d.Calculate Interpret this expression.

23. 23 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 7 Applying Composition to Sales Solution a.The MSRP is x dollars, rebate is $4000, so r (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 d(x) = x – 0.08x = 0.92x represents the price of the car after the dealer’s discount.

24. 24 © 2010 Pearson Education, Inc. All rights reserved 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.

25. 25 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 7 Applying Composition to Sales Solution continued This equation shows that it will cost $320 more for any car, regardless of its price, if you apply the rebate first and then the discount.