#### Transcript Chapter 2 Section 8 - Canton Local Schools

Chapter 2 Graphs and Functions © 2010 Pearson Education, Inc. All rights reserved © 2010 Pearson Education, Inc. All rights reserved 1 SECTION 2.8 Combining Functions; Composite Functions OBJECTIVES 1 2 3 4 5 Learn basic operations on functions. Form composite functions. Find the domain of a composite function. Decompose a function. Apply composition to practical problems. © 2010 Pearson Education, Inc. All rights reserved 2 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 f quotient are functions whose domains g consist of those values of x that are common to f the domains of f and g (except for where all g values x where g(x) = 0 must be excluded). These functions are defined as follows: © 2010 Pearson Education, Inc. All rights reserved 3 SUM, DIFFERENCE, PRODUCT, AND QUOTIENT OF FUNCTIONS (i) Sum f g x f x g x (ii) Difference f g x f x g x (iii) Product fg x f x g x (iv) Quotient f x f g x g x , g x 0. © 2010 Pearson Education, Inc. All rights reserved 4 EXAMPLE 1 Combining Functions 2 f x x 6 x 8, and g x x 2. Let Find each of the following functions. f a. f g 4 b. 3 c. f g x g d. f g x e. fg x © 2010 Pearson Education, Inc. All rights reserved f f. x g 5 EXAMPLE 1 Solution Combining Functions f x x 6x 8, and g x x 2. 2 © 2010 Pearson Education, Inc. All rights reserved 6 EXAMPLE 1 Combining Functions Solution continued f x x 2 6x 8, and g x x 2. c. f g x f x g x x 6 x 8 x 2 2 x 5x 6 2 d. f g x f x g x x2 6 x 8 x 2 x 2 7 x 10 © 2010 Pearson Education, Inc. All rights reserved 7 EXAMPLE 1 Combining Functions Solution continued f x x 6x 8 and g x x 2 2 e. 2 fg x x 6 x 8 x 2 x3 2 x 2 6 x 2 12 x 8 x 16 x3 8 x 2 20 x 16 f x f x 6x 8 f. x g x x2 g x 2 x 4 x 4, x 2 x2 2 © 2010 Pearson Education, Inc. All rights reserved 8 COMPOSITION OF FUNCTIONS If f and g are two functions, the composition of function f with function g is written as f g and is defined by the equation f g x f g x , where the domain of f g consists of those values x in the domain of g for which g(x) is in the domain of f. © 2010 Pearson Education, Inc. All rights reserved 9 COMPOSITION OF FUNCTIONS © 2010 Pearson Education, Inc. All rights reserved 10 EXAMPLE 2 Evaluating a Composite Function 3 f x x and g x x 1. Let Find each of the following. a. f g 1 Solution a. f b. g f 1 c. f f 1 g 1 f g 1 f 2 2 3 8 © 2010 Pearson Education, Inc. All rights reserved 11 EXAMPLE 2 Evaluating a Composite Function Solution continued f x x 3 and g x x 1 b. g f 1 g f 1 g 1 1 1 2 c. f f 1 f f 1 f 1 1 1 3 © 2010 Pearson Education, Inc. All rights reserved 12 EXAMPLE 3 Finding Composite Functions 2 f x 2x 1 and g x x 3. Let Find each composite function. a. f g x b. g f x c. f Solution a. f f x g x f g x f x 2 3 2 x 3 1 2 2x 6 1 2 2x 5 2 © 2010 Pearson Education, Inc. All rights reserved 13 EXAMPLE 3 Finding Composite Functions Solution continued f x 2x 1 and g x x 2 3. b. g f x g f x g 2 x 1 2 x 1 3 4 x 2 4 x 2 2 c. f f x f f x f 2 x 1 2 2 x 1 1 4 x 3 © 2010 Pearson Education, Inc. All rights reserved 14 EXAMPLE 4 Finding the Domain of a Composite Function 1 Let f x x 1 and g x . x a. Find f g x and its domain. b. Find g f x and its domain. Solution 1 1 a. f g x f g x f 1 x x Domain is (–∞, 0) U (0, ∞). © 2010 Pearson Education, Inc. All rights reserved 15 EXAMPLE 4 Finding the Domain of a Composite Function Solution continued 1 f x x 1 and g x x 1 b. g f x g f x g x 1 x 1 Domain is (–∞, –1) U (–1, ∞). © 2010 Pearson Education, Inc. All rights reserved 16 EXAMPLE 5 Decomposing a Function 1 Write H x 2x 1 functions f and g. 2 as f g for some Solution Step 1 Define g(x) as any expression in the defining equation for H. Let g(x) = 2x2 + 1. © 2010 Pearson Education, Inc. All rights reserved 17 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: © 2010 Pearson Education, Inc. All rights reserved 18 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. © 2010 Pearson Education, Inc. All rights reserved 19 EXAMPLE 6 Calculating the Area of an Oil Spill from a Tanker Solution The area of the oil slick is a function its radius. 2 A f r r The radius is a function time: increasing 2 mph r g t 2t a. The area is a composite function 2 2 A f g t f 2t 2t 4 t . b. Substitute t = 6. 2 A 4 6 4 36 144 square miles. The area of the oil slick is 144π square miles. © 2010 Pearson Education, Inc. All rights reserved 20 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. © 2010 Pearson Education, Inc. All rights reserved 21 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 r d x and d r x . What do they represent? d. Calculate d r x r d x . Interpret this expression. © 2010 Pearson Education, Inc. All rights reserved 22 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. © 2010 Pearson Education, Inc. All rights reserved 23 EXAMPLE 7 Applying Composition to Sales Solution continued c. (i) r d x r d x r 0.92 x 0.92 x 4000 represents the price when the dealer’s discount is is applied first. (ii) d r x d r x d x 4000 0.92 x 4000 0.92 x 3680 represents the price when the manufacturer’s rebate is applied first. © 2010 Pearson Education, Inc. All rights reserved 24 EXAMPLE 7 Applying Composition to Sales Solution continued d. d r x r d x d r x r d x 0.92 x 3680 0.92 x 4000 320 dollars 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. © 2010 Pearson Education, Inc. All rights reserved 25