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### Transcript of 9-4 Operations with Functions Holt Algebra2 Warm Up Warm Up Lesson Presentation Lesson Presentation...

• 9-4Operations with Functions Holt Algebra2Warm UpLesson PresentationLesson Quiz

• Warm UpSimplify. Assume that all expressions are defined.x2 x + 71. (2x + 5) (x2 + 3x 2)2. (x 3)(x + 1)23.x3 x2 5x 3

• Add, subtract, multiply, and divide functions. Write and evaluate composite functions.

Objectives

• composition of functions

Vocabulary

• You can perform operations on functions in much the same way that you perform operations on numbers or expressions. You can add, subtract, multiply, or divide functions by operating on their rules.

• Given f(x) = 4x2 + 3x 1 and g(x) = 6x + 2, find each function.Example 1A: Adding and Subtracting Functions(f + g)(x)Substitute function rules.(f + g)(x) = f(x) + g(x) Combine like terms.= (4x2 + 3x 1) + (6x + 2)= 4x2 + 9x + 1

• Given f(x) = 4x2 + 3x 1 and g(x) = 6x + 2, find each function.Example 1B: Adding and Subtracting Functions(f g)(x)Substitute function rules.(f g)(x) = f(x) g(x) Combine like terms.= (4x2 + 3x 1) (6x + 2)= 4x2 + 3x 1 6x 2Distributive Property= 4x2 3x 3

• Given f(x) = 5x 6 and g(x) = x2 5x + 6, find each function.(f + g)(x)Substitute function rules.(f + g)(x) = f(x) + g(x) Combine like terms.= (5x 6) + (x2 5x + 6)= x2 Check It Out! Example 1a

• (f g)(x)Substitute function rules.(f g)(x) = f(x) g(x) Combine like terms.= (5x 6) (x2 5x + 6)= 5x 6 x2 + 5x 6Distributive Property= x2 + 10x 12Check It Out! Example 1b Given f(x) = 5x 6 and g(x) = x2 5x + 6, find each function.

• When you divide functions, be sure to note any domain restrictions that may arise.

• Example 2A: Multiplying and Dividing Functions(fg)(x)Substitute function rules.Multiply.Combine like terms.Distributive Property= (6x2 x 12) (2x 3)Given f(x) = 6x2 x 12 and g(x) = 2x 3, find each function.(fg)(x) = f(x) g(x) = 6x2 (2x 3) x(2x 3) 12(2x 3)= 12x3 18x2 2x2 + 3x 24x + 36= 12x3 20x2 21x + 36

• Set up the division as a rational expression.Divide out common factors.Simplify.Example 2B: Multiplying and Dividing Functions

• Given f(x) = x + 2 and g(x) = x2 4, find each function.(fg)(x)Substitute function rules.(fg)(x) = f(x) g(x) = (x + 2)(x2 4)= x3 + 2x2 4x 8Multiply.Check It Out! Example 2a

• Set up the division as a rational expression.Divide out common factors.Simplify.Factor completely. Note that x 2. = x 2, where x 2 Check It Out! Example 2b

• Another function operation uses the output from one function as the input for a second function. This operation is called the composition of functions.

• The order of function operations is the same as the order of operations for numbers and expressions. To find f(g(3)), evaluate g(3) first and then substitute the result into f.

• Example 3A: Evaluating Composite FunctionsStep 1 Find g(4) g(x) = 7 xGiven f(x) = 2x and g(x) = 7 x, find each value.f(g(4))g(4) = 7 4 Step 2 Find f(3) = 3f(3) = 23= 8So f(g(4)) = 8.f(x) = 2x

• Example 3B: Evaluating Composite FunctionsStep 1 Find f(4) f(x) = 2xGiven f(x) = 2x and g(x) = 7 x, find each value.g(f(4))f(4) = 24 Step 2 Find g(16) = 16g(16) = 7 16= 9So g(f(4)) = 9.g(x) = 7 x.

• Step 1 Find g(3) g(x) = x2Given f(x) = 2x 3 and g(x) = x2, find each value.f(g(3))g(3) = 32 Step 2 Find f(9) = 9f(9) = 2(9) 3= 15So f(g(3)) = 15.f(x) = 2x 3Check It Out! Example 3a

• Step 1 Find f(3) f(x) = 2x 3 Given f(x) = 2x 3 and g(x) = x2, find each value.g(f(3))f(3) = 2(3) 3Step 2 Find g(3) = 3g(3) = 32= 9So g(f(3)) = 9.g(x) = x2Check It Out! Example 3b

• You can use algebraic expressions as well as numbers as inputs into functions. To find a rule for f(g(x)), substitute the rule for g into f.

• Example 4A: Writing Composite Functionsf(g(x))The domain of f(g(x)) is x 1 or {x|x 1} because g(1) is undefined. Use the rule for f. Note that x 1.Substitute the rule g into f.Simplify.

• Simplify. Note that x . The domain of g(f(x)) is x or {x|x } because f( ) = 1 and g(1) is undefined.g(f(x)) = g(x2 1) Example 4B: Writing Composite Functionsg(f(x))Use the rule for g.Substitute the rule f into g.

• f(g(x))Distribute. Note that x 0.Substitute the rule g into f.Simplify.Check It Out! Example 4a The domain of f(g(x)) is x 0 or {x|x 0}.

• g(f(x))Substitute the rule f into g.Check It Out! Example 4b

• Composite functions can be used to simplify a series of functions.

• A. Write a composite function to represent the total cost of a piece of furniture in dollars if the cost of the item is c pesos.Jake imports furniture from Mexico. The exchange rate is 11.30 pesos per U.S. dollar. The cost of each piece of furniture is given in pesos. The total cost of each piece of furniture includes a 15% service charge.Example 5: Business Application

• Step 1 Write a function for the total cost in U.S. dollars.Example 5 ContinuedP(c) = c + 0.15c= 1.15cStep 2 Write a function for the cost in dollars based on the cost in pesos.Use the exchange rate.

• Step 3 Find the composition D(P(c)).Example 5 ContinuedD(P(c)) = 1.15P(c) Substitute P(c) for c. Replace P(c) with its rule.B. Find the total cost of a table in dollars if it costs 1800 pesos.Evaluate the composite function for c = 1800. 183.19The table would cost \$183.19, including all charges.

• a. Write a composite function to represent the final cost of a kit for a preferred customer that originally cost c dollars. During a sale, a music store is selling all drum kits for 20% off. Preferred customers also receive an additional 15% off.Check It Out! Example 5 Step 1 Write a function for the final cost of a kit that originally cost c dollars.f(c) = 0.80cDrum kits are sold at 80% of their cost.

• Check It Out! Example 5 Continued Step 2 Write a function for the final cost if the customer is a preferred customer.Preferred customers receive 15% off.g(c) = 0.85c

• Step 3 Find the composition f(g(c)).f(g(c)) = 0.80(g(c))Substitute g(c) for c. Replace g(c) with its rule.Check It Out! Example 5 Continuedf(g(c)) = 0.80(0.85c)= 0.68cb. Find the cost of a drum kit at \$248 that a preferred customer wants to buy.Evaluate the composite function for c = 248.f(g(c) ) = 0.68(248)The drum kit would cost \$168.64.

• Lesson Quiz: Part IGiven f(x) = 4x2 1 and g(x) = 2x 1, find each function or value.1. (f + g)(x)4x2 + 2x 22. (fg)(x)8x3 4x2 2x + 13. 4. g(f(2))29

• Lesson Quiz: Part II5. f(g(x)) f(g(x)) = x 1;{x|x 1 or x 1}6. g(f(x)) {x|x 1}