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Holt Algebra Operations with Functions 9-4 Operations with Functions Holt Algebra2 Warm Up Warm Up Lesson Presentation Lesson Presentation Lesson Quiz Lesson Quiz

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Holt Algebra Operations with Functions Warm Up Simplify. Assume that all expressions are defined. –x 2 – x (2x + 5) – (x 2 + 3x – 2) 2. (x – 3)(x + 1) 2 3. x 3 – x 2 – 5x – 3 x – 3 x – 2

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Holt Algebra Operations with Functions Add, subtract, multiply, and divide functions. Write and evaluate composite functions. Objectives

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Holt Algebra Operations with Functions composition of functions Vocabulary

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Holt Algebra Operations with Functions 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.

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Holt Algebra Operations with Functions

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Holt Algebra Operations with Functions Given f(x) = 4x 2 + 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. = (4x 2 + 3x – 1) + (6x + 2) = 4x 2 + 9x + 1

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Holt Algebra Operations with Functions Given f(x) = 4x 2 + 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. = (4x 2 + 3x – 1) – (6x + 2) = 4x 2 + 3x – 1 – 6x – 2 Distributive Property = 4x 2 – 3x – 3

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Holt Algebra Operations with Functions Given f(x) = 5x – 6 and g(x) = x 2 – 5x + 6, find each function. (f + g)(x) Substitute function rules. (f + g)(x) = f(x) + g(x) Combine like terms. = (5x – 6) + (x 2 – 5x + 6) = x 2 Check It Out! Example 1a

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Holt Algebra Operations with Functions (f – g)(x) Substitute function rules. (f – g)(x) = f(x) – g(x) Combine like terms. = (5x – 6) – (x 2 – 5x + 6) = 5x – 6 – x 2 + 5x – 6 Distributive Property = –x x – 12 Check It Out! Example 1b Given f(x) = 5x – 6 and g(x) = x 2 – 5x + 6, find each function.

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Holt Algebra Operations with Functions When you divide functions, be sure to note any domain restrictions that may arise.

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

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Holt Algebra Operations with Functions Set up the division as a rational expression. Divide out common factors. Simplify. ( ) (x)(x) f g (x)(x) f g f(x) f(x) g(x)g(x) = 6x 2 – x –12 2x – 3 = Factor completely. Note that x ≠. 3 2 = (2x – 3)(3x + 4) 2x – 3 = (2x – 3)(3x +4) (2x – 3) = 3x + 4, where x ≠ 3 2 Example 2B: Multiplying and Dividing Functions

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Holt Algebra Operations with Functions Given f(x) = x + 2 and g(x) = x 2 – 4, find each function. (fg)(x) Substitute function rules. (fg)(x) = f(x) ● g(x) = (x + 2)(x 2 – 4) = x 3 + 2x 2 – 4x – 8 Multiply. Check It Out! Example 2a

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Holt Algebra Operations with Functions Set up the division as a rational expression. Divide out common factors. Simplify. ( ) (x)(x) g f (x)(x) g f g(x) g(x) f(x)f(x) = Factor completely. Note that x ≠ – 2. = (x – 2)(x + 2) x + 2 = (x – 2)(x + 2) (x + 2) = x – 2, where x ≠ – 2 Check It Out! Example 2b = x 2 – 4 x + 2

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Holt Algebra Operations with Functions Another function operation uses the output from one function as the input for a second function. This operation is called the composition of functions.

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Holt Algebra Operations with 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.

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Holt Algebra Operations with Functions The composition (f o g)(x) or f(g(x)) is read “f of g of x.” Reading Math

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Holt Algebra Operations with Functions Be careful not to confuse the notation for multiplication of functions with composition fg(x) ≠ f ( g(x) ) Caution!

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Holt Algebra Operations with Functions Example 3A: Evaluating Composite Functions Step 1 Find g(4) g(x) = 7 – x Given f(x) = 2 x and g(x) = 7 – x, find each value. f(g(4)) g(4) = 7 – 4 Step 2 Find f(3) = 3 f(3) = 2 3 = 8 So f(g(4)) = 8. f(x) = 2 x

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Holt Algebra Operations with Functions Example 3B: Evaluating Composite Functions Step 1 Find f(4) f(x) = 2 x Given f(x) = 2 x and g(x) = 7 – x, find each value. g(f(4)) f(4) = 2 4 Step 2 Find g(16) = 16 g(16) = 7 – 16 = –9 So g(f(4)) = –9. g(x) = 7 – x.

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Holt Algebra Operations with Functions Step 1 Find g(3) g(x) = x 2 Given f(x) = 2x – 3 and g(x) = x 2, find each value. f(g(3)) g(3) = 3 2 Step 2 Find f(9) = 9 f(9) = 2(9) – 3 = 15 So f(g(3)) = 15. f(x) = 2x – 3 Check It Out! Example 3a

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Holt Algebra Operations with Functions Step 1 Find f(3) f(x) = 2x – 3 Given f(x) = 2x – 3 and g(x) = x 2, find each value. g(f(3)) f(3) = 2(3) – 3 Step 2 Find g(3) = 3 g(3) = 3 2 = 9 So g(f(3)) = 9. g(x) = x 2 Check It Out! Example 3b

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Holt Algebra Operations with Functions 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.

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Holt Algebra Operations with Functions Example 4A: Writing Composite Functions Given f(x) = x 2 – 1 and g(x) =, write each composite function. State the domain of each. x 1 – x f(g(x)) f(g(x)) = f( ) x 1 – x –1 + 2x (1 – x) 2 = = ( ) 2 – 1 x 1 – 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.

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Holt Algebra Operations with Functions 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(x 2 – 1) Example 4B: Writing Composite Functions g(f(x)) Use the rule for g. Substitute the rule f into g. = x 2 – 1 2 – x 2 (x 2 – 1) 1 – (x 2 – 1) = Given f(x) = x 2 – 1 and g(x) =, write each composite function. State the domain of each. x 1 – x

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Holt Algebra Operations with Functions f(g(x)) Distribute. Note that x ≥ 0. Substitute the rule g into f. Simplify. Check It Out! Example 4a Given f(x) = 3x – 4 and g(x) = + 2, write each composite. State the domain of each. f(g(x)) = 3( + 2) – 4 = + 6 – 4 = + 2 The domain of f(g(x)) is x ≥ 0 or {x|x ≥ 0}.

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Holt Algebra Operations with Functions g(f(x)) Substitute the rule f into g. Check It Out! Example 4b g(f(x)) = = Note that x ≥. 4 3 The domain of g(f(x)) is x ≥ or {x|x ≥ } Given f(x) = 3x – 4 and g(x) = + 2, write each composite. State the domain of each.

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Holt Algebra Operations with Functions Composite functions can be used to simplify a series of functions.

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Holt Algebra Operations with 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 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

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Holt Algebra Operations with Functions Step 1 Write a function for the total cost in U.S. dollars. Example 5 Continued P(c) = c c = 1.15c Step 2 Write a function for the cost in dollars based on the cost in pesos. D (c) = c Use the exchange rate.

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Holt Algebra Operations with Functions Step 3 Find the composition D(P(c)). Example 5 Continued = 1.15 ( ) c D(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 = D(P(c) ) = 1.15 ( ) ≈ The table would cost $183.19, including all charges.

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Holt Algebra Operations with Functions 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) = 0.80c Drum kits are sold at 80% of their cost.

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Holt Algebra Operations with Functions 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

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Holt Algebra Operations with Functions 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 Continued f(g(c)) = 0.80(0.85c) = 0.68c b. 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 $

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Holt Algebra Operations with Functions Lesson Quiz: Part I Given f(x) = 4x 2 – 1 and g(x) = 2x – 1, find each function or value. 1. (f + g)(x) 4x 2 + 2x – 2 2. (fg)(x) 8x 3 – 4x 2 – 2x ( ) (x)(x) f g 4. g(f(2)) 29 2x + 1

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Holt Algebra Operations with Functions Lesson Quiz: Part II 5. f(g(x)) f(g(x)) = x – 1; {x|x ≤ – 1 or x ≥ 1} Given f(x) = x 2 and g(x) =, write each composite function. State the domain of each. 6. g(f(x)) {x|x ≥ 1}

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