9-4 with Functions 9-4 Operations Operations with Functions Warm Up Lesson Presentation Lesson Quiz Holt Algebra2 Holt Algebra 2 9-4 Operations with Functions Warm Up Simplify. Assume that all expressions are defined. 1. (2x + 5) – (x2 + 3x – 2) –x2 – x + 7 2. (x – 3)(x + 1)2 x3 – x2 – 5x – 3 3. x–3 x–2 Holt Algebra 2 9-4 Operations with Functions Objectives Add, subtract, multiply, and divide functions. Write and evaluate composite functions. Holt Algebra 2 9-4 Operations with Functions Vocabulary composition of functions Holt Algebra 2 9-4 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. Holt Algebra 2 9-4 Operations with Functions Holt Algebra 2 9-4 Operations with Functions Example 1A: Adding and Subtracting Functions Given f(x) = 4x2 + 3x – 1 and g(x) = 6x + 2, find each function. (f + g)(x) (f + g)(x) = f(x) + g(x) = (4x2 + 3x – 1) + (6x + 2) 2 = 4x + 9x + 1 Holt Algebra 2 Substitute function rules. Combine like terms. 9-4 Operations with Functions Example 1B: Adding and Subtracting Functions Given f(x) = 4x2 + 3x – 1 and g(x) = 6x + 2, find each function. (f – g)(x) (f – g)(x) = f(x) – g(x) = (4x2 + 3x – 1) – (6x + 2) Substitute function rules. = 4x2 + 3x – 1 – 6x – 2 Distributive Property = 4x2 – 3x – 3 Combine like terms. Holt Algebra 2 9-4 Operations with Functions Check It Out! Example 1a Given f(x) = 5x – 6 and g(x) = x2 – 5x + 6, find each function. (f + g)(x) (f + g)(x) = f(x) + g(x) = (5x – 6) + (x2 – 5x + 6) 2 =x Holt Algebra 2 Substitute function rules. Combine like terms. 9-4 Operations with Functions Check It Out! Example 1b Given f(x) = 5x – 6 and g(x) = x2 – 5x + 6, find each function. (f – g)(x) (f – g)(x) = f(x) – g(x) = (5x – 6) – (x2 – 5x + 6) 2 Substitute function rules. = 5x – 6 – x + 5x – 6 Distributive Property = –x2 + 10x – 12 Combine like terms. Holt Algebra 2 9-4 Operations with Functions When you divide functions, be sure to note any domain restrictions that may arise. Holt Algebra 2 9-4 Operations with Functions Example 2A: Multiplying and Dividing Functions Given f(x) = 6x2 – x – 12 and g(x) = 2x – 3, find each function. (fg)(x) (fg)(x) = f(x) ● g(x) = (6x2 – x – 12) (2x – 3) Substitute function rules. = 6x2 (2x – 3) – x(2x – 3) – 12(2x – 3) Distributive Property = 12x3 – 18x2 – 2x2 + 3x – 24x + 36 Multiply. = 12x3 – 20x2 – 21x + 36 Combine like terms. Holt Algebra 2 9-4 Operations with Functions Example 2B: Multiplying and Dividing Functions f g f g ( )(x) ( )(x) = f(x) g(x) 6x2 – x –12 = 2x – 3 (2x – 3)(3x + 4) = 2x – 3 (2x – 3)(3x +4) = (2x – 3) = 3x + 4, where x ≠ Holt Algebra 2 Set up the division as a rational expression. Factor completely. 3 Note that x ≠ 2 . Divide out common factors. 3 2 Simplify. 9-4 Operations with Functions Check It Out! Example 2a Given f(x) = x + 2 and g(x) = x2 – 4, find each function. (fg)(x) (fg)(x) = f(x) ● g(x) Holt Algebra 2 = (x + 2)(x2 – 4) Substitute function rules. = x3 + 2x2 – 4x – 8 Multiply. 9-4 Operations with Functions Check It Out! Example 2b g (x) f g g(x) (x) = f f(x) x2 – 4 = x+2 (x – 2)(x + 2) = x+2 (x – 2)(x + 2) = (x + 2) ( ) ( ) = x – 2, where x ≠ –2 Holt Algebra 2 Set up the division as a rational expression. Factor completely. Note that x ≠ –2. Divide out common factors. Simplify. 9-4 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. Holt Algebra 2 9-4 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. Holt Algebra 2 9-4 Operations with Functions Reading Math The composition (f g of x.” Holt Algebra 2 o g)(x) or f(g(x)) is read “f of 9-4 Operations with Functions Caution! Be careful not to confuse the notation for multiplication of functions with composition fg(x) ≠ f(g(x)) Holt Algebra 2 9-4 Operations with Functions Example 3A: Evaluating Composite Functions Given f(x) = 2x and g(x) = 7 – x, find each value. f(g(4)) Step 1 Find g(4) g(4) = 7 – 4 g(x) = 7 – x =3 Step 2 Find f(3) 3 f(3) = 2 =8 So f(g(4)) = 8. Holt Algebra 2 f(x) = 2x 9-4 Operations with Functions Example 3B: Evaluating Composite Functions Given f(x) = 2x and g(x) = 7 – x, find each value. g(f(4)) Step 1 Find f(4) f(4) = 24 = 16 Step 2 f(x) = 2x Find g(16) g(16) = 7 – 16 = –9 So g(f(4)) = –9. Holt Algebra 2 g(x) = 7 – x. 9-4 Operations with Functions Check It Out! Example 3a Given f(x) = 2x – 3 and g(x) = x2, find each value. f(g(3)) Step 1 Find g(3) g(3) = 32 g(x) = x2 =9 Step 2 Find f(9) f(9) = 2(9) – 3 = 15 So f(g(3)) = 15. Holt Algebra 2 f(x) = 2x – 3 9-4 Operations with Functions Check It Out! Example 3b Given f(x) = 2x – 3 and g(x) = x2, find each value. g(f(3)) Step 1 Find f(3) f(3) = 2(3) – 3 f(x) = 2x – 3 =3 Step 2 Find g(3) 2 g(3) = 3 =9 So g(f(3)) = 9. Holt Algebra 2 g(x) = x2 9-4 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. Holt Algebra 2 9-4 Operations with Functions Example 4A: Writing Composite Functions x 2 Given f(x) = x – 1 and g(x) = , write 1–x each composite function. State the domain of each. f(g(x)) f(g(x)) = f( =( = x 1–x x 1–x ) )2 – 1 –1 + 2x (1 – x)2 Substitute the rule g into f. Use the rule for f. Note that x ≠ 1. Simplify. The domain of f(g(x)) is x ≠ 1 or {x|x ≠ 1} because g(1) is undefined. Holt Algebra 2 9-4 Operations with Functions Example 4B: Writing Composite Functions x 2 Given f(x) = x – 1 and g(x) = , write 1–x each composite function. State the domain of each. g(f(x)) 2 g(f(x)) = g(x – 1) = (x2 – 1) 2 1 – (x – 1) x2 – 1 = 2 – x2 Substitute the rule f into g. Use the rule for g. Simplify. Note that x ≠ The domain of g(f(x)) is x ≠ or {x|x ≠ because f( ) = 1 and g(1) is undefined. Holt Algebra 2 . } 9-4 Operations with Functions 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)) f(g(x)) = 3( + 2) – 4 Substitute the rule g into f. = +6–4 Distribute. Note that x ≥ 0. = +2 Simplify. The domain of f(g(x)) is x ≥ 0 or {x|x ≥ 0}. Holt Algebra 2 9-4 Operations with Functions Check It Out! Example 4b Given f(x) = 3x – 4 and g(x) = + 2 , write each composite. State the domain of each. g(f(x)) g(f(x)) = = Substitute the rule f into g. Note that x ≥ The domain of g(f(x)) is x ≥ Holt Algebra 2 4 3 4 3 . or {x|x ≥ 4 3 }. 9-4 Operations with Functions Composite functions can be used to simplify a series of functions. Holt Algebra 2 9-4 Operations with Functions Example 5: Business Application 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. 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. Holt Algebra 2 9-4 Operations with Functions Example 5 Continued Step 1 Write a function for the total cost in U.S. dollars. P(c) = c + 0.15c = 1.15c Step 2 Write a function for the cost in dollars based on the cost in pesos. D(c) = Holt Algebra 2 c 11.30 Use the exchange rate. 9-4 Operations with Functions Example 5 Continued Step 3 Find the composition D(P(c)). D(P(c)) = 1.15P(c) = 1.15 ( c 11.30 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. D(P(c) ) = 1.15 ( 1800 11.30 ) ≈ 183.19 The table would cost $183.19, including all charges. Holt Algebra 2 9-4 Operations with Functions Check It Out! Example 5 During a sale, a music store is selling all drum kits for 20% off. Preferred customers also receive an additional 15% off. a. Write a composite function to represent the final cost of a kit for a preferred customer that originally cost c dollars. Step 1 Write a function for the final cost of a kit that originally cost c dollars. f(c) = 0.80c Holt Algebra 2 Drum kits are sold at 80% of their cost. 9-4 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. g(c) = 0.85c Holt Algebra 2 Preferred customers receive 15% off. 9-4 Operations with Functions Check It Out! Example 5 Continued Step 3 Find the composition f(g(c)). f(g(c)) = 0.80(g(c)) Substitute g(c) for c. f(g(c)) = 0.80(0.85c) Replace g(c) with its rule. = 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 $168.64. Holt Algebra 2 9-4 Operations with Functions Lesson Quiz: Part I Given f(x) = 4x2 – 1 and g(x) = 2x – 1, find each function or value. 1. (f + g)(x) 4x2 + 2x – 2 2. (fg)(x) 8x3 – 4x2 – 2x + 1 3. f g ( )(x) 4. g(f(2)) Holt Algebra 2 2x + 1 29 9-4 Operations with Functions Lesson Quiz: Part II Given f(x) = x2 and g(x) = , write each composite function. State the domain of each. 5. f(g(x)) f(g(x)) = x – 1; {x|x ≥ 1} 6. g(f(x)) {x|x ≤ – 1 or x ≥ 1} Holt Algebra 2