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**Composition of Functions**

Pg. 60 – 61 February 17, 2015

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**Warm-Up 1) Find the inverse function of y = –3x – 6**

Simplify the expression… ½ (2x + 4) – 2

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Title Composition of Functions In the composition function f(g(x)), the ________ of g(x) is used as the ________ of the function f(x). In the composition g(f(x)), the _______ of f(x) is used as the _______ of g(x). NOTE: f(g(x)) can also be written as (f ᵒ g)(x)

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**What is function composition ?**

Essential Question What is function composition ?

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What is it? A composition of two functions uses one function as the INPUT for the other function. It is written as either f(g(x)) or as f◦g(x). Both mean to use the “g” function as the input for the “f” function. So, the expression for “g” replaces the x in function “f.” It reads as “the f of g of x”

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*NOTE* g(f(x)) & g◦f(x) mean to use the “f” as the input. It is read as “the g of f of x.” Performing a composition with values: If f(x) = 3x – 11 and g(x) = 4x + 7, find the f(g(6)). Work from inside-out: f(4(6)+7) = f(31) = 3(31) – 11 = 93 – 11 = 82 …PLUG 6 INTO g(x) and SIMPLIFY… NOW USE that value as the INPUT for “f” and SIMPLIFY

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**Find the composition function… g(f(x)) when f(x) = 3x – 11 and g(x) = 4x + 7.**

f(g(x)) = f (4x + 7) … use function “g” as your INPUT = 3( ) – 11 … it plugs INTO function “f” = 3(4x + 7) – 11 … simplify to find the composite function = 12x + 21 – 11 = x + 10 find the g(f(x)) = g(3x – 11 )… now “f” is the INPUT = 4(3x – 11) + 7 … plug into “g” = 12x – … simplify = 12x – 37

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Example 3 Find the f◦g(x) when f(x) = 2/3 x – 5 and g(x) = 6x – 18. f(g(x)) = f(6x – 18) = (2/3)(6x – 18) – 5 = 4x – 12 – 5 = 4x – 17 Find the g◦f(x): g(f(x)) = g(2/3 x – 5) = 6(2/3 x – 5) – 18 = 4x – 30 – 18 = 4x - 48

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Example 4 Find the f◦g(x) when f(x) = x2 + 3 and g(x) = 2x – 1. f(g(x)) = f(2x – 1) = (2x – 1)2 + 3 = (2x – 1)(2x – 1) + 3 = 4x2 – 4x = 4x2 – 4x + 4 Find the g◦f(x): g(f(x)) = g(x2 + 3) = 2(x2 + 3) – 1 = 2x2 + 6 – 1 = 2x2 + 5

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HOMEWORK Problems Given: f(x) = 4x – 8, g(x) = 2x2 – 5 & h(x) = ¼ x + 2 Find the following: f(h(3)) g(f(5)) h(g(–2)) f(g(x)) h(f(x)) h(g(x))

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Warm-up Arithmetic Combinations (f+g)(x) = f(x) + g(x) (f-g)(x) = f(x) – g(x) (fg)(x) = f(x) ∙ g(x) (f/g)(x) = f(x) ; g(x) ≠0 g(x) The domain for these.

Warm-up Arithmetic Combinations (f+g)(x) = f(x) + g(x) (f-g)(x) = f(x) – g(x) (fg)(x) = f(x) ∙ g(x) (f/g)(x) = f(x) ; g(x) ≠0 g(x) The domain for these.

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