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Warm Up Determine the domain of the function.
f(x) = 7π₯ β42 7x β 42 β₯ 0 x β₯ 6
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Operations and Compositions of Functions
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Operations with Functions Sum Difference Product Quotient
(f + g)(x) = f(x) + g(x) (f β g)(x) = f(x) β g(x) (f β’ g)(x) = f(x) β’ g(x) π(π₯) π(π₯) ( π π )(x) =
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f(x) + g(x) f(x) β g(x) Examples 1. f(x) = 2x + 1 g(x) = x β 2
Find each: a. (f + g)(x) b. (f β g)(x) f(x) + g(x) (2x + 1) + (x β 2) 3x β 1 f(x) β g(x) (2x + 1) β (x β 2) 2x + 1 β x + 2 x + 3
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f(x) β g(x) (2x + 1) β (x β 2) 2 π₯ 2 β 4x + x β 2 2 π₯ 2 β 3x β 2
c. (f β’ g)(x) d. ( π π )(x) f(x) β g(x) (2x + 1) β (x β 2) 2 π₯ 2 β 4x + x β 2 2 π₯ 2 β 3x β 2 π(π₯) π(π₯) 2π₯+1 π₯β2 ; x β 2
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Examples 2. f(x) = 2x + 1 g(x) = x β 2 Find each: a. (f + g)(6)
f(x) + g(x) = 3x β 1 13 + 4 3(6) β 1 17 17
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b. (f β g)(6) f(x) β g(x) = x + 3 f(6) β g(6) 6 + 3 13 β 4 9 9
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f(x) β’ g(x) = 2 π₯ 2 β 3x β 2 f(6) β’ g(6) 2 (6) 2 β 3(6) β 2
c. (f β’ g)(6) f(x) β’ g(x) = 2 π₯ 2 β 3x β 2 f(6) β’ g(6) 2 (6) 2 β 3(6) β 2 2(36) β 18 β 2 13 β’ 4 72 β 18 β 2 52 52
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π(π₯) π(π₯) 2π₯+1 π₯β2 = π(6) π(6) = 2(6)+1 6β2 13 4 ; x β 2 = 12+1 6β2
d. π π (6) 2π₯+1 π₯β2 = π(6) π(6) = 2(6)+1 6β2 13 4 ; x β 2 = β2 = 13 4 ; x β 2
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Examples 3. f(x) = x + 3 Find each: a. (f + g)(x)
f(x) + g(x) (x + 3) + (2π₯) (π₯ β 5) (π₯+3) (2π₯) (π₯ β 5) β (π₯ β 5) (π₯ β 5) 1 π₯ 2 β15 π₯ β5 π₯ 2 β 5π₯ + 3π₯ β15 π₯ β5 + (2π₯) (π₯ β 5) ; x β 5
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(π₯+3) β (2π₯) (π₯ β 5) f(x) β g(x) (x + 3) β (2π₯) (π₯ β 5)
b. (f β g)(x) f(x) β g(x) (x + 3) β (2π₯) (π₯ β 5) (π₯+3) β (2π₯) (π₯ β 5) β (π₯ β 5) (π₯ β 5) 1 π₯ 2 β 5π₯ + 3π₯ β15 π₯ β5 β (2π₯) (π₯ β 5) π₯ 2 β 5π₯ + 3π₯ β15 β2π₯ π₯ β5 π₯ 2 β4π₯ β15 π₯ β5 ; x β 5
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f(x) β g(x) f(x) = x + 3 (π₯+ 3) 1 β (2π₯) (π₯ β 5) g(x) = ππ π β π
(π₯+ 3) 1 β (2π₯) (π₯ β 5) g(x) = ππ π β π 2 π₯ 2 +6π₯ π₯ β5 c. (f β’ g)(x) d. ( π π )(x) ; x β 5 (π₯ + 3) 1 Γ· (2π₯) (π₯ β5) (π₯+3) (2π₯) (π₯ β5) (π₯ + 3) 1 β (π₯ β 5) (2π₯) π₯ 2 β2π₯ β15 2π₯ π₯ 2 β5π₯ +3π₯ β15 2π₯ ; x β 0, 5
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Homework: Function Operation WS #1 β 7, 9 β 15, 17, 20
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Warm Up Given the graph, state the domain and range and determine whether or not it represents a function: π«πππππ:βπβ€πβ€π πΉππππ:βπβ€πβ€π Not a function
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(f ΠΎ g)(x) = f(g(x)) (g ΠΎ f)(x) = g(f(x)) Composition of Functions
- the process of combining two functions where one function is performed first and the result of which is substituted in place of eachΒ xΒ in the other function. read as βf of g of xβ read as βg of f of xβ (f ΠΎ g)(x) = f(g(x)) (g ΠΎ f)(x) = g(f(x))
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Examples 3. f(x) = 2x + 1 g(x) = x β 2 Find each: a. (f ΠΎ g)(x)
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= g(f(x)) = g(2x + 1) = (2x + 1) β 2 = 2x β 1
3. f(x) = 2x + 1 g(x) = x β 2 b. (g ΠΎ f)(x) = g(f(x)) = g(2x + 1) = (2x + 1) β 2 = 2x β 1
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Examples 3. f(x) = x2 + 4 g(x) = x β 4 a. (f (g(x))
= x2 β 4x β 4x = x2 β 8x + 20
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= g(f(x)) = g(x2 + 4) = (x2 + 4) β 4 = x2 b. (g(f(x)) 3. f(x) = x2 + 4
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Homework: Function Operation WS #8, 16, 18, 19, 21, 22
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