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Section 6.1: Composite functions 1) f(x) = 3x - 6 g(x) = 2x + 10 1a) (π β π)(π₯) = 3(2x+10) β 6 = 6x + 30 β 6 1b) the domain of (π β π)(π₯) The domain of both f and g are all real numbers, so there is no work to find the domain of part a. Answer: (π β π)(π₯) = 6x+24 Answer: domain (ββ, β) 1c) (π β π)(π₯) = 2(3x β 6) + 10 = 6x β 12 + 10 1d) the domain of (π β π)(π₯) The domain of both f and g are all real numbers, so there is no work to find the domain of part c. Answer: (π β π)(π₯) 3) f(x) = x2 + 5 = 6x -2 Answer: domain (ββ, β) g(x) = 3x β 4 3a) (π β π)(π₯) = (3x-4)2 +5 = (3x β 4)(3x β 4) + 5 = 9x2 β 12x β 12x + 16 + 5 = 9x2 β 24x + 21 = 3(3x2 - 8x + 7) (this doesnβt factor more) 3b) the domain of (π β π)(π₯) The domain of both f and g are all real numbers, so there is no work to find the domain of part a. Answer: domain (ββ, β) Answer: (π β π)(π₯) = 3(3x2 β 8x + 7) 3c) (π β π)(π₯) 3d) the domain of (π β π)(π₯) The domain of both f and g are all real numbers, so there is no work to find the domain of part c. = 3(x2 + 5) β 4 = 3x2 + 15 β 4 Answer: (π β π)(π₯) Answer: domain (ββ, β) = 3x2 + 11 g(x) = x2 + 2x- 1 5) f(x) = x β 4 5a) (π β π)(π₯) = (x2 + 2x β 1) β 4 = x2 + 2x β 1 β 4 Answer: (π β π)(π₯) = x2 + 2x β 5 5b) the domain of (π β π)(π₯) The domain of both f and g are all real numbers, so there is no work to find the domain of part a. Answer: domain (ββ, β) 5c) (π β π)(π₯) = (x-4)2 + 2(x-4) β 1 = x2 β 4x β 4x + 16 + 2x β 8 β 1 = x2 β 6x +7 (this is prime and canβt be factored) Answer: (π β π)(π₯) = x β 6x + 7 5d) the domain of (π β π)(π₯) The domain of both f and g are all real numbers, so there is no work to find the domain of part c. Answer: domain (ββ, β) 2 2 7) π(π₯) = π₯+4 3 g(x) = π₯β7 7a) (π β π)(π₯) 2 = 3 7b) the domain of (π β π)(π₯) = f(g(x)) +4 First note that the domain of g(x) is all real numbers except 7, so we exclude 7 from the domain of (π β π)(π₯). π₯β7 2 = 3 4(π₯β7) + π₯β7 π₯β7 = = To find the number to exclude from the domain of g(x) just set the denominator equal to zero and solve for x. x-7 =0 X=7 2 3+4π₯β28 π₯β7 2(π₯β7) 4π₯β25 In addition the answer to part a has a domain of all real numbers except, so 25/4 needs to be excluded from the domain as well. 2(π₯β7) = 4π₯β25 2(π₯β7) Answer: (π β π)(π₯) = 4π₯β25 To find the number to exclude from the domain of (π β π)(π₯) just set the denominator of the answer equal to zero and solve for x. 4x β 25 = 0 4x = 25 x = 25/4 Answer: domain of (π β π)(π₯) is all real numbers except 7 and 25/4. 7c) (π β π)(π₯) = = = = 7d) the domain of (π β π)(π₯) the domain of (π β π)(π₯) First note that the domain of f(x) is all real numbers except -4, so we exclude -4 from the domain of (π β π)(π₯) . 3 2 β7 π₯+4 3 2 7(π₯+4) β π₯+4 π₯+4 To find the number to exclude from the domain of f(x) just set the denominator equal to zero and solve for x. 3 2β7π₯β28 π₯+4 x+4=0 x = -4 3(π₯+4) β7π₯+26 Answer: (π β π)(π₯)= 3(π₯+4) β7π₯+26 In addition the answer to part c has a domain of all real numbers except 26/7, so 26/7 needs to be excluded from the domain as well. To find the number to exclude from the domain of (π β π)(π₯) just set the denominator of the answer equal to zero and solve for x. -7x + 26 = 0 26 = 7x 26/7 = x Answer: domain of (π β π)(π₯) is all real numbers except -4 and 26/7 9) π(π₯) = 1 π₯β3 π(π₯) = 1 π₯ 9a) (π β π)(π₯) = f(g(x)) = 9b) the domain of (π β π)(π₯) First note that the domain of g(x) is all real numbers except 0, so we exclude 0 from the domain of (π β π)(π₯). 1 1 β3 π₯ =1 1 To find the number to exclude from the domain of g(x) just set the denominator equal to zero and solve for x. x =0 π₯ π₯ π₯ = β β3 1 1β3π₯ π₯ π₯ Answer: (π β π)(π₯) = β3π₯+1 In addition the answer to part a has a domain of all real numbers except 1/3, so 1/3 needs to be excluded from the domain as well. To find the number to exclude from the domain of (π β π)(π₯) just set the denominator of the answer equal to zero and solve for x. -3x+ 1 = 0 1 = 3x 1/3 = x Answer: domain of (π β π)(π₯) is all real numbers except 0 and 1/3 9c) (π β π)(π₯) = = 1 1 π₯β3 1(π₯β3) 1 Answer: (π β π)(π₯) = x - 3 9d) the domain of (π β π)(π₯) the domain of (π β π)(π₯) First note that the domain of f(x) is all real numbers except 3, so we exclude 3 from the domain of (π β π)(π₯) . To find the number to exclude from the domain of f(x) just set the denominator equal to zero and solve for x. x-3 =0 x=3 In addition the answer to part c has a domain of all real numbers, so nothing more needs to be excluded from the domain.. Answer: domain of (π β π)(π₯) is all real numbers except 3 11) f(x) = 7x + 1 g(x) = π₯β1 7 11a) show (π β π)(π₯) = x 11b) show (π β π)(π₯) (π β π)(π₯) = 7(π(π₯)) π₯β1 (π β π)(π₯) = 7 ( ) + 1 (π β π)(π₯) = =xβ1+1 =x (π β π)(π₯) = 7 13) π(π₯) = π₯β5 2 =x π(π₯)β1 7 (7π₯+1)β1 7 = 7π₯+1β1 7π₯ = π₯ 7 =x π(π₯) = 2π₯ + 5 13a) show (π β π)(π₯) = x (π β π)(π₯) = π(π₯)β5 2 (π β π)(π₯) = 2π₯+5β5 2π₯ = 2 2 =x 13b) π βππ€ (π β π)(π₯) =x (π β π)(π₯) = 2(π(π₯)) + 5 (π β π)(π₯) = 2( π₯β5 )+ 2 5 = x-5+5 = x