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# hmk3sol - COT 3100 Homework#3 Solutions Fall 00 1 Find the...

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COT 3100 Homework #3 Solutions Fall ’00 1) Find the inverse of each of the following functions: (The domain is the set of real number unless otherwise specified.) a) f(x) = 12x –7 x = 12f -1 (x) – 7 12f -1 (x) = (x + 7) f -1 (x) = (x + 7)/12, domain and range here are all reals. b) f(x) = (3x+1)/(x-2), for x 2 x = (3f -1 (x) + 1)/( f -1 (x) – 2) x( f -1 (x) – 2) = 3f -1 (x) + 1 x f -1 (x) – 2x = 3f -1 (x) + 1 x f -1 (x) – 3f -1 (x) = (1 + 2x) f -1 (x)[x – 3] = (1 + 2x) f -1 (x) = (2x + 1)/(x – 3), domain is all reals except x=3 and the range is all reals except y=2. (Drawing out the graph shows that y=2 is an asymptote.) c) f(x) = 2 x+4 x = 2 f-1(x) + 4 log 2 x = f -1 (x) + 4 f -1 (x) = log 2 x – 4, domain is all reals x > 0, and the range is all reals. 2) Compute each of the following function compositions: a) f(x) = 3x+7, g(x)=x 2 , compute g(f(x)). g(f(x)) = g(3x+7) = (3x+7) 2 = 9x 2 + 42x + 49 b) f(x) = 2 3x – 2 , g(x)=(x+2)/3, compute f(g(x)). f(g(x) = f((x+2)/3) = 2 3(x+2)/3 – 2 = 2 x + 2 – 2 = 2 x

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c) f(x) = (4x 2 – 5)/(3x+2), g(x) = x – 4, compute f(g(x)). f(g(x)) = f(x – 4) = (4(x – 4) 2 – 5)/(3(x – 4) +2) = (4x 2 – 32x + 64 – 5)/(3x – 12 + 2) = (4x 2 – 32x + 59)/(3x – 10) 3) Define f
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hmk3sol - COT 3100 Homework#3 Solutions Fall 00 1 Find the...

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