(i) Let f: R+ - R+ be a function such that (f(x))² = x*(f(y))² for all positive x and y, and f(2) = 6. Find f(50). (ii) Let f be a function such that f(3) = 1 and f(3x) = x + f(3x - 3) for all x. Find f(300).

Question

(i) Let f: R+ -> R+ be a function such that (f(x))² = x*(f(y))² for all positive x and y, and f(2) = 6. Find f(50). (ii) Let f be a function such that f(3) = 1 and f(3x) = x + f(3x - 3) for all x. Find f(300).

Accepted Answer

(i) 30; (ii) 5050. (i) (f(x))² = x*(f(y))². Taking y fixed, (f(x))² is proportional to x, so f(x) = k*sqrt(x). f(2) = k*sqrt(2) = 6 -> k = 6/sqrt(2). Then f(50) = k*sqrt(50) = (6/sqrt(2))*sqrt(50) = 6*sqrt(50/2) = 6*sqrt(25) = 6*5 = 30. (ii) f(3x) = x + f(3x - 3) = x + f(3(x-1)). Let aₙ = f(3n). Then aₙ = n + aₙ₋₁, a₁ = f(3) = 1. So aₙ = 1 + 2 +... + n = n(n+1)/2. f(300) = a₁₀₀ = 100*101/2 = 5050.

(i) Let f: R+ -> R+ be a function such that (f(x))² = x*(f(y))² for all positive x and y, and f(2) = 6. Find f(50). (ii) Let f be a function such that f(3) = 1 and f(3x) = x + f(3x - 3) for all x. Find f(300).

Solution

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