BMAD Posted June 1, 2022 Report Share Posted June 1, 2022 (f(x+y)-f(xy))/(3x) = f(y/(3x))-11-y Find f(x) where f(x) is a polynomial. 1 Quote Link to comment Share on other sites More sharing options...

0 EventHorizon Posted June 1, 2022 Report Share Posted June 1, 2022 Alternatively, Spoiler x,0 (f(x)-f(0))/(3x) = f(0)-11 f(x)-f(0) = 3x * (f(0)-11) = ax, for some constant a. clearly f(x) is linear. f(x) = ax+b (f(x+y)-f(xy))/(3x) = f(y/(3x))-11-y (a(x+y)+b-axy-b)/3x = ay/3x+b-11-y (ax+ay-axy)/3x = ay/3x +b-11-y a/3 + ay/3x - ay/3 = ay/3x +b-11-y a/3 - ay/3 = b-11-y ay/3 = y since the variable terms must be equal a = 3 a/3 = b-11 1 = b-11 b=12 f(x) = 3x+12 Quote Link to comment Share on other sites More sharing options...

0 EventHorizon Posted June 1, 2022 Report Share Posted June 1, 2022 Spoiler 1,0 (f(1)-f(0))/3 = f(0)-11 f(1) = 4*f(0)-33 -1,0 (f(-1)-f(0))/(-3) = f(0)-11-y f(-1) = -2*f(0)+33 -1,1 (f(0)-f(-1))/(-3) = f(-1/3)-12 f(0)-f(-1) = 36 - 3*f(-1/3) 1,-1 (f(0)-f(-1))/(3) = f(-1/3)-10 f(0)-f(-1) = 3*f(-1/3) - 30 3*f(-1/3) - 30 = 36 - 3*f(-1/3) 6*f(-1/3) = 66 f(-1/3) = 11 f(0)-f(-1) = 3 (f(0)-3) = -2*f(0)+33 3*f(0) = 36 f(0) = 12 f(-1) = 9 f(1) = 4*(12)-33 f(1) = 48-33 = 15 f(x) = 3x+12?? (f(x+y)-f(xy))/(3x) = f(y/(3x))-11-y (3(x+y)+12 - (3xy+12))/3x = (y/x +12)-11-y (3x+3y+12-3xy-12)/3x = y/x +1 -y (3x+3y-3xy)/3x = y/x +1 -y 1+y/x-y = y/x +1 -y yup. f(x) = 3x+12 Quote Link to comment Share on other sites More sharing options...

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## BMAD

(f(x+y)-f(xy))/(3x) = f(y/(3x))-11-y

Find f(x) where f(x) is a polynomial.

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