HEEEELP MATHS

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Let f(x) be a polynomial such that

f(f(x))−x^2=xf(x).

Find f(−100).

  • HEEEELP MATHS -

    It is clear that f(x) is linear. If it were, say, quadratic,

    f(x) = ax^2+bx+c with a≠0
    f(f) = af^2+bf+c
    = a(ax^2+bx+c)^2 + b(ax^2+bx+c) + c
    is a 4th-degree polynomial.
    xf(x)+x^2 = x(ax^2+bx+c) + x^2
    is a 3rd-degree polynomial. That would mean a=0.

    So, let
    f(x) = ax+b
    f(f) = af+b = a(ax+b)+b = a^2x+(ab+b)
    xf(x)+x^2 = x(ax+b)+x^2 = (a+1)x^2 + bx

    So, a+1=0, and a = -1
    That makes b=1, and so

    f(x) = 1-x
    check:
    f(f(x)) = 1-f = 1-(1-x) = x
    xf(x)+x^2 = x(1-x)+x^2 = x

  • HEEEELP MATHS -

    101

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