Posted by **Anubhav** on Tuesday, May 14, 2013 at 10:33am.

Let f(x) be a polynomial such that

f(f(x))−x^2=xf(x).

Find f(−100).

- HEEEELP MATHS -
**Steve**, Tuesday, May 14, 2013 at 11:30am
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 -
**intermediate player PLEASE HELP**, Tuesday, May 14, 2013 at 12:19pm
101

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