Question

Let $f:R \rightarrow R$ be a polynomial function satisfying the equation $f(f(x)-2y)=2x-3y+f(f(y)-x)$, $\forall x, y \in R$, then the value of $f(9)-f(3)$ is equal to

• A. 5
• B. 4
• C. 6

Answer 1

Answer: (C)

6

Answer 2

Assume $f(x) = ax + b$.

Substitute into the equation:

$$f(f(x)-2y) = a(ax + b - 2y) + b = a^2x + ab - 2ay + b,$$ $$f(f(y)-x) = a(ay + b - x) + b = a^2y + ab - ax + b.$$

So the equation becomes:

$$a^2x + ab - 2ay + b = 2x - 3y + a^2y + ab - ax + b.$$

Matching coefficients for $x$ and $y$:

• For $x$: $a^2 = 2 - a$ $\Rightarrow$ $a^2 + a - 2 = 0$ $\Rightarrow$ $(a+2)(a-1)=0$. Thus, $a = 1$ (rejecting $a=-2$ as it doesn't satisfy the $y$ coefficient below).
• For $y$: $-2a = a^2 - 3$. With $a = 1$: $-2 = 1 - 3 \Rightarrow -2 = -2$.

Thus, $f(x) = x + b$. Therefore,

$$f(9)-f(3) = (9+b) - (3+b) = 6.$$