Question

If $f:R^{+}\to R^{+}$ is a polynomial function satisfying the functional equation $f\left(f\right(x\left)\right)=6x-f\left(x\right),$ then $f\left(17\right)$ is equal to

• A. $17$
• B. $51$
• C. $34$
• D. $-51$

Answer 1

The correct option is C $34$

For equal degree in both sides, $f\left(x\right)$ must be a linear function.
Let $f\left(x\right)=ax+b$
$\Rightarrow f(ax+b)=6x-ax-b$
Given that $f\left(f\right(x\left)\right)=6x-f\left(x\right)$
$\Rightarrow a(ax+b)+b=6x-ax-b$
On comparing both sides,
$a^2=6-a$ and $ab+b=-b$
$\Rightarrow a^2+a-6=0$ and $ab+2b=0$
$\Rightarrow (a+3)(a-2)$ and $b(a+2)=0$
$\Rightarrow a=2$ or $-3$ and $b=0$ or $a=-2$
$\Rightarrow a=2,b=0$ or $a=-3,b=0$
$\Rightarrow f\left(x\right)=2x$ or $-3x$
But codomain of $f$ is $R^{+}$.
So, $f\left(x\right)=2x$
$\Rightarrow f\left(17\right)=34$