Question

If $f\left(x\right)$ is a real function such that $f(x+y)=f\left(xy\right)$ and $f\left(-2\right)=\frac{2}{3}$ for all $x,y$ then $f\left(2001\right)=$

• A. $2001$
• B. $\frac{-2001}{3}$
• C. $\frac{2001}{3}$
• D. $\frac{2}{3}$

Answer 1

Answer: (D)

$\frac{2}{3}$

$f(x+y)=f\left(xy\right)$

Let $y=1⟹f\left(x\right)=f(x+1)$

So period is $1⟹1\times \left(2003\right)$ is alos the period of the function

$f\left(x\right)=f(x+2003)$

Put $x=-2f(-2)=f(-2+2003)=f\left(2001\right)=\frac{2}{3}$