# Given f(x) is a polynomial function of x, satisfying f(x)*f(y)= f(x) +f(y)+f(x y)-2 and that f(2)=5. Then f(3) is equal to? A) 10 B) 24 C) 15 D) none

Mar 6, 2017

option d)

#### Explanation:

$f \left(1\right) f \left(3\right) = f \left(1\right) + f \left(3\right) + f \left(3\right) - 2$ then

$f \left(3\right) \left(f \left(1\right) - 2\right) = f \left(1\right) - 2 \to f \left(3\right) = \frac{f \left(1\right) - 2}{f \left(1\right) - 2}$

To avoid indeterminations

$f \left(1\right) \cdot f \left(1\right) = f \left(1\right) + f \left(1\right) + f \left(1\right) - 2$ or

$f {\left(1\right)}^{2} - 3 f \left(1\right) + 2 = 0$

solving for $f \left(1\right)$ we have

$f \left(1\right) = \left\{1 , 2\right\}$ so taking $f \left(1\right) = 1$ we have finally

$f \left(3\right) = 1$