Question

Given that $f\left(x\right)$
is a differentiable function of $x$ and that $f\left(x\right)$ . $f\left(y\right)$ = $f\left(x\right)$+ $f\left(y\right)$ + $f\left(xy\right)-2$ and that
$f\left(2\right)=5$.

Then $f\left(3\right)$ is equal to?

• A. $6$
• B. $24$
• C. $15$
• D. $19$

Answer 1

The correct option is A $6$

Given, $f\left(x\right).f\left(y\right)=f\left(x\right)+f\left(y\right)+f\left(xy\right)-2-\left(1\right)$

$f\left(2\right)=5$

$f\left(x\right)=x^2+1$

Equation $1\Rightarrow$ $(x^2+1)(y^2+1)=x^2+1+y^2+1+x^2{y}^2+1-2$

$x^2{y}^2+x^2+y^2+1=x^2+y^2+x^2{y}^2+1$

$\therefore f\left(3\right)=x^2+1=3^2+1=10$