Question

If $g\left(x\right)$ is a polynomial satisfying $g\left(x\right)g\left(y\right)=g\left(x\right)+g\left(y\right)+g\left(xy\right)-2$ for all real $x$ and $y$ and $g\left(2\right)=5$ then $g\left(3\right)$ is equal to -

• A. $10$
• B. $24$
• C. $21$
• D. none of these

Answer 1

The correct option is A $10$

$g\left(x\right)g\left(y\right)=g\left(x\right)+g\left(y\right)+g\left(xy\right)-2$

Substitute $x=2$ & $y=1$

$g\left(2\right)g\left(1\right)=g\left(2\right)+g\left(1\right)+g\left(2\right)-2$

$\Rightarrow 4g\left(1\right)=8$

$\Rightarrow g\left(1\right)=2$

$g\left(x\right)g\left(y\right)=g\left(x\right)+g\left(y\right)+g\left(xy\right)-2$, now substitute $y=\frac{1}{x}$

Now $g\left(x\right)g\left(\frac{1}{x}\right)=g\left(x\right)+g\left(\frac{1}{x}\right)$

$g\left(x\right)=1\pm x^n$

$\therefore 5=1\pm 2^n(\because g(2)=5)$

$2^n=4$

So, $n=2$

Now $g\left(3\right)=1$ + $3^2=10$