Question

For any function $f\left(x\right):$
$f\left(x\right)f\left(\frac{1}{x}\right)=f\left(x\right)+f\left(\frac{1}{x}\right)\forall x\in R-\left\{0\right\},$ where $f\left(x\right)$ be a polynomial function and $f\left(5\right)=126$ then $f\left(3\right)=$

• A. $27$
• B. $25$
• C. $26$
• D. $28$

Answer 1

The correct option is D $28$

Given:
$f\left(x\right)f\left(\frac{1}{x}\right)=f\left(x\right)+f\left(\frac{1}{x}\right)\Rightarrow f\left(x\right)(f\left(\frac{1}{x}\right)-1)=f\left(\frac{1}{x}\right)\Rightarrow f\left(x\right)=\frac{f\left(\frac{1}{x}\right)}{(f\left(\frac{1}{x}\right)-1)}\Rightarrow f\left(x\right)=\frac{f\left(\frac{1}{x}\right)-1+1}{(f\left(\frac{1}{x}\right)-1)}\Rightarrow f\left(x\right)=1+\frac{1}{(f\left(\frac{1}{x}\right)-1)}$

$\Rightarrow \left(f\left(x\right)-1\right)\left(f\left(\frac{1}{x}\right)-1\right)=1$
Also, $f\left(x\right)$ is a polynomial function:
$\Rightarrow f\left(x\right)=1\pm x^n$
$f\left(5\right)=1\pm 5^n\Rightarrow 126=\pm 5^n\Rightarrow \pm 5^n=125\Rightarrow \pm 5^n=5^3\Rightarrow f\left(3\right)=1+3^3=28$