Question

If $f\left(x\right)$ is a polynomial satisfying $f\left(x\right)f\left(\frac{1}{x}\right)=f\left(x\right)+f\left(\frac{1}{x}\right)$ and $f\left(3\right)=28$, then $f\left(4\right)$ is given by

• A. $63$
• B. $64$
• C. $65$
• D. $66$

Answer 1

The correct option is C

$65$

Explanation for the correct option:

Find the value of $f\left(4\right)$:

Let the function is,

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

Now,

$\begin{array}{rcl}f\left(3\right)& =& 28\\ & \Rightarrow & 3^n+1=28\\ & \Rightarrow & 3^n=27\\ & \Rightarrow & 3^n=3^3\\ & \Rightarrow & n=3\end{array}$

Now, substitute $x=4$ and $n=3$, in the function, we get

$$\begin{gathered} f\left(4\right)=4^3+1 \\ =65 \end{gathered}$$

Hence, the correct option is C.