Question

If $f\left(x\right)·f\left(\frac{1}{x}\right)=f\left(x\right)+f\left(\frac{1}{x}\right)$ and $f\left(3\right)=28$, then $f\left(4\right)$ is equal to

• A. $65$
• B. $217$
• C. $215$
• D. $64$

Answer 1

The correct option is A

$65$

Explanation for the correct option.

Step 1. Find the functions $f\left(x\right)$ and $f\left(\frac{1}{x}\right)$.

Solve the given equation $f\left(x\right)·f\left(\frac{1}{x}\right)=f\left(x\right)+f\left(\frac{1}{x}\right)$ for $f\left(x\right)$.

$$\begin{gathered} f\left(x\right)·f\left(\frac{1}{x}\right)-f\left(x\right)=f\left(\frac{1}{x}\right) \\ \Rightarrow f\left(x\right)\left[f\left(\frac{1}{x}\right)-1\right]=f\left(\frac{1}{x}\right) \\ \Rightarrow f\left(x\right)=\frac{f\left({\displaystyle \frac{1}{x}}\right)}{f\left({\displaystyle \frac{1}{x}}\right)-1}...\left(1\right) \end{gathered}$$

Again, solve the given equation $f\left(x\right)·f\left(\frac{1}{x}\right)=f\left(x\right)+f\left(\frac{1}{x}\right)$ for $f\left(\frac{1}{x}\right)$.

$$\begin{gathered} f\left(x\right)·f\left(\frac{1}{x}\right)-f\left(\frac{1}{x}\right)=f\left(x\right) \\ \Rightarrow f\left(\frac{1}{x}\right)\left[f\left(x\right)-1\right]=f\left(x\right) \\ \Rightarrow f\left(\frac{1}{x}\right)=\frac{f\left(x\right)}{f\left(x\right)-1}...\left(2\right) \end{gathered}$$

Step 2. Find the function $f\left(x\right)$.

Multiply the equations $1$ and $2$.

$$\begin{gathered} f\left(x\right)f\left(\frac{1}{x}\right)=\frac{f\left({\displaystyle \frac{1}{x}}\right)}{f\left(\frac{1}{x}\right)-1}\frac{f\left(x\right)}{f\left(x\right)-1} \\ \Rightarrow \left\{f\left(\frac{1}{x}\right)-1\right\}\left\{f\left(x\right)-1\right\}=1 \end{gathered}$$

So, the functions $f\left(x\right)-1$ and $f\left(\frac{1}{x}\right)-1$ are reciprocal functions.

So, let $f\left(x\right)-1=x^n$, then $f\left(x\right)=x^n+1$.

It is given that, $f\left(3\right)=28$, so

$$\begin{gathered} 28=3^n+1 \\ \Rightarrow 3^3+1=3^n+1 \end{gathered}$$

On comparing both sides it is found that $n=3$, so the function becomes $f\left(x\right)=x^3+1$.

Step 3. Find the value of $f\left(4\right)$.

For the function $f\left(x\right)=x^3+1$ the value of $f\left(4\right)$ is given as:

$\begin{array}{rcl}f\left(4\right)& =& 4^3+1\\ & =& 64+1\\ & =& 65\end{array}$

So the value of$f\left(4\right)$ is $65$.

Hence, the correct option is A.