Mathematics - Functions Question with Solution | TestHub

Given $f\left(3x\right)-f\left(x\right)=x$

Now replance $x \text{by }\frac{x}{3}$  infinite times,

$f\left(x\right)-f\left(\frac{x}{3}\right)=\frac{x}{3}$

$$\begin{gathered} f\left(\frac{x}{3}\right)-f\left(\frac{x}{3^2}\right)=\frac{x}{3^2} \\ . \\ . \end{gathered}$$

On adding, we get

$f\left(x\right)-\underset{n\to \infty }{\lim }f\left(\frac{x}{3^n}\right)=x\left(\frac{1}{3}+\frac{1}{3^2}+\cdots +\infty \right)$

$f\left(x\right)-\underset{n\to \infty }{\lim }f\left(\frac{x}{3^n}\right)=x\left(\frac{{\displaystyle \frac{1}{3}}}{1-{\displaystyle \frac{1}{3}}}\right)=\frac{x}{2}$

$\Rightarrow f\left(x\right)-f\left(0\right)=\frac{x}{2}$

Given $f\left(8\right)=7$

so $f\left(8\right)-f\left(0\right)=4$

i.e. $f\left(0\right)=3$

$\therefore f\left(x\right)=\frac{x}{2}+3$

Hence $f\left(14\right)=10$