Let f : R [] R be a continuous function such that f(3x) - f(x) = x.

Question

Let f : R $$\to$$ R be a continuous function such that $f(3x) - f(x) = x$. If $f(8) = 7$, then $f(14)$ is equal to :

Accepted Answer

$f(3x) - f(x) = x$ ...... (1)

$$x \to {x \over 3}$$

$$f(x) - f\left( {{x \over 3}} \right) = {x \over 3}$$ ....... (2)

Again $$x \to {x \over 3}$$

$$f\left( {{x \over 3}} \right) - f\left( {{x \over 9}} \right) = {x \over {{3^2}}}$$ ...... (3)

Similarly

$$f\left( {{x \over {{3^{n - 2}}}}} \right) - f\left( {{x \over {{3^{n - 1}}}}} \right) = {x \over {{3^{n - 1}}}}\,.....\,(n)$$
Adding all these and applying $$n \to \infty $$

$$\mathop {\lim }\limits_{n \to \infty } \left( {f(3x) - f\left( {{x \over {{3^{n - 1}}}}} \right)} \right) = x\left( {1 + {1 \over 3} + {1 \over {{3^2}}}\, + \,....} \right)$$

$$f(3x) - f(0) = {{3x} \over 2}$$

Putting $$x = {8 \over 3}$$

$f(8) - f(0) = 4$

$$ \Rightarrow f(0) = 3$$

Putting $$x = {{14} \over 3}$$

$$f(14) - 3 = 7 \Rightarrow f(14) = 10$$

Solutions · 1