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

JEE Main · 2022 · 26 Jul 2022 · Shift 1 · Sets Relations and Functions

Let

f : R \to R

be a continuous function such that

f (3 x) - f (x) = x

. If

f (8) = 7

, then

f (14)

is equal to :

Answer: (B)

From the functional equation

f (3 x) - f (x) = x

, we can write

f (x) - f (x /3) = x /3

f (x /3) - f (x /9) = x /9

, and so on. Adding these infinitely many equations:

f (x) - n \to \infty lim f (\frac{x}{3 ^{n}}) = x (\frac{1}{3} + \frac{1}{9} + \dots) = x \cdot \frac{1/3}{1 - 1/3} = \frac{x}{2}

Since

f

is continuous,

lim_{n \to \infty} f (x / 3^{n}) = f (0)

. Thus

f (x) = \frac{x}{2} + f (0)

.

Using

f (8) = 7

4 + f (0) = 7

, so

f (0) = 3

.

Therefore

f (x) = \frac{x}{2} + 3

, and

f (14) = 7 + 3 = 10

JEE Main · 2022 · 26 Jul 2022 · Shift 1 · Single correct option