JEE Main 2022 · 26 July · morning · Single correct · +4 / −1
Let f : R → R be a continuous function such that f(3x) - f(x) = x. If f(8) = 7, then f(14) is equal to :
- (a) 4
- (b) 10
- (c) 11
- (d) 16
Answer and solution
Answer (b) 10
Solution
Maths sir · 4 steps
Keep replacing x by x/3 and add the equations. The sum telescopes, and continuity closes it at f(0).
- Shrink x repeatedly. Put x/3, x/9, …, x/3ⁿ in place of x. This gives f(x) - f(x/3) = x/3, then f(x/3) - f(x/9) = x/9, and so on.
- Add and telescope. On the left every middle term cancels, so only f(x) - f(x/3ⁿ) remains. As n → ∞, x/3ⁿ → 0. Since f is continuous, f(x/3ⁿ) → f(0). On the right you get a geometric series.
f(x) - f(0) = x/3 + x/9 + x/27 + … = (x/3)/(1 - 1/3) = x/2
- Find the constant. So f(x) = x/2 + c. Check it: 3x/2 - x/2 = x. From f(8) = 4 + c = 7 you get c = 3.
- So. f(14) = 14/2 + 3 = 10, which is option (b).
The idea to keep. For f(kx) and f(x) equations, keep replacing x by x/k and add. The left side telescopes, and continuity lets you take the limit at f(0).
Answer from the NTA final answer key. Solution by PrepWiser.