Let f(x) = (x - 1)/(x + 1), x ∈ R - { 0, - 1,1}. If fⁿ⁺¹(x) = f(fⁿ(x)) for all n ∈ N, then f⁶(6) + f⁷(7) is equal to :

Solution

Maths sir · 4 steps

Do not compose six times. Find when fⁿ comes back to x, then use that cycle.

1. Compose once. Put (x-1)/(x+1) into f. The top becomes (x-1-(x+1))/(x+1) = -2/(x+1) and the bottom becomes (x-1+x+1)/(x+1) = 2x/(x+1), so f²(x) = -1/x.
2. Find the cycle. Applying f² twice gives f⁴(x) = -1/(-1/x) = x. So the powers of f repeat every 4 steps.

   n	fⁿ(x)
   1	(x-1)/(x+1)
   2 (and 6)	-1/x
   3 (and 7)	f(-1/x) = (-1-x)/(-1+x) = (x+1)/(1-x)
   4	x

3. Use the cycle. f⁶ = f², so f⁶(6) = -1/6. Also f⁷ = f³, so f⁷(7) = (7+1)/(1-7) = -4/3.
4. So. f⁶(6) + f⁷(7) = -1/6 - 8/6 = -9/6 = -3/2, which is option (b).

The idea to keep. When a question asks for a high power like f⁶ or f²⁰²⁴, compute f², f³, … until you get back x. Then reduce the power by the cycle length.

Answer from the NTA final answer key. Solution by PrepWiser.