Question

Let
\(f(x)=\frac{x−1}{x+1},x∈R− \left\{0,−1,1\right\}\)
If ƒⁿ⁺¹(x) = ƒ(ƒⁿ(x)) for all n∈N, then ƒ⁶(6) + ƒ⁷(7) is equal to :

• A. \(\frac{7}{6}\)
• B. \(-\frac{3}{2}\)
• C. \(\frac{7}{12}\)
• D. \(-\frac{11}{12}\)

Answer 1

The Correct Option is B

 To solve this problem, we need to find the expression for \( f^6(6) + f^7(7) \), where the function \( f(x) = \frac{x-1}{x+1} \). We are dealing with a functional iteration question where \( f^{n+1}(x) = f(f^n(x)) \).

1. Let's first calculate the first few iterations of the function to observe any pattern:
2. Calculate \( f(x) \):
3. Calculate \( f^2(x) = f(f(x)) \):
4. Calculate \( f^3(x) = f(f^2(x)) = f\left(\frac{-1}{x}\right) \):
5. Calculate \( f^4(x) = f(f^3(x)) = f\left(\frac{x+1}{x-1}\right) \):
6. Calculate \( f^5(x) = f(f^4(x)) = f\left(\frac{1}{x}\right) \):
7. Calculate \( f^6(x) = f(f^5(x)) = f\left(\frac{1-x}{1+x}\right) \):
8. We note that \( f^6(x) = -x \), suggesting that when you reach \( f^6 \), the function becomes a negation of \( x \), and similarly, for \( f^7(x) = f^6(f(x)) = f(-x) \). Therefore:
   • \( f^6(6) = -6 \)
   • \( f^7(7) = f(-7) \)
9. Therefore, the required sum is:
10. Calculate the final expression:

Thus, the correct option is \(-\frac{3}{2}\).