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Iterates of a function: f composed with itself n times

This type of question, asked 3 times: 26 Jun 2022, 30 Jan 2023, 4 Apr 2024. All 3 questions below, oldest first.

Try each one first, then open its answer.

  1. Question 1

    26 Jun 2022

    61
    Let f(x)=x−1x+1,x∈R−{0,−1,1)f(x) =\frac{x - 1}{x + 1}\mathbf{,x \in R - \{ 0, - 1,1)}. If fn+1(x)=f(fn(x))f^{n + 1}(x) = f\left( f^{n}(x) \right) for all n∈Nn\in N, then f6(6)+f7(7)f^{6}(6) +f^{7}(7) is equal to:

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  2. Question 2

    30 Jan 2023

    87
    Let f1(x)=3x+22x+3,x∈R−{−32}f^{1}\left( x \right)=\frac{3x + 2}{2x + 3},x \in R -\left\{ \frac{- 3}{2} \right\}For n≥2n \geq 2, define fn(x)=f1f^{n}(x) =f^{1} of fn−1(x)f^{n - 1}\left( x \right)If f5(x)=ax+bbx+a,gcd(a,b)=1f^{5}(x) =\frac{ax + b}{bx + a},gcd(a,b) = 1, then a+ba + b is equal to

  3. Question 3

    4 Apr 2024

    174
    Consider the function f:R→Rf:R \rightarrow R defined by f(x)=2x1+9x2f(x) =\frac{2x}{\sqrt{1 + 9x^{2}}}. If the composition of f,( f o f o f o...o f)10 times (x)=210x1+9αx2,f,\underset{10\text{~times~}}{\overset{(\text{~f o f o f o...o~}f)}{}}(x) =\frac{2^{10}x}{\sqrt{1 + 9\alpha x^{2}}}, then the value of 3α+1\sqrt{3\alpha + 1} is equal to .....