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Self-inverse fractional linear function: f(f(x)) = x

This type of question, asked 3 times: 2021 Paper 12, 10 Apr 2023, 31 Jan 2024. All 3 questions below, oldest first.

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  1. Question 1

    2021 Paper 12

    46
    Let f:R−{α6}→Rf:\mathbb{R}-\left\{ \frac{\alpha}{6} \right\}\rightarrow R be defined by f(x)=5x+36x−αf(x) =\frac{5x + 3}{6x -\alpha}. Then the value of α\alpha for which (fof) (x)=x(x) =x, for all x∈R−{α6}x \in R -\left\{ \frac{\alpha}{6} \right\}, is:

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  2. Question 2

    10 Apr 2023

    63
    If f(x)=(tan⁡1∘)x+log⁡e(123)xlog⁡e(1234)−(tan⁡1∘),x>0f(x) =\frac{\left( \tan1^{\circ} \right)x +\log_{e}(123)}{x\log_{e}(1234) -\left( \tan1^{\circ} \right)},x > 0, then the least value of f(f(x))+f(f(4x))f(f(x)) + f\left( f\left( \frac{4}{x} \right) \right) is

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  3. Question 3

    31 Jan 2024

    66
    If f(x)=4x+36x−4,x≠23f(x) =\frac{4x + 3}{6x - 4},x \neq\frac{2}{3} and (fof) (x)=g(x)(x) = g(x), where g:R−{23}→R−{23}g:\mathbb{R}-\left\{ \frac{2}{3} \right\}\rightarrow\mathbb{R}-\left\{ \frac{2}{3} \right\}, then (gogog) (4)(4) is equal to

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