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Solve a functional equation in f(x) and f(c/x) or f(1/(1 - x))

This type of question, asked 4 times: 2021 Paper 2, 1 Feb 2023, 3 Apr 2025, 24 Jan 2026 S2. All 4 questions below, oldest first.

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  1. Question 1

    2021 Paper 2

    83
    If a+α=1,b+β=2a+\alpha= 1,b+\beta= 2 and af(x)+αf(1x)=bx+βx,x≠0af(x) +\alpha f\left( \frac{1}{x} \right)=bx+\frac{\beta}{x},x\neq 0, then the value of the expression
    f(x)+f(1x)x+1x is \frac{f(x) +f\left( \frac{1}{x} \right)}{x+\frac{1}{x}}\text{~is~}

  2. Question 2

    1 Feb 2023

    155
    Let f:R−{0,1}→Rf:R - \{0,1\} \rightarrow R be a function such that f(x)+f(11−x)=1+xf(x) + f\left( \frac{1}{1 - x} \right)= 1 + x. Then f(2)f(2) is equal to

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  3. Question 3

    3 Apr 2025

    135
    Let ff be a function such that f(x)+3f(24x)f(x) + 3f\left( \frac{24}{x} \right) =4x,x≠0= 4x,x\neq 0. Then f(3)+f(8)f(3) +f(8) is equal to.

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  4. Question 4

    24 Jan 2026 S2

    67
    Let f be a function such that 3f(x)+2f(m19x)=5x3f(x) + 2f\left( \frac{m}{19x} \right)= 5x, x≠0x \neq 0, where m=∑i=19(i)2m =\sum_{i = 1}^{9} (i)^{2}. Then f(5)−f(2)f(5) - f(2) is equal to

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