If-f-x-ax-1-4x-3-x-R-x-3-4-and-ff-x-x-find-the-values-of-a-

Question Number 123356 by ZiYangLee last updated on 25/Nov/20

If f (x) = \frac{a x + 1}{4 x - 3}, x \in R, x \neq \frac{3}{4}

and f f (x) = x, find the values of a

Answered by liberty last updated on 25/Nov/20

\to {\begin{cases} f (x) = \frac{a x + 1}{4 x - 3} \\ f^{- 1} (x) = \frac{3 x + 1}{4 x - a} \end{cases}

\Rightarrow f (f (x)) = x; f^{- 1} (x) = f (x)

\Rightarrow \frac{3 x + 1}{4 x - a} = \frac{a x + 1}{4 x - 3}; (3 x + 1) (4 x - 3) = (a x + 1) (4 x - a)

12 x^{2} - 5 x - 3 = 4 {a x}^{2} + (4 - a^{2}) x - a

{\begin{cases} 12 = 4 a; a = 3 \\ - 5 = 4 - a^{2}; a = 3, a = - 3 (r e j e c t e d) \end{cases}

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