Let f ( x ) = x x + 1 , x - 1 . Then, for what value of is f ( f ( x ) ) = x ? [2001]

Question

Let $f (x) = \frac{α x}{x + 1},$ $x \neq - 1 .$ Then, for what value of $α$ is $f (f (x)) = x ?$ [2001]

Smart Achievers · Accepted Answer

(4)

$f (x) = \frac{α x}{x + 1},$ $x \neq - 1$

Now, $f (f (x)) = x$ $\Rightarrow \frac{α (\frac{α x}{x + 1})}{\frac{α x}{x + 1} + 1} = x$

$\Rightarrow \frac{α^{2} x}{(α + 1) x + 1} = x$ $\Rightarrow (α + 1) x^{2} + (1 - α^{2}) x = 0$

$\Rightarrow α + 1 = 0$ and $1 - α^{2} = 0$

$∴$ $α = - 1$

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