Consider the function f : R ? R defined by f ( x ) = 2 x 1 + 9 x 2 . If the composition of f , ( f o f o f o . . . o f ) ? 10 times ( x ) = 2 10 x 1 + 9 ? x 2 , then the value of 3 ? + 1 is equal to _________ . [2024]

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Solution

Q.
Consider the function $f : R \to R$ defined by $f (x) = \frac{2 x}{\sqrt{1 + 9 x^{2}}} .$ If the composition of $\underset{10 times}{f, \underset{⏟}{(f o f o f o . . . o f)}} (x)$ $= \frac{2^{10} x}{\sqrt{1 + 9 α x^{2}}},$ then the value of $\sqrt{3 α + 1}$ is equal to _________ . [2024]

Ans.
(1024)

We have, $f (x) = \frac{2 x}{\sqrt{1 + 9 x^{2}}}$

   $∴$    $(f o f) (x) = \frac{2 f (x)}{\sqrt{1 + 9 (f {(x))}^{2}}} = \frac{\frac{4 x}{\sqrt{1 + 9 x^{2}}}}{\sqrt{1 + 9 \times \frac{4 x^{2}}{1 + 9 x^{2}}}} = \frac{4 x}{\sqrt{1 + 45 x^{2}}} = \frac{2^{2} x}{\sqrt{1 + 5 \times 9 x^{2}}}$

$(f o f o f) (x) = \frac{4 \times \frac{2 x}{\sqrt{1 + 9 x^{2}}}}{\sqrt{1 + 45 \times \frac{4 x^{2}}{1 + 9 x^{2}}}} = \frac{2^{3} x}{\sqrt{1 + 21 \times 9 x^{2}}}$

   $(f o f o f o f) (x) = \frac{2^{4} x}{\sqrt{1 \times 85 \times 9 x^{2}}}$

   $∴$    $\underset{n times}{\underset{⏟}{(f o f o f o . . . . . o f)}} (x)$ $= \frac{2^{n} x}{\sqrt{1 + 9 (\frac{4^{n} - 1}{3}) x^{2}}}$

   $∴$    $\underset{10 times}{\underset{⏟}{(f o f o f o . . . . . o f)}} (x)$ $= \frac{2^{10} x}{\sqrt{1 + 9 (\frac{4^{10} - 1}{3}) x^{2}}} = \frac{2^{10} x}{\sqrt{1 + 9 α x^{2}}}$

( $∵$ Given)

On comparing, we get

   $α = \frac{4^{10} - 1}{3} \Rightarrow 3 α + 1 = 4^{10}$

$\Rightarrow \sqrt{3 α + 1} = \sqrt{4^{10}} = 4^{5} = 1024 .$

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