If f ( x ) = 4 x + 3 6 x - 4 , x ? 2 3 and ( f o f ) ( x ) = g ( x ) , where g : R - { 2 3 } ? R - { 2 3 } , then ( g o g o g ) ( 4 ) is equal to [2024]

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Solution

Q.
If $f (x) = \frac{4 x + 3}{6 x - 4}, x \neq \frac{2}{3}$ and $(f o f) (x) = g (x)$ , where $g : R - {\frac{2}{3}} \to R - {\frac{2}{3}},$ then $(g o g o g) (4)$ is equal to [2024]

1 $\frac{19}{20}$

2 $- \frac{19}{20}$

3 $- 4$

4 $4$

Ans.
(4)

Here, $f (x) = \frac{4 x + 3}{6 x - 4}$

   $f [f (x)] = \frac{4 f (x) + 3}{6 f (x) - 4} = \frac{4 (\frac{4 x + 3}{6 x - 4}) + 3}{6 (\frac{4 x + 3}{6 x - 4}) - 4}$

   $= \frac{16 x + 12 + 18 x - 12}{6 x - 4} = \frac{34 x}{34} = x$

   $∴$    $f [f (x)] = x \Rightarrow g (x) = x$

Now, $(g o g o g) (4) = g o g [g (4)] = g o g (4) = g (4) = 4$

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