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  4. For x ∈ R - 0, 1 , let f1 (x) = (1/x) , f2 (x) = 1 -x and f3 (x) = (1/1-x) be three given functions. If a function, J(x) satisfies (f2 °J °f1 )(x) = f3 (x) then J(x) is equal to :-

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Q. For $x \in R - \{ 0, 1\} $ , let $f_1 (x) = \frac{1}{x} , f_2 (x) = 1 -x$ and $f_3 (x) = \frac{1}{1-x}$ be three given functions. If a function, J(x) satisfies $(f_2 {^{\circ}J} {^{\circ}f}_1 )(x) = f_3 (x)$ then J(x) is equal to :-

JEE MainJEE Main 2019Relations and Functions - Part 2

Solution:

Given $f_{1} \left(x\right) = \frac{1}{x} , f_{2}\left(x\right)=1-x $ and $ f_{3}\left(x\right)= \frac{1}{1-x}$
$ \left(f_{2} \circ J \circ f_{1}\right)\left(x\right)=f_{3}\left(x\right) $
$ f_{2} \circ\left(J\left(f_{1}\left(x\right)\right)\right) = f_{3}\left(x\right) $
$ f_{2} \circ\left(J\left(\frac{1}{x}\right)\right) = \frac{1}{1-x} $
$ 1 -J \left(\frac{1}{x}\right) = \frac{1}{1-x} $
$ J \left(\frac{1}{x}\right) = 1 - \frac{1}{1-x} = \frac{-x}{1-x} = \frac{x}{x-1} $
Now $ x \to \frac{1}{x}$
$ J\left(x\right) = \frac{\frac{1}{x}}{\frac{1}{x}-1} = \frac{1}{1-x} =f_{3}\left(x\right) $