For $x \in \mathbf{R}-\{0,1\}$, 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, $\mathrm{J}(x)$
satisfies $\left(f_{2} o J o f_{1}\right)(x)=f_{3}(x)$ then $J(x)$ is equal to:
Correct Option: 1
The given relation is
$\left(f_{2} o J o f_{1}\right)(x)=f_{3}(x)=\frac{1}{1-x}$
$\Rightarrow\left(f_{2} o J\right)\left(f_{1}(x)\right)=\frac{1}{1-x}$
$\Rightarrow\left(f_{2} o J\right)\left(\frac{1}{x}\right)=\frac{1}{1-\frac{1}{\frac{1}{x}}}=\frac{\frac{1}{x}}{\frac{1}{x}-1} \quad\left[\because f_{1}(x)=\frac{1}{x}\right]$
$\Rightarrow\left(f_{2} o J\right)(x)=\frac{x}{x-1}$ $\left[\frac{1}{x}\right.$ is replaced by $\left.x\right]$
$\Rightarrow f_{2}(J(x))=\frac{x}{x-1}$
$\Rightarrow 1-J(x)=\frac{x}{x-1}$ $\left[\because f_{2}(x)=1-x\right]$
$\therefore J(x)=1-\frac{x}{x-1}=\frac{1}{1-x}=f_{3}(x)$
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