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For $x\in \mathbb R - \{0,1\}$ , 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:
Option 1)
$f_3 (x)$
Option 2)
$\frac{1}{x}f_3 (x)$
Option 3)
$f_2 (x)$
Option 4)
$f_1 (x)$

Answers (1)

best_answer

COMPOSITION OF FUNCTIONS -

Let f? A → B and g? B → C be two functions.
composition of f and g, denoted by g o f, then g o f (x) = g (f (x)), ∀ x ∈ A.

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Given that

$f_{1}\left ( x \right )=\frac{1}{x},\; \; \; f_{2}\left ( x \right )=1-x\; \; \; \; and\; \; \; \; \; f_{3}\left ( x \right )=\frac{1}{1-x}$

$\left ( f_{2}0J0f_{1}\right )\left ( x \right )=f_{3}\left ( x \right )$

from the concept of composition.

$f_{2}o\left ( J\left ( f_{1}\left ( x \right ) \right ) \right )=f_{3}\left ( x \right )$

$f_{2}o\left ( J\left ( \frac{1}{x} \right ) \right )=\frac{1}{1-x}$

$1-J\left ( \frac{1}{x} \right )=\frac{1}{1-x}$

$J\left ( \frac{1}{x} \right )=\frac{x}{x-1}$

Now, $x\rightarrow \frac{1}{x}$

$J\left ( x \right )=\frac{\frac{1}{x}}{\frac{1}{x}-1}=\frac{1}{1-x}=f_{3}\left ( x \right )$

Option 1)
$f_3 (x)$
Option 2)

$\frac{1}{x}f_3 (x)$

Option 3)

$f_2 (x)$

Option 4)

$f_1 (x)$

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