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  2. Help me answer: - For, letandbe three given functions. If a function,satisfiesthenis equal to: - Sets, Relations and Functions - JEE Main
# Engineering

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)
A admin

 

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.
 

-

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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