Question

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 (f₂oJof₁)(x) = f₃(x) then J (x) is equal to: (A)f₃(x) (B) (1/x)f₃(x)
(C) f₂(x) (D) f₁(x)
[JEE Main 2019]

Answer 1

J(x) is equal to f3(x)

• f₁(x) = 1/x , f₂(x) = (1 - x) and f₃(x) = 1/(1 - x) , be three given functions.

• Also, J(x) satisfies  (f₂oJof₁)(x) = f₃(x)

i.e.,

• f2( J ( f1 (x) ) ) = f3 (x) = 1/(1 - x)
• f2( J ( 1/x) ) = 1/( 1 - x )
• f2(x ) = 1 - x
• 1/(1 -x) = 1 + x/(1 - x ) =1 +  $\frac{1}{\frac{1}{x} - 1 }$ = 1 -  $\frac{1}{1 - \frac{1}{x} }$  ==>

• 1 -  $\frac{1}{1 - \frac{1}{x} }$ = f2( $\frac{1}{1 - \frac{1}{x} }$) = >
• J(1/x) = 1/(1 - 1/x) ==>
• J(x) = 1/(1 - x) ==>
• J(x) = f3(x)

Hence proved that J(x) is equal to f3(x)