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For x in R-{0,1}, " let " f(1)(x)=(1)/(x...

For `x in R-{0,1}, " let " f_(1)(x)=(1)/(x), f_(2)(x)=1-x and f_(3)(x)=(1)/(1-x)` be three given functions. If a function, J(x) satisfies `(f_(2) @J@f_(1))(x)=f_(3)(x), " then " J(x)` is equal to

A

`f_(2)(x)`

B

`f_(3)(x)`

C

`f_(1)(x)`

D

`(1)/(x)f_(3)(x)`

Text Solution

Verified by Experts

The correct Answer is:
B
We have,
`f_(1)(x)=(1)/(x), f_(2)(x)=1-x and f_(3)(x)=(1)/(1-x)`
Also, we have `(f_(2)@ J @f_(1))(x)=f_(3)(x)`
`rArr f_(2)((J@f_(1)(x))=f_(3)(x)`
`rArr f_(2)(J(f_(1)(x))=f_(3)(x)`
`rArr 1-J(f_(1)(x))=(1)/(1-x)`
` " "[ because f_(2)(x)=1-x and f_(3)(x)=(1)/(1-x)]`
`rArr 1-J((1)/(x))=(1)/(1-x) " [ because f_(1)(x)=(1)/(x)]`
`rArr J((1)/(x))=1-(1)/(1-x)`
`=(1-x-1)/(1-x)= (-x)/(1-x)`
Now, put `(1)/(x)=X,` then
`J(X)=((1)/(X))/(1-(1)/(X)) " "[ because x=(1)/(X)]`
`=(-1)/(X-1)=(1)/(1-X)`
` rArr J(X)=f_(3)(X) or J(x)=f_(3)(x)`
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