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Let a function f(x), x ne 0 be such that...

Let a function `f(x), x ne 0` be such that
`f(x)+f((1)/(x))=f(x)*f((1)/(x))" then " f(x)` can be

A

`1-x^(2013)`

B

`sqrt(|x|)+1`

C

`(pi)/(2tan^(-1)|x|)`

D

`(2)/(1+k" In "|x|)`

Text Solution

Verified by Experts

The correct Answer is:
A, B, C, D
`(f((1)/(x))-1)=(1)/((f(x)-1))i.e., (f((1)/(x))-1)`
is reciprocal of `(f(x)-1).`
Now, for `f(x)=((pi)/(2))/(tan^(-1)|x|)`
`f(x)-1=(cot^(-1)|x|)/(tan^(-1)|x|),f((1)/(x))-1=(tan^(-1)|x|)/(cot^(-1)|x|)`
Also for `f(x)=(2)/(1+k" In " |x|)`
`f(x)-1=(1-k" In " |x|)/(1+k" In " |x|),f((1)/(x))-1=(1+k" In "|x|)/(1-k " In "|x|)`
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