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If f(x)+2f((1)/(x))=3x,x ne 0, and S={x ...

If `f(x)+2f((1)/(x))=3x,x ne 0, and S={x in R: f(x)=f(-x)},` then S

A

is an empty set

B

contains exactly one elements

C

contains exactly two elements

D

contains more than two elements

Text Solution

Verified by Experts

The correct Answer is:
C
We have, `f(x)+2f((1)/(x))=3x, x ne 0 " …(i)" `
On replacing x by `(1)/(x)` in the above equation, we get
`f((1)/(x))+2f(x)=(3)/(x)`
`rArr 2f(x)+f((1)/(x))=(3)/(x) " ….(ii)" `
On multiplying Eq. (ii) by 2 and subtracting Eq. (i) from Eq. (ii), we get
`4f(x)+2f((1)/(x))=(6)/(x)`
`(underset (-)(f)(x)underset(-)(+)2f((1)/(x))underset(-)(=)3x)/(3f(x)=(6)/(x)-3x)`
`rArr f(x)=(2)/(x) -x`
Now, consider ` f(x)=f(-x)`
`rArr (2)/(x)-x=-(2)/(x)+x rArr (4)/(x)=2x`
`rArr 2x^(2)=4 rArr x^(2) =2`
`rArr x = pm sqrt(2)`
Hence, S contains exactly two elements.
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