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

If `f(x)+2f(1/x)=3x , x!=0,` and `S={x in R :f(x)=f(-x)}` ; then S: (1) is an empty set. (2) contains exactly one element. (3) contains exactly two elements. (4) contains more than two elements

A

contains exactly one element

B

contains exactly two elements

C

contains more than two elements

D

is an empty set

Text Solution

Verified by Experts

The correct Answer is:
B
`f(x)+2f((1)/(x))=3x " (1)" `
Replacing x by `(1)/(x)`, we get
`f((1)/(x)) +2f(x)=(3)/(x) " (2)" `
Solving (1) and (2) for `f(x)`, we get
`-3f(x)=3x-(6)/(x) or f(x)=(2)/(x)-x`
Now given `f(x)=f(-x)`
` :. (2)/(x)-x= -(2)/(x)+x`
` :. (4)/(x)=2x`
` :. (2)/(x)=x`
` :. x=+-sqrt(2)`
Hence set S contains two elements.
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