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

Options

A. contains exactly one element
B. contains exactly two elements
C. contains more than two elements
D. is an empty set

Answer

B

Explanation

`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.