Question

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

Answer 1

Correct Answer - 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.