Let `f: R^+vecR` be a function which satisfies `f(x)dotf(y)=f(x y)+2(1/x+1/y+1)` for `x , y > 0.` Then find `f(x)dot`

Explanation

We have `f(x)xxf(y)=f(xy)+2((1)/(x)+(1)/(y)+1) " …(1)" `
To get `f(x)` we put `y=1`,
` :. f(x)xxf(1)=f(x)+2((1)/(x)+2)`
`=f(x)+2((2x+1)/(x))`
`impliesf(x)(f(1)-1)=(2(2x+1))/(x)`
`implies f(x)=(2(2x+1))/(x(f(1)-1)) " ...(2)" `
Now, we need the value of f(1).
Put `x=1` and `y=1` in (1), we get
`(f(1))^(2)-f(1)-6=0`
`implies f(1)=3 " or "f(1)= -2`
For `f(1)=3,f(x)=(2x+1)/(x)`
and for `f(1)= -2, f(x)=(2(2x+1))/(-3x)`