Find all the functions $f:$ $\Bbb R^*\rightarrow \Bbb R $ that prove: $$ \forall x\in \Bbb R^*: f(x) +3f(\cfrac{1}{x}) = x^2 $$

My answers or what I tried to do is :

I put $f$ as a solution to the basic equation and I followed the substitution method by giving $x=0$ which gives us $f(0)=0 $ and it led me nothing since this method has usually two variables

So, I tried next both of injectivity and surjectivity but it's still a dead end .

Note: This is not a homework(since school is over). I just wanted a more valable or strategic method to this kind of equations since we never studied it I just found in a book but I couldn't understand it very well alone.

  • 2
    $\begingroup$ Notice that 0 does not belong in your domain. What is f(1/0)? $\endgroup$ – Francisco José Letterio Jun 27 '18 at 23:36
  • 2
    $\begingroup$ substitute $\frac 1x$ for $x$ in your equations. $\endgroup$ – Doug M Jun 27 '18 at 23:42

Substitute $1/x$ for $x$ and we have \begin{eqnarray*} f(1/x)+3f(x)=\frac{1}{x^2}. \end{eqnarray*} Now multiply this by $3$ and subtract the orignal eqaution & we have ...

  • 1
    $\begingroup$ Yes Thnak you I just solved it, I found $$ f(x) = \cfrac{3-x^2}{8x^2} $$ $\endgroup$ – Manuela NIEVES Jun 27 '18 at 23:47
  • $\begingroup$ \begin{eqnarray*} f(x) = \cfrac{3-\color{red}{x^4}}{8x^2} \end{eqnarray*} $\endgroup$ – Donald Splutterwit Jun 27 '18 at 23:49
  • $\begingroup$ Sorry, I didn't notice $\endgroup$ – Manuela NIEVES Jun 27 '18 at 23:53

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.