Finding all functions that verify a functional equation

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.

• Notice that 0 does not belong in your domain. What is f(1/0)? – Francisco José Letterio Jun 27 '18 at 23:36
• substitute $\frac 1x$ for $x$ in your equations. – 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 ...
• Yes Thnak you I just solved it, I found $$f(x) = \cfrac{3-x^2}{8x^2}$$ – Manuela NIEVES Jun 27 '18 at 23:47