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Find all the functions $f: \mathbb R^*\to \mathbb R $ such that: $$ \forall x\in \mathbb R^*: f(x) +3f\left(\frac{1}{x}\right) = x^2 $$

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.

Mohsen Shahriari
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Manuela NIEVES
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2 Answers2

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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 ...

Donald Splutterwit
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Substituting $\frac{1}{x} \mapsto x$ and then multiplying both sides by $3$, we have that:

$$3 f \left(\frac{1}{x} \right) + 9f(x) = \frac{3}{x^2}$$ $$-3 f \left(\frac{1}{x} \right) - f(x) = -x^2$$

which implies $8f(x) = \frac{3}{x^2} - x^2$, or $f(x) = \frac{3-x^4}{8x^2}$.

Toby Mak
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