3. determine all functions $f: \mathbb{R} \backslash\{0\} \rightarrow \mathbb{R}$ for which the following applies

$$
\frac{1}{x} f(-x)+f\left(\frac{1}{x}\right)=x \quad \forall x \in \mathbb{R} \backslash\{0\}
$$

## Solution
For $x \neq 0$ we can replace $x$ in the equation with $-1 / x$ and obtain

$$
-x f\left(\frac{1}{x}\right)+f(-x)=-\frac{1}{x}
$$

If you multiply the original equation by $x$ and add it to (1), you get $2 f(-x)=x^{2}-1 / x$. If you finally replace $x$ with $-x$, you get

$$
f(x)=\frac{1}{2}\left(x^{2}+\frac{1}{x}\right), \quad \forall x \in \mathbb{R} \backslash\{0\}
$$

It is easy to calculate that this is actually a solution to the original equation.