## Exercise 4

Find all functions $f: \mathbb{R}^{*} \rightarrow \mathbb{R}^{*}$ that satisfy $x f\left(\frac{x}{2}\right)-f\left(\frac{2}{x}\right)=1$ for all non-zero real numbers $x$.
## Corrected

Let $x>0$. Apply the relation from the statement at $x$ and $\frac{4}{x}$ to obtain the system

$$
\left\{\begin{aligned}
x f\left(\frac{x}{2}\right)-f\left(\frac{2}{x}\right) & =1 \\
-f\left(\frac{x}{2}\right)+\frac{4}{x} f\left(\frac{2}{x}\right) & =1
\end{aligned}\right.
$$

By adding the first line to $x$ times the second, we obtain the condition $3 f\left(\frac{2}{x}\right)=1+x$. Applying this relation at $\frac{2}{x}$, we deduce

$$
f(x)=\frac{2+x}{3 x}
$$

However, $f(-2)=0$. Since $f$ must take values in $\mathbb{R}^{*}$, we conclude that there are no functions satisfying the conditions of the statement.