3. If $2 f(1-x)+1=x f(x)$, then $f(x)=$
3. $\frac{x-3}{x^{2}-x+4}$.

Let there exist a function $f(x)$ that satisfies the given equation
$$
2 f(1-x)+1=x f(x)
$$

Substitute $1-x$ for $x$ to get
$$
2 f(x)+1=(1-x) f(1-x) .
$$

From (1), we have $f(1-x)=\frac{1}{2}[x f(x)-1]$.
Substitute (3) into (2), to get
$$
2 f(x)+1=(1-x) \cdot \frac{1}{2}[x f(x)-1] .
$$

This leads to
$$
f(x)=\frac{x-3}{x^{2}-x+4} .
$$

Upon verification, the obtained function satisfies equation (1).