3. Let $[x]$ denote the greatest integer not exceeding the real number $x$. Then the range of the function
$$
f(x)=[2 \sin x \cdot \cos x]+[\sin x+\cos x]
$$

is $\qquad$ .
3. $\{-2,-1,1,2\}$.

From $f(x+2 \pi)=f(x)$, and $f\left(\frac{\pi}{2}-x\right)=f(x)$, we know that the period of $f(x)$ is $2 \pi$, and the graph is symmetric about the line $x=\frac{\pi}{4}$.

Therefore, we only need to discuss the values of $f(x)$ when $x \in\left[\frac{\pi}{4}, \frac{5 \pi}{4}\right]$.
It is easy to see that when $x=\frac{\pi}{4}$, $f(x)$ reaches its maximum value of 2;
When $x \in\left(\frac{3 \pi}{4}, \pi\right) \cup\left(\pi, \frac{5 \pi}{4}\right)$, $f(x)$ reaches its minimum value of -2.
Thus, the range of $f(x)$ is $\{-2,-1,1,2\}$.