3. The range of the function $f(x)=\sqrt{2 x-x^{2}}+x$ is $\qquad$
$$
f(x)=\sqrt{1-(x-1)^{2}}+x \text {, let } x=\sin \theta+1, \theta \in\left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \text {, }
$$

then $f(x)=\cos \theta+\sin \theta+1=\sqrt{2} \sin \left(\theta+\frac{\pi}{4}\right)+1$, since $\theta+\frac{\pi}{4} \in\left[-\frac{\pi}{4}, \frac{3 \pi}{4}\right]$, so the range of $f(x)=\sqrt{2 x-x^{2}}+x$ is $[0, \sqrt{2}+1]$.