9. Given the function
$$
f(x)=\sin \left(\omega x+\frac{\pi}{2}\right) \cdot \sin \left(\omega x+\frac{\pi}{4}\right)(\omega>0)
$$

has the smallest positive period of $\pi$. Then the range of $f(x)$ on the interval $\left[0, \frac{\pi}{2}\right]$ is . $\qquad$
9. $\left[0, \frac{2+\sqrt{2}}{4}\right]$