(3) Given the function $f(x)=\left\{\begin{array}{lr}(3-a) x-3 & (x \leqslant 7), \\ a^{x-6} & (x>7),\end{array}\right.$ the sequence $\left\{a_{n}\right\}$ satisfies $a_{n}=$ $f(n)\left(n \in \mathbf{N}^{*}\right)$, and $\left\{a_{n}\right\}$ is an increasing sequence, then the range of the real number $a$ is ( ).
(A) $\left(\frac{9}{4}, 3\right)$
(B) $\left[\frac{9}{4}, 3\right)$
(C) $(1,3)$
(D) $(2,3)$
(3) From $\left\{a_{n}\right\}$ being an increasing sequence, we get $\left\{\begin{array}{l}3-a>0, \\ a>1,\end{array}\right.$ which means $12$. Therefore, the range of real number $a$ is $(2,3)$. Hence, the correct choice is D.