解:∵点$A(a,$$b)$与点$B$关于$x$轴对称,将点$A$向左平移$3$个单位长度得到点$C$
∴$B(a,$$-b),$$C(a - 3,$$b)$
∵$B,$$C$两点都在函数$y = 2x + 1$的图象上
∴可得方程组$\begin {cases}2a + 1 = -b\\2(a - 3)+1 = b\end {cases},$解得$\begin {cases}{a = 1}\\{b=-3}\end {cases}$
∴点$A$的坐标为$(1,$$-3)$