jmc

Set $$y=\frac{2x+7}{x-3}.$$Solving for $x,$ we find $$x=\frac{3y+7}{y-2}.$$Thus, for any value of $y,$ we can find a corresponding value of $x,$ except $y=2.$ Therefore, the range of the function is $\boxed{(-\infty ,2)\cup (2,\infty )}.$