[tex]f(x)=3^x\\\\g(x)=3^{x+a}+b\\\\\text{We have the points (2, 0) and (3, 2). Substitute to the equation of f(x)}:\\\\0=3^{2+a}+b\to b=-3^{2+a}\\\\2=3^{3+a}+b\qquad\text{substitute b}\\\\2=3^{3+a}-3^{2+a}\qquad\text{use}\ a^n\cdot a^m=a^{n+m}\\\\3^3\cdot3^a-3^2\cdot3^a=2\\\\27\cdot3^a-9\cdot3^a=2\\\\18\cdot2^a=2\qquad\text{divide both sides by 2}\\\\9\cdot3^a=1\\\\3^2\cdot3^a=3^0\\\\3^{2+a}=3^0\iff2+a=0\qquad\text{subtract 2 from both sides}\\\\\boxed{a=-2}[/tex]
[tex]\text{Substitute the value of a to the equation of b}\\\\b=-3^{2+(-2)}\\\\b=-3^0\\\\\boxed{b=-1}\\\\Answer:\ \boxed{\boxed{g(x)=3^{x-2}-1}}[/tex]
