[tex]f(x)=3^x\\\\g(x)=3^{x+a}+b\\\\\text{We have the points (-2, 2) and (-1, 4). Substitute:}\\\\\underline{+\left\{\begin{array}{ccc}2=3^{-2+a}+b\\4=3^{-1+a}+b\end{array}\right}\qquad\text{subtract both sides of the equations}\\.\ \ -2=3^{-2+a}-3^{-1+a}\\-2=3^{-2}\cdot3^a-3^{-1}\cdot3^a\\-2=\dfrac{1}{3^2}\cdot3^a-\dfrac{1}{3}\cdot3^a\\-2=\left(\dfrac{1}{9}-\dfrac{1}{3}\right)\cdot3^a\\-2=\left(\dfrac{1}{9}-\dfrac{3}{9}\right)\cdot3^a\\-2=-\dfrac{2}{9}\cdot3^a\qquad\text{multiply both sides by}\ -\dfrac{9}{2}[/tex]
[tex]3=3^a\\3^a=3^2\to a=2\\\\\text{substitute value of a to the second equation}\\\\3^{-1+2}+b=4\\3^1+b=4\\3+b=4\qquad\text{subtract 3 from both sides}\\b=1\\\\Answer:\ \boxed{g(x)=3^{x+2}+1}[/tex]
