[tex]\boxed{g(x)=3^{x-2}-1}[/tex]
Explanation:
We can write the graph of g(x) as:
[tex]g(x)=3^{x+c}+k[/tex]
Where:
[tex]c:Represents \ an \ horizontal \ shift \\ \\ k:Represents \ a \ vertical \ shift[/tex]
From the graph (2, 0) and (3, 2) are points that line on the graph, therefore:
[tex]For \ (2,0): \\ \\ 0=3^{2+c}+k \\ \\ k=-3^23^c \\ \\ \\ For \ (3,2): \\ \\ 2=3^{3+c}+k \\ \\ k=2-3^33^c[/tex]
Equating k:
[tex]-3^23^c=2-3^33^c \\ \\ Solving \ for \ c: \\ \\ 3^33^c-3^23^c=2 \\ \\ 3^c(3^3-3^2)=2 \\ \\ 3^c(18)=2 \\ \\ 3^c=\frac{1}{9} \\ \\ 3^c=\frac{1}{3^2} \\ \\ 3^c=3^{-2} \\ \\ So, \ by \ property: \\ \\ a^x=a^y \ then \ x=y \\ \\ \\ So: \\ \\ c=-2[/tex]
Solving for k:
[tex]k=-3^23^c \\ \\ k=-3^23^{-2} \\ \\ k=-3^{2-2} \\ \\ k=-3^{0} \\ \\ k=1[/tex]
Finally, our equation is:
[tex]\boxed{g(x)=3^{x-2}-1}[/tex]
Learn more:
Even functions: https://brainly.com/question/11309886
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