Based on the graph,
At,
f(2) = 1 (1)
f(9) = 6 (2)
[tex]\text{ f(x) = ab}^{\text{x}}[/tex]
At f(2) = 1,
[tex]\text{ 1 = ab}^2[/tex][tex]\text{ a = }\frac{\text{ 1}}{b^2}[/tex]
At f(9) = 6 ,
[tex]\text{ 6 = ab}^9[/tex][tex]\text{ a = }\frac{\text{ 6}}{b^9}[/tex]
Equate them to find b, a = a,
[tex]\text{ }\frac{\text{ 1}}{b^2}\text{ = }\frac{\text{ 6}}{b^9}\text{ }\rightarrow\text{ }\frac{b^9}{b^2}\text{ = }\frac{6}{1}[/tex][tex]\text{ b}^7\text{ = 6}[/tex][tex]\text{ b = }\sqrt[7]{6}[/tex]
Let's find a at f(2) = 1 and the seventh root of 6,
[tex]\text{ f(x) = ab}^{\text{x}}[/tex][tex]\text{ 1 = a(}\sqrt[7]{6})^2[/tex][tex]\text{ a = }\frac{\text{ 1}}{\sqrt[7]{36}}[/tex]
Let's now complete the equation.
[tex]\text{ f(x) = ab}^{\text{x}}[/tex]
[tex]\text{ f(x) =}\frac{1}{\sqrt[7]{36}}\text{ x (}\sqrt[7]{6})^x=\frac{\text{(}\sqrt[7]{6})^x}{\sqrt[7]{36}}[/tex][tex]\text{ f(x) }=\frac{\text{(}\sqrt[7]{6})^x}{\sqrt[7]{36}}[/tex]
