Answer:
[tex]f(x) = \frac{24}{25} * \frac{5}{6}^x[/tex]
Step-by-step explanation:
Given
[tex](x_1,y_1) = (2,\frac{2}{3})[/tex]
[tex](x_2,y_2) = (3,\frac{5}{9})[/tex]
Required
Write the equation of the function [tex]f(x) = ab^x[/tex]
Express the function as:
[tex]y = ab^x[/tex]
In: [tex](x_1,y_1) = (2,\frac{2}{3})[/tex]
[tex]y = ab^x[/tex]
[tex]\frac{2}{3} = a * b^2[/tex] --- (1)
In [tex](x_2,y_2) = (3,\frac{5}{9})[/tex]
[tex]y = ab^x[/tex]
[tex]\frac{5}{9} = a * b^3[/tex] --- (2)
Divide (2) by (1)
[tex]\frac{5}{9}/\frac{2}{3} = \frac{a*b^3}{a*b^2}[/tex]
[tex]\frac{5}{9}/\frac{2}{3} = b[/tex]
[tex]\frac{5}{9}*\frac{3}{2} = b[/tex]
[tex]\frac{5}{3}*\frac{1}{2} = b[/tex]
[tex]\frac{5}{6} = b[/tex]
[tex]b = \frac{5}{6}[/tex]
Substitute 5/6 for b in (1)
[tex]\frac{2}{3} = a * b^2[/tex]
[tex]\frac{2}{3} = a * \frac{5}{6}^2[/tex]
[tex]\frac{2}{3} = a * \frac{25}{36}[/tex]
[tex]a = \frac{2}{3} * \frac{36}{25}[/tex]
[tex]a = \frac{2}{1} * \frac{12}{25}[/tex]
[tex]a = \frac{24}{25}[/tex]
The function: [tex]f(x) = ab^x[/tex]
[tex]f(x) = \frac{24}{25} * \frac{5}{6}^x[/tex]
