Answer:
[tex]f(x)=2(4)^x[/tex]
Explanations:
The standard exponential function is given as:
[tex]y=ab^x[/tex]
Given the coordinate points (2, 32) and (3, 128)
The two exponential functions will be;
[tex]\begin{gathered} y_1=ab^{x_1} \\ y_2=ab^{x_2} \\ \end{gathered}[/tex]
Substitute the given coordinates to have:
[tex]\begin{gathered} 32=ab^2 \\ 128=ab^3 \end{gathered}[/tex]
Divide both expressions to have:
[tex]\begin{gathered} \frac{128}{32}=\frac{ab^3}{ab^2} \\ b^{3-2}=4 \\ b=4 \end{gathered}[/tex]
Determine the value of "a"
[tex]\begin{gathered} 32=ab^2 \\ 32=a(4)^2 \\ 32=16a \\ a=2 \end{gathered}[/tex]
Substitute b = 4 and a = 2 into the original equation to have;
[tex]\begin{gathered} y=ab^x \\ y=2(4)^x \\ f(x)=2(4)^x \end{gathered}[/tex]
This gives the required exponential function
