[tex]\begin{gathered} P1(1,2) \\ f(1) \\ 2=ab^{}\text{ (1)} \\ P2(3,32) \\ f(2) \\ 32=ab^3(2) \\ from\text{ (1)} \\ a=\frac{2}{b} \\ \text{now in 2 } \\ 32=(\frac{2}{b})\cdot b^3 \\ 32=2b^2 \\ \frac{32}{2}=b^2 \\ 16=b^2 \\ b=\sqrt[]{16} \\ b=4 \\ f(x)=a\cdot4^x \\ U\sin g\text{ point 2 (3,32)} \\ 32=a\cdot4^3 \\ 32=64a \\ a=\frac{32}{64} \\ \\ a=\frac{1}{2} \\ \\ f(x)=\frac{1}{2}\cdot4^x \\ \text{The exponential function is }f(x)=\frac{1}{2}\cdot4^x \end{gathered}[/tex]
