So we must find an exponential function. The general form of these functions is:
[tex]f(x)=a\cdot b^x[/tex]
By using the table provided by the question we can find the values of a and b and thus the expression of the function. In this case we can take two pairs of values from the table. On one hand we have that x=1 and f(1)=3, on the other hand we have x=2 and f(2)=9. Using the former formula we can build two equations:
[tex]\begin{gathered} f(1)=3=a\cdot b^1 \\ f(2)=9=a\cdot b^2 \\ \text{Then we have a system of two equations:} \\ 3=a\cdot b \\ 9=a\cdot b^2 \end{gathered}[/tex]
Let's use the first one:
[tex]\begin{gathered} 3=a\cdot b \\ b=\frac{3}{a} \end{gathered}[/tex]
Now we substitute 3/a in place of b in the second equation:
[tex]\begin{gathered} 9=a\cdot b^2 \\ 9=a\cdot(\frac{3}{a})^2 \\ 9=a\cdot\frac{9}{a^2}=\frac{a\cdot9}{a\cdot a}=\frac{9}{a} \\ 9=\frac{9}{a} \\ a\cdot9=9 \\ a=\frac{9}{9}=1 \end{gathered}[/tex]
So a=1 and since we found that b=3/a then b=3. Then the exponential function that we are looking for is:
[tex]f(x)=3^x[/tex]
