Question

What is the exponential function modeled by the following table?

x f(x)
2 9
3 27
4 81

Answer 1

3^2 = 9
3^3 = 27
4^3 = 81
so f(x) = 3^x

Answer 2

`f(x)=9^(x-1)`

Explanation:

The general form of the exponential function is,

`y=ab^x`

The points on the graph will be,

`(2,9),(3,27),(4,81)`

Putting the points in the function,

`9=ab^2` -----------------------------------1

`27=ab^3` ---------------------------------2

Dividing equation 2 by equation 1,

`\Rightarrow (27)/(3)=(ab^3)/(ab^2)`

`\Rightarrow 9=(b^3)/(b^2)`

`\Rightarrow b=9`

Putting the value of b in equation 1,

`\Rightarrow a(9)^2=9`

`\Rightarrow a=(9)/(9^2)`

`\Rightarrow a=(1)/(9)`

Putting all the values,

`\Rightarrow y=(1)/(9)\cdot 9^x`

`\Rightarrow y=9^(x-1)`