Question

If f(x) is an exponential function where f(3) = 18 and f(7.5) = 60, then find the

value of f(12), to the nearest hundredth.

Answer 1

Given:

f(x) is an exponential function where f(3) = 18 and f(7.5) = 60.

To find:

The value of f(12), to the nearest hundredth.

Solution:

Let the exponential function be

`f(x)=ab^x` ...(i)

It is given that
`f(3) = 18`. Substitute
`x=3` in (i).

`f(3)=ab^3`

`18=ab^3` ...(ii)

It is given that
`f(7.5) = 60`. Substitute
`x=7.5` in (i).

`f(7.5)=ab^(7.5)`

`60=ab^(7.5)` ...(iii)

Divide (iii) by (ii).

`(60)/(18)=(ab^(7.5))/(ab^3)`

`(10)/(3)=b^(4.5)`

`\left((10)/(3)\right)^(1)/(4.5)=b`

`b\approx 1.30676`

Putting
`b=1.30676` in (ii), we get

`18=a(1.30676)^3`

`(18)/((1.30676)^3)=a`

`a\approx 8.0665`

Putting
`a=8.0665` and
`b=1.30676` in (i), we get

`f(x)=8.0665(1.30676)^x`

Putting x=12, we get

`f(12)=8.0665(1.30676)^(12)`

`f(12)=200.0024`

`f(12)\approx 200.00`

Therefore, the value of f(12) is about 200.00.