Question

If f(x) is an exponential function where f(2.5)=26 and f(6.5)=81, then find the value of f(12), to the nearest hundredth.

Answer 1

Given:

f(x) is an exponential function.

`f(2.5)=26`

`f(6.5)=81`

To find:

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

Solution:

The general exponential function is

`f(x)=ab^x`

For, x=2.5,

`f(2.5)=ab^(2.5)`

`26=ab^(2.5)` ...(i)

For, x=6.5,

`f(6.5)=ab^(6.5)`

`81=ab^(6.5)` ...(ii)

Divide (ii) by (i).

`(81)/(26)=(ab^(6.5))/(ab^(2.5))`

`(81)/(26)=b^4`

Taking 4th root on both sides, we get

`\sqrt[4]{(81)/(26)}=b`

`b\approx 1.32855`

Putting b=1.32855 in (i), we get

`26=a(1.32855)^(2.5)`

`26=a(2.03444)`

`(26)/(2.03444)=a`

`a\approx 12.7799`

Now, the required function is

`f(x)=12.7799(1.32855)^x`

Putting x=12, we get

`f(12)=12.7799(1.32855)^(12)`

`f(12)=386.4224`

`f(12)\approx 386.422`

Therefore, the value of f(12) is 386.422.