If f ( x ) f(x) is an exponential function where f ( − 3.5 ) = 25 f(−3.5)=25 and f ( 6 ) = 33 f(6)=33, then find the value of f ( 6.5 ) f(6.5), to the nearest hundredth.

Given:

f(x) is an exponential function.

To find:

The value of f(6.5).

Solution:

Let the exponential function is

        ...(i)

Where, a is the initial value and b is the growth factor.

We have, . So, put x=-3.5 and f(x)=25 in (i).

        ...(ii)

We have, . So, put x=6 and f(x)=33 in (i).

        ...(iii)

On dividing (iii) by (ii), we get

Putting b=1.03 in (iii), we get

Putting a=27.63 and b=1.03 in (i), we get

Therefore, the required exponential function is .

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