Given:
f(x) is an exponential function.
![f(-3.5)=25, f(6)=33](https://tex.z-dn.net/?f=f%28-3.5%29%3D25%2C%20f%286%29%3D33)
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
![\dfrac{33}{25}=\dfrac{ab^{6}}{ab^{-3.5}}](https://tex.z-dn.net/?f=%5Cdfrac%7B33%7D%7B25%7D%3D%5Cdfrac%7Bab%5E%7B6%7D%7D%7Bab%5E%7B-3.5%7D%7D)
![1.32=b^{9.5}](https://tex.z-dn.net/?f=1.32%3Db%5E%7B9.5%7D)
![(1.32)^{\frac{1}{9.5}}=b](https://tex.z-dn.net/?f=%281.32%29%5E%7B%5Cfrac%7B1%7D%7B9.5%7D%7D%3Db)
![1.0296556=b](https://tex.z-dn.net/?f=1.0296556%3Db)
![b\approx 1.03](https://tex.z-dn.net/?f=b%5Capprox%201.03)
Putting b=1.03 in (iii), we get
![33=a(1.03)^{6}](https://tex.z-dn.net/?f=33%3Da%281.03%29%5E%7B6%7D)
![33=a(1.194)](https://tex.z-dn.net/?f=33%3Da%281.194%29)
![\dfrac{33}{1.194}=a](https://tex.z-dn.net/?f=%5Cdfrac%7B33%7D%7B1.194%7D%3Da)
![a\approx 27.63](https://tex.z-dn.net/?f=a%5Capprox%2027.63)
Putting a=27.63 and b=1.03 in (i), we get
![f(x)=27.63(1.03)^x](https://tex.z-dn.net/?f=f%28x%29%3D27.63%281.03%29%5Ex)
Therefore, the required exponential function is
.