Given:
f(x) is an exponential function.
[tex]f(-3.5)=25, f(6)=33[/tex]
To find:
The value of f(6.5).
Solution:
Let the exponential function is
[tex]f(x)=ab^x[/tex] ...(i)
Where, a is the initial value and b is the growth factor.
We have, [tex]f(-3.5)=25[/tex]. So, put x=-3.5 and f(x)=25 in (i).
[tex]25=ab^{-3.5}[/tex] ...(ii)
We have, [tex]f(6)=33[/tex]. So, put x=6 and f(x)=33 in (i).
[tex]33=ab^{6}[/tex] ...(iii)
On dividing (iii) by (ii), we get
[tex]\dfrac{33}{25}=\dfrac{ab^{6}}{ab^{-3.5}}[/tex]
[tex]1.32=b^{9.5}[/tex]
[tex](1.32)^{\frac{1}{9.5}}=b[/tex]
[tex]1.0296556=b[/tex]
[tex]b\approx 1.03[/tex]
Putting b=1.03 in (iii), we get
[tex]33=a(1.03)^{6}[/tex]
[tex]33=a(1.194)[/tex]
[tex]\dfrac{33}{1.194}=a[/tex]
[tex]a\approx 27.63[/tex]
Putting a=27.63 and b=1.03 in (i), we get
[tex]f(x)=27.63(1.03)^x[/tex]
Therefore, the required exponential function is [tex]f(x)=27.63(1.03)^x[/tex].
