Given:
f(x) is an exponential function.
[tex]f(2.5)=26[/tex]
[tex]f(6.5)=81[/tex]
To find:
The value of f(12), to the nearest hundredth.
Solution:
The general exponential function is
[tex]f(x)=ab^x[/tex]
For, x=2.5,
[tex]f(2.5)=ab^{2.5}[/tex]
[tex]26=ab^{2.5}[/tex] ...(i)
For, x=6.5,
[tex]f(6.5)=ab^{6.5}[/tex]
[tex]81=ab^{6.5}[/tex] ...(ii)
Divide (ii) by (i).
[tex]\dfrac{81}{26}=\dfrac{ab^{6.5}}{ab^{2.5}}[/tex]
[tex]\dfrac{81}{26}=b^4[/tex]
Taking 4th root on both sides, we get
[tex]\sqrt[4]{\dfrac{81}{26}}=b[/tex]
[tex]b\approx 1.32855[/tex]
Putting b=1.32855 in (i), we get
[tex]26=a(1.32855)^{2.5}[/tex]
[tex]26=a(2.03444)[/tex]
[tex]\dfrac{26}{2.03444}=a[/tex]
[tex]a\approx 12.7799[/tex]
Now, the required function is
[tex]f(x)=12.7799(1.32855)^x[/tex]
Putting x=12, we get
[tex]f(12)=12.7799(1.32855)^{12}[/tex]
[tex]f(12)=386.4224[/tex]
[tex]f(12)\approx 386.422[/tex]
Therefore, the value of f(12) is 386.422.
