Given:
f(x) is an exponential function.
[tex]f(-0.5)=27[/tex]
[tex]f(1.5)=21[/tex]
To find:
The value of f(0.5), to the nearest hundredth.
Solution:
The general exponential function is
[tex]f(x)=ab^x[/tex]
For, x=-0.5,
[tex]f(-0.5)=ab^{-0.5}[/tex]
[tex]27=ab^{-0.5}[/tex] ...(i)
For, x=1.5,
[tex]f(1.5)=ab^{1.5}[/tex]
[tex]21=ab^{1.5}[/tex] ...(ii)
Divide (ii) by (i).
[tex]\dfrac{21}{27}=\dfrac{ab^{1.5}}{ab^{-0.5}}[/tex]
[tex]\dfrac{7}{9}=b^2[/tex]
Taking square root on both sides, we get
[tex]\dfrac{\sqrt{7}}{3}=b[/tex]
[tex]b\approx 0.882[/tex]
Putting b=0.882 in (i), we get
[tex]27=a(0.882)^{-0.5}[/tex]
[tex]27=a(1.0648)[/tex]
[tex]\dfrac{27}{1.0648}=a[/tex]
[tex]a\approx 25.357[/tex]
Now, the required function is
[tex]f(x)=25.357(0.882)^x[/tex]
Putting x=0.5, we get
[tex]f(0.5)=25.357(0.882)^{0.5}[/tex]
[tex]f(0.5)=23.81399[/tex]
[tex]f(0.5)\approx 23.81[/tex]
Therefore, the value of f(0.5) is 23.81.
