Answer :
The value of [tex]$f(0.5)$[/tex] is [tex]$23.81$[/tex]
What is exponential function?
- F(x)=exp or e(x) is a mathematical symbol for the exponential function. Unless otherwise stated, the term normally refers to the positive-valued function of a real variable, though it can be extended to the complex numbers or adapted to other mathematical objects like matrices or Lie algebras.
- Calculating the exponential growth or decay of a set of given data is done using an exponential function. Exponential functions can be used, for instance, to estimate population changes, loan interest rates, bacterial growth, radioactive decay, or the spread of disease.
[tex]$f(x)$[/tex] is an exponential function.
[tex]$f(-0.5)=27$[/tex]
[tex]$f(1.5)=21$[/tex]
To find:
The value of[tex]$f(0.5)$[/tex], to the nearest hundredth.
Solution:
The general exponential function is [tex]$f(x)=a b^x$[/tex]
For,[tex]$\mathrm{x}=-0.5$[/tex],
[tex]$f(-0.5)=a b^{-0.5}$[/tex]
[tex]$27=a b^{-0.5}$[/tex]
For,[tex]$\mathrm{x}=1.5$,[/tex]
[tex]$f(1.5)=a b^{1.5}$[/tex]
[tex]$21=a b^{1.5}$[/tex]
Divide (ii) by (i).
[tex]& \frac{21}{27}=\frac{a b^{1.5}}{a b^{-0.5}} \\[/tex]
[tex]& \frac{7}{9}=b^2[/tex]
Taking square root on both sides, we get
[tex]& \frac{\sqrt{7}}{3}=b \\[/tex]
[tex]& b \approx 0.882[/tex]
Putting [tex]$b=0.882$[/tex] in (i), we get
[tex]& 27=a(0.882)^{-0.5} \\[/tex]
[tex]& 27=a(1.0648) \\[/tex]
[tex]& \frac{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 [tex]$x=0.5$[/tex] 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 [tex]$f(0.5)$[/tex] is [tex]$23.81$[/tex]
The complete question is,
If f(x)f(x) is an exponential function where f(-0.5)=27f(−0.5)=27 and f(1.5)=21f(1.5)=21, then find the value of f(0.5)f(0.5), to the nearest hundredth.
To learn more about exponential function refer to:
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