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Answer :
Answer: f(0) ≈ 4.37
Step-by-step explanation:
We can use the information given to find the base and the equation of the exponential function.
Let the exponential function be of the form f(x) = a*b^x, where a is the initial value of f(0) and b is the base of the exponential function.
From the given information, we have:
f(-3) = 27 = ab^(-3)
f(0.5) = 83 = ab^(0.5)
To eliminate the variable a, we can divide the second equation by the first equation:
f(0.5)/f(-3) = (ab^(0.5))/(ab^(-3))
83/27 = b^(0.5+3)
b^3 = (83/27)^2
b = (83/27)^(2/3)
Substituting this value of b in the first equation, we can solve for a:
27 = a*(83/27)^(-1)
a = 27*(83/27)
Therefore, the equation of the exponential function is f(x) = (83/27)^(2/3)*((83/27)^(-1))^x.
To find f(0), we substitute x = 0 in the above equation:
f(0) = (83/27)^(2/3)*((83/27)^(-1))^0
f(0) = (83/27)^(2/3)
f(0) ≈ 4.37
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