To find an exponential function that fits the given values, we can use the general form of an exponential function:
F(x) = a * b^x
Given that F(3) = 8 and F(5) = 20, we can set up the following equations:
8 = a * b^3 (1)
20 = a * b^5 (2)
We can solve this system of equations to find the values of a and b.
Divide equation (2) by equation (1):
(20 / 8) = (a * b^5) / (a * b^3)
Simplify:
5/2 = b^2
Taking the square root of both sides:
b = sqrt(5/2)
Now, substitute b = sqrt(5/2) into equation (1) to solve for a:
8 = a * (sqrt(5/2))^3
8 = a * (5/2) * sqrt(5/2)
8 = a * (5/2) * sqrt(10/4)
8 = a * (5/2) * (sqrt(10) / 2)
8 = a * (5 * sqrt(10)) / 4
a = (8 * 4) / (5 * sqrt(10))
a = 32 / (5 * sqrt(10))
Therefore, the exponential function that fits the given values is:
F(x) = (32 / (5 * sqrt(10))) * (sqrt(5/2))^x
