Answer:
We do not have enough information to find the exact value of f(2.5) without additional assumptions about the nature of the exponential function. However, we can make an estimate using the given data and the properties of exponential functions.
First, we can write the general form of an exponential function as:
f(x) = a * b^x
where a is the initial value or y-intercept, and b is the base or growth factor. We can use the two given data points to set up a system of equations and solve for a and b:
f(-1) = a * b^(-1) = 18
f(5) = a * b^5 = 75
Dividing the second equation by the first equation, we get:
f(5) / f(-1) = (a * b^5) / (a * b^(-1)) = b^6 = 75 / 18 = 25 / 6
Taking the sixth root of both sides, we get:
b = (25 / 6)^(1/6) ≈ 1.472
Substituting this value of b into the first equation, we get:
a = f(-1) / b^(-1) = 18 / 1.472 ≈ 12.223
Therefore, we have the exponential function:
f(x) ≈ 12.223 * 1.472^x
Using this function, we can estimate the value of f(2.5) as:
f(2.5) ≈ 12.223 * 1.472^(2.5) ≈ 34.311
Note that this is only an estimate, and the exact value of f(2.5) may be different depending on the specific nature of the exponential function.