Answer: We know that the function f(x) is an exponential function. An exponential function can be written in the form:
f(x) = a * b^x
where:
a = the initial value (when x = 0)
b = the base of the exponential function
x = the independent variable
To find the values of a and b, we can use the given information:
f(1) = 8, so:
8 = a * b^1
8 = a * b
f(10.5) = 72, so:
72 = a * b^10.5
We can now use the first equation to solve for a:
a = 8/b
Substituting this value into the second equation, we get:
72 = (8/b) * b^10.5
Simplifying:
72 = 8 * b^9.5
9 = b^9.5
b = 9^(1/9.5)
b = 1.4608 (approx.)
Substituting the values of a and b into the exponential function, we get:
f(x) = 8 * 1.4608^x
Now we can find f(3.5):
f(3.5) = 8 * 1.4608^3.5
f(3.5) = 8 * 6.2797
f(3.5) = 50.2376 (approx.)
Therefore, f(3.5) is approximately equal to 50.24.
Explanation: