Final answer:
The exponential function that goes through the points (0,125) and (3,216) is f(x) = 125 × 1.2^x, after determining the initial value 'a' from the first point and solving for the base 'b' using the second point.
Step-by-step explanation:
Finding the Exponential Function for Specific Points
To find the exponential function that passes through the points (0,125) and (3,216), we can use the general form of an exponential function, which is f(x) = ab^x, where a is the initial value when x is 0, and b is the base of the exponential function.
First, we substitute the first point (0,125) into the equation:
f(0) = ab^0 = a × 1 = a. Thus, we have a = 125.
Then, we substitute the second point (3,216) and the value of a into the equation:
f(3) = 125b^3 = 216. To find b, we need to solve for b^3 = 216/125, which gives b = ∛(216/125). Computing this, we get b ≈ 1.2.
Therefore, the exponential function that goes through the points (0,125) and (3,216) is f(x) = 125 × 1.2^x.