Final answer:
The exponential function represents rapid growth where a constant base is raised to a variable exponent. Understanding its properties, such as the initial value, domain, and range, helps in analyzing growth patterns and calculating probabilities in exponential distributions.
Step-by-step explanation:
The exponential function is a mathematical expression where a constant base is raised to a variable exponent, and commonly takes the form f(x) = b^x. The initial value, often denoted as f(0), is the starting point of the function when x is zero. In your function, if the initial value is 8 and the base is 64, the function can be expressed as f(x) = 64^x, where x defines the exponent and f(x) the resulting value.
To understand exponential growth, consider the doubling sequence: Starting at 1, after each time interval you multiply by the base, which is 2 in a doubling sequence. After n intervals, you would have 2^n. The domain of a function is all the possible inputs or 'x' values, while the range is all possible outputs or 'y' values (f(x) values).
Using natural logarithms and the number e, the inverse of the exponential function, we can compute various attributes, such as growth rates or the probability within a certain range for an exponential distribution. Similarly, by understanding the properties of exponents, like any base raised to the power of zero equals one, we can better grasp how these functions behave.